118.89/30.87	YES
118.89/30.88	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
118.89/30.88	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
118.89/30.88	
118.89/30.88	
118.89/30.88	Termination of the given RelTRS could be proven:
118.89/30.88	
118.89/30.88	(0) RelTRS
118.89/30.88	(1) FlatCCProof [EQUIVALENT, 0 ms]
118.89/30.88	(2) RelTRS
118.89/30.88	(3) RootLabelingProof [EQUIVALENT, 0 ms]
118.89/30.88	(4) RelTRS
118.89/30.88	(5) RelTRSRRRProof [EQUIVALENT, 447 ms]
118.89/30.88	(6) RelTRS
118.89/30.88	(7) SIsEmptyProof [EQUIVALENT, 0 ms]
118.89/30.88	(8) QTRS
118.89/30.88	(9) RFCMatchBoundsTRSProof [EQUIVALENT, 0 ms]
118.89/30.88	(10) YES
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(0)
118.89/30.88	Obligation:
118.89/30.88	Relative term rewrite system:
118.89/30.88	The relative TRS consists of the following R rules:
118.89/30.88	
118.89/30.88	   a(b(b(x1))) -> c(b(a(x1)))
118.89/30.88	   a(a(c(x1))) -> b(c(a(x1)))
118.89/30.88	   a(a(b(x1))) -> a(c(b(x1)))
118.89/30.88	
118.89/30.88	The relative TRS consists of the following S rules:
118.89/30.88	
118.89/30.88	   b(a(a(x1))) -> c(a(a(x1)))
118.89/30.88	   c(b(b(x1))) -> b(a(a(x1)))
118.89/30.88	   c(b(a(x1))) -> b(c(c(x1)))
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(1) FlatCCProof (EQUIVALENT)
118.89/30.88	We used flat context closure [ROOTLAB]
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(2)
118.89/30.88	Obligation:
118.89/30.88	Relative term rewrite system:
118.89/30.88	The relative TRS consists of the following R rules:
118.89/30.88	
118.89/30.88	   a(a(b(x1))) -> a(c(b(x1)))
118.89/30.88	   a(a(b(b(x1)))) -> a(c(b(a(x1))))
118.89/30.88	   b(a(b(b(x1)))) -> b(c(b(a(x1))))
118.89/30.88	   c(a(b(b(x1)))) -> c(c(b(a(x1))))
118.89/30.88	   a(a(a(c(x1)))) -> a(b(c(a(x1))))
118.89/30.88	   b(a(a(c(x1)))) -> b(b(c(a(x1))))
118.89/30.88	   c(a(a(c(x1)))) -> c(b(c(a(x1))))
118.89/30.88	
118.89/30.88	The relative TRS consists of the following S rules:
118.89/30.88	
118.89/30.88	   a(b(a(a(x1)))) -> a(c(a(a(x1))))
118.89/30.88	   b(b(a(a(x1)))) -> b(c(a(a(x1))))
118.89/30.88	   c(b(a(a(x1)))) -> c(c(a(a(x1))))
118.89/30.88	   a(c(b(b(x1)))) -> a(b(a(a(x1))))
118.89/30.88	   b(c(b(b(x1)))) -> b(b(a(a(x1))))
118.89/30.88	   c(c(b(b(x1)))) -> c(b(a(a(x1))))
118.89/30.88	   a(c(b(a(x1)))) -> a(b(c(c(x1))))
118.89/30.88	   b(c(b(a(x1)))) -> b(b(c(c(x1))))
118.89/30.88	   c(c(b(a(x1)))) -> c(b(c(c(x1))))
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(3) RootLabelingProof (EQUIVALENT)
118.89/30.88	We used plain root labeling [ROOTLAB] with the following heuristic:
118.89/30.88	LabelAll: All function symbols get labeled
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(4)
118.89/30.88	Obligation:
118.89/30.88	Relative term rewrite system:
118.89/30.88	The relative TRS consists of the following R rules:
118.89/30.88	
118.89/30.88	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	
118.89/30.88	The relative TRS consists of the following S rules:
118.89/30.88	
118.89/30.88	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(5) RelTRSRRRProof (EQUIVALENT)
118.89/30.88	We used the following monotonic ordering for rule removal:
118.89/30.88	Polynomial interpretation [POLO]:
118.89/30.88	
118.89/30.88	   POL(a_{a_1}(x_1)) = 2*x_1
118.89/30.88	   POL(a_{b_1}(x_1)) = 2*x_1
118.89/30.88	   POL(a_{c_1}(x_1)) = x_1
118.89/30.88	   POL(b_{a_1}(x_1)) = 4 + 2*x_1
118.89/30.88	   POL(b_{b_1}(x_1)) = 4 + 2*x_1
118.89/30.88	   POL(b_{c_1}(x_1)) = x_1
118.89/30.88	   POL(c_{a_1}(x_1)) = 4*x_1
118.89/30.88	   POL(c_{b_1}(x_1)) = 4*x_1
118.89/30.88	   POL(c_{c_1}(x_1)) = 2*x_1
118.89/30.88	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
118.89/30.88	Rules from R:
118.89/30.88	
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1))))
118.89/30.88	Rules from S:
118.89/30.88	
118.89/30.88	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
118.89/30.88	   a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
118.89/30.88	   b_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
118.89/30.88	   c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
118.89/30.88	
118.89/30.88	
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(6)
118.89/30.88	Obligation:
118.89/30.88	Relative term rewrite system:
118.89/30.88	The relative TRS consists of the following R rules:
118.89/30.88	
118.89/30.88	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	
118.89/30.88	S is empty.
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(7) SIsEmptyProof (EQUIVALENT)
118.89/30.88	The TRS S is empty. Hence, termination of R/S is equivalent to termination of R.
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(8)
118.89/30.88	Obligation:
118.89/30.88	Q restricted rewrite system:
118.89/30.88	The TRS R consists of the following rules:
118.89/30.88	
118.89/30.88	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	
118.89/30.88	Q is empty.
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(9) RFCMatchBoundsTRSProof (EQUIVALENT)
118.89/30.88	Termination of the TRS R could be shown with a Match Bound [MATCHBOUNDS1,MATCHBOUNDS2] of 1. This implies Q-termination of R.
118.89/30.88	The following rules were used to construct the certificate:
118.89/30.88	
118.89/30.88	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{c_1}(c_{b_1}(b_{a_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{c_1}(c_{b_1}(b_{b_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{c_1}(c_{b_1}(b_{c_1}(x1)))
118.89/30.88	   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{b_1}(b_{a_1}(a_{c_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
118.89/30.88	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
118.89/30.88	
118.89/30.88	The certificate found is represented by the following graph.
118.89/30.88	The certificate consists of the following enumerated nodes:
118.89/30.88	163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174
118.89/30.88	
118.89/30.88	Node 163 is start node and node 164 is final node.
118.89/30.88	
118.89/30.88	Those nodes are connected through the following edges:
118.89/30.88	
118.89/30.88	* 163 to 165 labelled a_{c_1}_1(0), c_{c_1}_1(0)* 163 to 167 labelled a_{b_1}_1(0), b_{b_1}_1(0), c_{b_1}_1(0)* 164 to 164 labelled #_1(0), a_{c_1}_1(0), a_{c_1}_1(1)* 165 to 166 labelled c_{b_1}_1(0)* 166 to 164 labelled b_{a_1}_1(0), b_{b_1}_1(0), b_{c_1}_1(0)* 166 to 170 labelled b_{b_1}_1(1)* 167 to 168 labelled b_{c_1}_1(0)* 168 to 169 labelled c_{a_1}_1(0)* 168 to 170 labelled c_{b_1}_1(1)* 168 to 173 labelled c_{c_1}_1(1)* 169 to 164 labelled a_{a_1}_1(0), a_{b_1}_1(0), a_{c_1}_1(0)* 169 to 173 labelled a_{c_1}_1(1)* 169 to 170 labelled a_{b_1}_1(1)* 170 to 171 labelled b_{c_1}_1(1)* 171 to 172 labelled c_{a_1}_1(1)* 171 to 170 labelled c_{b_1}_1(1)* 171 to 173 labelled c_{c_1}_1(1)* 172 to 164 labelled a_{a_1}_1(1), a_{b_1}_1(1), a_{c_1}_1(1)* 172 to 173 labelled a_{c_1}_1(1)* 172 to 170 labelled a_{b_1}_1(1)* 173 to 174 labelled c_{b_1}_1(1)* 174 to 164 labelled b_{a_1}_1(1), b_{b_1}_1(1), b_{c_1}_1(1)* 174 to 170 labelled b_{b_1}_1(1)
118.89/30.88	
118.89/30.88	
118.89/30.88	----------------------------------------
118.89/30.88	
118.89/30.88	(10)
118.89/30.88	YES
119.12/30.93	EOF