31.48/8.80	YES
31.86/8.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
31.86/8.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
31.86/8.99	
31.86/8.99	
31.86/8.99	Termination of the given RelTRS could be proven:
31.86/8.99	
31.86/8.99	(0) RelTRS
31.86/8.99	(1) FlatCCProof [EQUIVALENT, 0 ms]
31.86/8.99	(2) RelTRS
31.86/8.99	(3) RootLabelingProof [EQUIVALENT, 17 ms]
31.86/8.99	(4) RelTRS
31.86/8.99	(5) RelTRSRRRProof [EQUIVALENT, 82 ms]
31.86/8.99	(6) RelTRS
31.86/8.99	(7) RelTRSRRRProof [EQUIVALENT, 1826 ms]
31.86/8.99	(8) RelTRS
31.86/8.99	(9) RelTRSRRRProof [EQUIVALENT, 982 ms]
31.86/8.99	(10) RelTRS
31.86/8.99	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
31.86/8.99	(12) YES
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(0)
31.86/8.99	Obligation:
31.86/8.99	Relative term rewrite system:
31.86/8.99	The relative TRS consists of the following R rules:
31.86/8.99	
31.86/8.99	   a(c(b(x1))) -> b(a(b(x1)))
31.86/8.99	   a(b(c(x1))) -> b(b(c(x1)))
31.86/8.99	
31.86/8.99	The relative TRS consists of the following S rules:
31.86/8.99	
31.86/8.99	   a(b(c(x1))) -> c(c(b(x1)))
31.86/8.99	   b(a(c(x1))) -> c(a(c(x1)))
31.86/8.99	   a(a(c(x1))) -> a(a(b(x1)))
31.86/8.99	   b(c(a(x1))) -> a(c(b(x1)))
31.86/8.99	   a(a(a(x1))) -> c(a(b(x1)))
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(1) FlatCCProof (EQUIVALENT)
31.86/8.99	We used flat context closure [ROOTLAB]
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(2)
31.86/8.99	Obligation:
31.86/8.99	Relative term rewrite system:
31.86/8.99	The relative TRS consists of the following R rules:
31.86/8.99	
31.86/8.99	   a(a(c(b(x1)))) -> a(b(a(b(x1))))
31.86/8.99	   c(a(c(b(x1)))) -> c(b(a(b(x1))))
31.86/8.99	   b(a(c(b(x1)))) -> b(b(a(b(x1))))
31.86/8.99	   a(a(b(c(x1)))) -> a(b(b(c(x1))))
31.86/8.99	   c(a(b(c(x1)))) -> c(b(b(c(x1))))
31.86/8.99	   b(a(b(c(x1)))) -> b(b(b(c(x1))))
31.86/8.99	
31.86/8.99	The relative TRS consists of the following S rules:
31.86/8.99	
31.86/8.99	   a(a(c(x1))) -> a(a(b(x1)))
31.86/8.99	   a(a(b(c(x1)))) -> a(c(c(b(x1))))
31.86/8.99	   c(a(b(c(x1)))) -> c(c(c(b(x1))))
31.86/8.99	   b(a(b(c(x1)))) -> b(c(c(b(x1))))
31.86/8.99	   a(b(a(c(x1)))) -> a(c(a(c(x1))))
31.86/8.99	   c(b(a(c(x1)))) -> c(c(a(c(x1))))
31.86/8.99	   b(b(a(c(x1)))) -> b(c(a(c(x1))))
31.86/8.99	   a(b(c(a(x1)))) -> a(a(c(b(x1))))
31.86/8.99	   c(b(c(a(x1)))) -> c(a(c(b(x1))))
31.86/8.99	   b(b(c(a(x1)))) -> b(a(c(b(x1))))
31.86/8.99	   a(a(a(a(x1)))) -> a(c(a(b(x1))))
31.86/8.99	   c(a(a(a(x1)))) -> c(c(a(b(x1))))
31.86/8.99	   b(a(a(a(x1)))) -> b(c(a(b(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(3) RootLabelingProof (EQUIVALENT)
31.86/8.99	We used plain root labeling [ROOTLAB] with the following heuristic:
31.86/8.99	LabelAll: All function symbols get labeled
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(4)
31.86/8.99	Obligation:
31.86/8.99	Relative term rewrite system:
31.86/8.99	The relative TRS consists of the following R rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
31.86/8.99	
31.86/8.99	The relative TRS consists of the following S rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(5) RelTRSRRRProof (EQUIVALENT)
31.86/8.99	We used the following monotonic ordering for rule removal:
31.86/8.99	Polynomial interpretation [POLO]:
31.86/8.99	
31.86/8.99	   POL(a_{a_1}(x_1)) = 1 + x_1
31.86/8.99	   POL(a_{b_1}(x_1)) = x_1
31.86/8.99	   POL(a_{c_1}(x_1)) = x_1
31.86/8.99	   POL(b_{a_1}(x_1)) = 1 + x_1
31.86/8.99	   POL(b_{b_1}(x_1)) = x_1
31.86/8.99	   POL(b_{c_1}(x_1)) = x_1
31.86/8.99	   POL(c_{a_1}(x_1)) = 1 + x_1
31.86/8.99	   POL(c_{b_1}(x_1)) = x_1
31.86/8.99	   POL(c_{c_1}(x_1)) = x_1
31.86/8.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
31.86/8.99	Rules from R:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1))))
31.86/8.99	Rules from S:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(6)
31.86/8.99	Obligation:
31.86/8.99	Relative term rewrite system:
31.86/8.99	The relative TRS consists of the following R rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	The relative TRS consists of the following S rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(7) RelTRSRRRProof (EQUIVALENT)
31.86/8.99	We used the following monotonic ordering for rule removal:
31.86/8.99	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(a_{a_1}(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(a_{b_1}(x_1)) =  	[[0], [1]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 2]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
31.86/8.99	Rules from R:
31.86/8.99	none
31.86/8.99	Rules from S:
31.86/8.99	
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(8)
31.86/8.99	Obligation:
31.86/8.99	Relative term rewrite system:
31.86/8.99	The relative TRS consists of the following R rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	The relative TRS consists of the following S rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(9) RelTRSRRRProof (EQUIVALENT)
31.86/8.99	We used the following monotonic ordering for rule removal:
31.86/8.99	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(a_{a_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(c_{b_1}(x_1)) =  	[[1], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(b_{a_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(c_{a_1}(x_1)) =  	[[2], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	   <<<
31.86/8.99	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
31.86/8.99	>>>
31.86/8.99	
31.86/8.99	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
31.86/8.99	Rules from R:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
31.86/8.99	Rules from S:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
31.86/8.99	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
31.86/8.99	
31.86/8.99	
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(10)
31.86/8.99	Obligation:
31.86/8.99	Relative term rewrite system:
31.86/8.99	R is empty.
31.86/8.99	The relative TRS consists of the following S rules:
31.86/8.99	
31.86/8.99	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
31.86/8.99	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	   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))))
31.86/8.99	
31.86/8.99	
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(11) RIsEmptyProof (EQUIVALENT)
31.86/8.99	The TRS R is empty. Hence, termination is trivially proven.
31.86/8.99	----------------------------------------
31.86/8.99	
31.86/8.99	(12)
31.86/8.99	YES
32.20/9.03	EOF