35.11/10.33	YES
35.11/10.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
35.11/10.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.11/10.35	
35.11/10.35	
35.11/10.35	Termination of the given RelTRS could be proven:
35.11/10.35	
35.11/10.35	(0) RelTRS
35.11/10.35	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
35.11/10.35	(2) RelTRS
35.11/10.35	(3) FlatCCProof [EQUIVALENT, 0 ms]
35.11/10.35	(4) RelTRS
35.11/10.35	(5) RootLabelingProof [EQUIVALENT, 0 ms]
35.11/10.35	(6) RelTRS
35.11/10.35	(7) RelTRSRRRProof [EQUIVALENT, 81 ms]
35.11/10.35	(8) RelTRS
35.11/10.35	(9) RelTRSRRRProof [EQUIVALENT, 34 ms]
35.11/10.35	(10) RelTRS
35.11/10.35	(11) RelTRSRRRProof [EQUIVALENT, 1777 ms]
35.11/10.35	(12) RelTRS
35.11/10.35	(13) RelTRSRRRProof [EQUIVALENT, 5 ms]
35.11/10.35	(14) RelTRS
35.11/10.35	(15) RIsEmptyProof [EQUIVALENT, 0 ms]
35.11/10.35	(16) YES
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(0)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   a(a(c(x1))) -> a(a(a(x1)))
35.11/10.35	   c(a(c(x1))) -> a(a(a(x1)))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   b(b(a(x1))) -> c(c(c(x1)))
35.11/10.35	   c(b(a(x1))) -> a(c(b(x1)))
35.11/10.35	   b(b(b(x1))) -> b(c(b(x1)))
35.11/10.35	   c(a(a(x1))) -> c(c(a(x1)))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(1) RelTRS Reverse (EQUIVALENT)
35.11/10.35	We have reversed the following relative TRS [REVERSE]:
35.11/10.35	The set of rules R is 
35.11/10.35	   a(a(c(x1))) -> a(a(a(x1)))
35.11/10.35	   c(a(c(x1))) -> a(a(a(x1)))
35.11/10.35	
35.11/10.35	The set of rules S is 
35.11/10.35	   b(b(a(x1))) -> c(c(c(x1)))
35.11/10.35	   c(b(a(x1))) -> a(c(b(x1)))
35.11/10.35	   b(b(b(x1))) -> b(c(b(x1)))
35.11/10.35	   c(a(a(x1))) -> c(c(a(x1)))
35.11/10.35	
35.11/10.35	We have obtained the following relative TRS:
35.11/10.35	The set of rules R is 
35.11/10.35	   c(a(a(x1))) -> a(a(a(x1)))
35.11/10.35	   c(a(c(x1))) -> a(a(a(x1)))
35.11/10.35	
35.11/10.35	The set of rules S is 
35.11/10.35	   a(b(b(x1))) -> c(c(c(x1)))
35.11/10.35	   a(b(c(x1))) -> b(c(a(x1)))
35.11/10.35	   b(b(b(x1))) -> b(c(b(x1)))
35.11/10.35	   a(a(c(x1))) -> a(c(c(x1)))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(2)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   c(a(a(x1))) -> a(a(a(x1)))
35.11/10.35	   c(a(c(x1))) -> a(a(a(x1)))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   a(b(b(x1))) -> c(c(c(x1)))
35.11/10.35	   a(b(c(x1))) -> b(c(a(x1)))
35.11/10.35	   b(b(b(x1))) -> b(c(b(x1)))
35.11/10.35	   a(a(c(x1))) -> a(c(c(x1)))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(3) FlatCCProof (EQUIVALENT)
35.11/10.35	We used flat context closure [ROOTLAB]
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(4)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   c(c(a(a(x1)))) -> c(a(a(a(x1))))
35.11/10.35	   a(c(a(a(x1)))) -> a(a(a(a(x1))))
35.11/10.35	   b(c(a(a(x1)))) -> b(a(a(a(x1))))
35.11/10.35	   c(c(a(c(x1)))) -> c(a(a(a(x1))))
35.11/10.35	   a(c(a(c(x1)))) -> a(a(a(a(x1))))
35.11/10.35	   b(c(a(c(x1)))) -> b(a(a(a(x1))))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   b(b(b(x1))) -> b(c(b(x1)))
35.11/10.35	   a(a(c(x1))) -> a(c(c(x1)))
35.11/10.35	   c(a(b(b(x1)))) -> c(c(c(c(x1))))
35.11/10.35	   a(a(b(b(x1)))) -> a(c(c(c(x1))))
35.11/10.35	   b(a(b(b(x1)))) -> b(c(c(c(x1))))
35.11/10.35	   c(a(b(c(x1)))) -> c(b(c(a(x1))))
35.11/10.35	   a(a(b(c(x1)))) -> a(b(c(a(x1))))
35.11/10.35	   b(a(b(c(x1)))) -> b(b(c(a(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(5) RootLabelingProof (EQUIVALENT)
35.11/10.35	We used plain root labeling [ROOTLAB] with the following heuristic:
35.11/10.35	LabelAll: All function symbols get labeled
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(6)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
35.11/10.35	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
35.11/10.35	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(7) RelTRSRRRProof (EQUIVALENT)
35.11/10.35	We used the following monotonic ordering for rule removal:
35.11/10.35	Polynomial interpretation [POLO]:
35.11/10.35	
35.11/10.35	   POL(a_{a_1}(x_1)) = x_1
35.11/10.35	   POL(a_{b_1}(x_1)) = 1 + x_1
35.11/10.35	   POL(a_{c_1}(x_1)) = x_1
35.11/10.35	   POL(b_{a_1}(x_1)) = x_1
35.11/10.35	   POL(b_{b_1}(x_1)) = 1 + x_1
35.11/10.35	   POL(b_{c_1}(x_1)) = x_1
35.11/10.35	   POL(c_{a_1}(x_1)) = x_1
35.11/10.35	   POL(c_{b_1}(x_1)) = 1 + x_1
35.11/10.35	   POL(c_{c_1}(x_1)) = x_1
35.11/10.35	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
35.11/10.35	Rules from R:
35.11/10.35	none
35.11/10.35	Rules from S:
35.11/10.35	
35.11/10.35	   b_{b_1}(b_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{b_1}(b_{c_1}(x1)))
35.11/10.35	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{b_1}(b_{a_1}(x1)))
35.11/10.35	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{b_1}(b_{b_1}(x1)))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(8)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(9) RelTRSRRRProof (EQUIVALENT)
35.11/10.35	We used the following monotonic ordering for rule removal:
35.11/10.35	Polynomial interpretation [POLO]:
35.11/10.35	
35.11/10.35	   POL(a_{a_1}(x_1)) = x_1
35.11/10.35	   POL(a_{b_1}(x_1)) = 1 + x_1
35.11/10.35	   POL(a_{c_1}(x_1)) = x_1
35.11/10.35	   POL(b_{a_1}(x_1)) = x_1
35.11/10.35	   POL(b_{b_1}(x_1)) = x_1
35.11/10.35	   POL(b_{c_1}(x_1)) = 1 + x_1
35.11/10.35	   POL(c_{a_1}(x_1)) = x_1
35.11/10.35	   POL(c_{b_1}(x_1)) = 1 + x_1
35.11/10.35	   POL(c_{c_1}(x_1)) = x_1
35.11/10.35	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
35.11/10.35	Rules from R:
35.11/10.35	
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	Rules from S:
35.11/10.35	
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(10)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(11) RelTRSRRRProof (EQUIVALENT)
35.11/10.35	We used the following monotonic ordering for rule removal:
35.11/10.35	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(c_{c_1}(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(a_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	   <<<
35.11/10.35	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 2], [0, 0]] 	* 	x_1
35.11/10.35	>>>
35.11/10.35	
35.11/10.35	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
35.11/10.35	Rules from R:
35.11/10.35	none
35.11/10.35	Rules from S:
35.11/10.35	
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(12)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	The relative TRS consists of the following R rules:
35.11/10.35	
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	The relative TRS consists of the following S rules:
35.11/10.35	
35.11/10.35	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(13) RelTRSRRRProof (EQUIVALENT)
35.11/10.35	We used the following monotonic ordering for rule removal:
35.11/10.35	Knuth-Bendix order [KBO] with precedence:c_{c_1}_1 > c_{a_1}_1 > c_{b_1}_1 > a_{a_1}_1 > a_{b_1}_1 > a_{c_1}_1 > b_{c_1}_1
35.11/10.35	
35.11/10.35	and weight map:
35.11/10.35	
35.11/10.35	   c_{c_1}_1=2
35.11/10.35	   c_{a_1}_1=3
35.11/10.35	   a_{a_1}_1=2
35.11/10.35	   a_{c_1}_1=3
35.11/10.35	   a_{b_1}_1=1
35.11/10.35	   c_{b_1}_1=2
35.11/10.35	   b_{c_1}_1=1
35.11/10.35	
35.11/10.35	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
35.11/10.35	Rules from R:
35.11/10.35	
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
35.11/10.35	Rules from S:
35.11/10.35	
35.11/10.35	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
35.11/10.35	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
35.11/10.35	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(14)
35.11/10.35	Obligation:
35.11/10.35	Relative term rewrite system:
35.11/10.35	R is empty.
35.11/10.35	S is empty.
35.11/10.35	
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(15) RIsEmptyProof (EQUIVALENT)
35.11/10.35	The TRS R is empty. Hence, termination is trivially proven.
35.11/10.35	----------------------------------------
35.11/10.35	
35.11/10.35	(16)
35.11/10.35	YES
35.44/10.48	EOF