111.85/29.53	YES
111.85/29.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
111.85/29.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
111.85/29.53	
111.85/29.53	
111.85/29.53	Termination of the given RelTRS could be proven:
111.85/29.53	
111.85/29.53	(0) RelTRS
111.85/29.53	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
111.85/29.53	(2) RelTRS
111.85/29.53	(3) FlatCCProof [EQUIVALENT, 0 ms]
111.85/29.53	(4) RelTRS
111.85/29.53	(5) RootLabelingProof [EQUIVALENT, 0 ms]
111.85/29.53	(6) RelTRS
111.85/29.53	(7) RelTRSRRRProof [EQUIVALENT, 573 ms]
111.85/29.53	(8) RelTRS
111.85/29.53	(9) RelTRSRRRProof [EQUIVALENT, 3 ms]
111.85/29.53	(10) RelTRS
111.85/29.53	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
111.85/29.53	(12) YES
111.85/29.53	
111.85/29.53	
111.85/29.53	----------------------------------------
111.85/29.53	
111.85/29.53	(0)
111.85/29.53	Obligation:
111.85/29.53	Relative term rewrite system:
111.85/29.53	The relative TRS consists of the following R rules:
111.85/29.53	
111.85/29.53	   a(a(a(x1))) -> b(c(b(x1)))
111.85/29.53	   b(b(b(x1))) -> a(a(c(x1)))
111.85/29.53	   a(b(c(x1))) -> a(c(a(x1)))
111.85/29.53	   b(a(a(x1))) -> c(a(c(x1)))
111.85/29.53	   c(a(b(x1))) -> c(a(c(x1)))
111.85/29.53	
111.85/29.53	The relative TRS consists of the following S rules:
111.85/29.53	
111.85/29.53	   a(c(c(x1))) -> a(a(c(x1)))
111.85/29.53	   b(a(c(x1))) -> a(a(b(x1)))
111.85/29.53	
111.85/29.53	
111.85/29.53	----------------------------------------
111.85/29.53	
111.85/29.53	(1) RelTRS Reverse (EQUIVALENT)
111.85/29.53	We have reversed the following relative TRS [REVERSE]:
111.85/29.54	The set of rules R is 
111.85/29.54	   a(a(a(x1))) -> b(c(b(x1)))
111.85/29.54	   b(b(b(x1))) -> a(a(c(x1)))
111.85/29.54	   a(b(c(x1))) -> a(c(a(x1)))
111.85/29.54	   b(a(a(x1))) -> c(a(c(x1)))
111.85/29.54	   c(a(b(x1))) -> c(a(c(x1)))
111.85/29.54	
111.85/29.54	The set of rules S is 
111.85/29.54	   a(c(c(x1))) -> a(a(c(x1)))
111.85/29.54	   b(a(c(x1))) -> a(a(b(x1)))
111.85/29.54	
111.85/29.54	We have obtained the following relative TRS:
111.85/29.54	The set of rules R is 
111.85/29.54	   a(a(a(x1))) -> b(c(b(x1)))
111.85/29.54	   b(b(b(x1))) -> c(a(a(x1)))
111.85/29.54	   c(b(a(x1))) -> a(c(a(x1)))
111.85/29.54	   a(a(b(x1))) -> c(a(c(x1)))
111.85/29.54	   b(a(c(x1))) -> c(a(c(x1)))
111.85/29.54	
111.85/29.54	The set of rules S is 
111.85/29.54	   c(c(a(x1))) -> c(a(a(x1)))
111.85/29.54	   c(a(b(x1))) -> b(a(a(x1)))
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(2)
111.85/29.54	Obligation:
111.85/29.54	Relative term rewrite system:
111.85/29.54	The relative TRS consists of the following R rules:
111.85/29.54	
111.85/29.54	   a(a(a(x1))) -> b(c(b(x1)))
111.85/29.54	   b(b(b(x1))) -> c(a(a(x1)))
111.85/29.54	   c(b(a(x1))) -> a(c(a(x1)))
111.85/29.54	   a(a(b(x1))) -> c(a(c(x1)))
111.85/29.54	   b(a(c(x1))) -> c(a(c(x1)))
111.85/29.54	
111.85/29.54	The relative TRS consists of the following S rules:
111.85/29.54	
111.85/29.54	   c(c(a(x1))) -> c(a(a(x1)))
111.85/29.54	   c(a(b(x1))) -> b(a(a(x1)))
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(3) FlatCCProof (EQUIVALENT)
111.85/29.54	We used flat context closure [ROOTLAB]
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(4)
111.85/29.54	Obligation:
111.85/29.54	Relative term rewrite system:
111.85/29.54	The relative TRS consists of the following R rules:
111.85/29.54	
111.85/29.54	   a(a(a(a(x1)))) -> a(b(c(b(x1))))
111.85/29.54	   b(a(a(a(x1)))) -> b(b(c(b(x1))))
111.85/29.54	   c(a(a(a(x1)))) -> c(b(c(b(x1))))
111.85/29.54	   a(b(b(b(x1)))) -> a(c(a(a(x1))))
111.85/29.54	   b(b(b(b(x1)))) -> b(c(a(a(x1))))
111.85/29.54	   c(b(b(b(x1)))) -> c(c(a(a(x1))))
111.85/29.54	   a(c(b(a(x1)))) -> a(a(c(a(x1))))
111.85/29.54	   b(c(b(a(x1)))) -> b(a(c(a(x1))))
111.85/29.54	   c(c(b(a(x1)))) -> c(a(c(a(x1))))
111.85/29.54	   a(a(a(b(x1)))) -> a(c(a(c(x1))))
111.85/29.54	   b(a(a(b(x1)))) -> b(c(a(c(x1))))
111.85/29.54	   c(a(a(b(x1)))) -> c(c(a(c(x1))))
111.85/29.54	   a(b(a(c(x1)))) -> a(c(a(c(x1))))
111.85/29.54	   b(b(a(c(x1)))) -> b(c(a(c(x1))))
111.85/29.54	   c(b(a(c(x1)))) -> c(c(a(c(x1))))
111.85/29.54	
111.85/29.54	The relative TRS consists of the following S rules:
111.85/29.54	
111.85/29.54	   c(c(a(x1))) -> c(a(a(x1)))
111.85/29.54	   a(c(a(b(x1)))) -> a(b(a(a(x1))))
111.85/29.54	   b(c(a(b(x1)))) -> b(b(a(a(x1))))
111.85/29.54	   c(c(a(b(x1)))) -> c(b(a(a(x1))))
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(5) RootLabelingProof (EQUIVALENT)
111.85/29.54	We used plain root labeling [ROOTLAB] with the following heuristic:
111.85/29.54	LabelAll: All function symbols get labeled
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(6)
111.85/29.54	Obligation:
111.85/29.54	Relative term rewrite system:
111.85/29.54	The relative TRS consists of the following R rules:
111.85/29.54	
111.85/29.54	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1))))
111.85/29.54	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
111.85/29.54	   a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
111.85/29.54	   a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
111.85/29.54	   b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
111.85/29.54	   b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
111.85/29.54	   b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
111.85/29.54	   c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
111.85/29.54	   c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
111.85/29.54	   c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	
111.85/29.54	The relative TRS consists of the following S rules:
111.85/29.54	
111.85/29.54	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
111.85/29.54	   a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(7) RelTRSRRRProof (EQUIVALENT)
111.85/29.54	We used the following monotonic ordering for rule removal:
111.85/29.54	Polynomial interpretation [POLO]:
111.85/29.54	
111.85/29.54	   POL(a_{a_1}(x_1)) = 1 + 2*x_1
111.85/29.54	   POL(a_{b_1}(x_1)) = 4 + 2*x_1
111.85/29.54	   POL(a_{c_1}(x_1)) = 4*x_1
111.85/29.54	   POL(b_{a_1}(x_1)) = 1 + 2*x_1
111.85/29.54	   POL(b_{b_1}(x_1)) = 4 + 2*x_1
111.85/29.54	   POL(b_{c_1}(x_1)) = 4*x_1
111.85/29.54	   POL(c_{a_1}(x_1)) = x_1
111.85/29.54	   POL(c_{b_1}(x_1)) = x_1
111.85/29.54	   POL(c_{c_1}(x_1)) = 1 + 2*x_1
111.85/29.54	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
111.85/29.54	Rules from R:
111.85/29.54	
111.85/29.54	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1))))
111.85/29.54	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
111.85/29.54	   a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
111.85/29.54	   a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
111.85/29.54	   b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
111.85/29.54	   b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
111.85/29.54	   b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
111.85/29.54	   c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
111.85/29.54	   c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
111.85/29.54	   c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	Rules from S:
111.85/29.54	
111.85/29.54	   a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
111.85/29.54	
111.85/29.54	
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(8)
111.85/29.54	Obligation:
111.85/29.54	Relative term rewrite system:
111.85/29.54	The relative TRS consists of the following R rules:
111.85/29.54	
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	
111.85/29.54	The relative TRS consists of the following S rules:
111.85/29.54	
111.85/29.54	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(9) RelTRSRRRProof (EQUIVALENT)
111.85/29.54	We used the following monotonic ordering for rule removal:
111.85/29.54	Knuth-Bendix order [KBO] with precedence:a_{c_1}_1 > a_{a_1}_1 > a_{b_1}_1 > c_{a_1}_1 > b_{a_1}_1 > c_{b_1}_1 > b_{c_1}_1 > c_{c_1}_1
111.85/29.54	
111.85/29.54	and weight map:
111.85/29.54	
111.85/29.54	   c_{a_1}_1=3
111.85/29.54	   a_{a_1}_1=1
111.85/29.54	   a_{b_1}_1=3
111.85/29.54	   b_{c_1}_1=1
111.85/29.54	   c_{c_1}_1=2
111.85/29.54	   a_{c_1}_1=1
111.85/29.54	   c_{b_1}_1=4
111.85/29.54	   b_{a_1}_1=1
111.85/29.54	
111.85/29.54	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
111.85/29.54	Rules from R:
111.85/29.54	
111.85/29.54	   c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
111.85/29.54	   c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
111.85/29.54	Rules from S:
111.85/29.54	
111.85/29.54	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{a_1}(a_{a_1}(x1)))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{a_1}(a_{b_1}(x1)))
111.85/29.54	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{a_1}(a_{c_1}(x1)))
111.85/29.54	
111.85/29.54	
111.85/29.54	
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(10)
111.85/29.54	Obligation:
111.85/29.54	Relative term rewrite system:
111.85/29.54	R is empty.
111.85/29.54	S is empty.
111.85/29.54	
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(11) RIsEmptyProof (EQUIVALENT)
111.85/29.54	The TRS R is empty. Hence, termination is trivially proven.
111.85/29.54	----------------------------------------
111.85/29.54	
111.85/29.54	(12)
111.85/29.54	YES
111.85/29.56	EOF