4.12/1.74	YES
4.22/1.75	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
4.22/1.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.22/1.75	
4.22/1.75	
4.22/1.75	Termination of the given RelTRS could be proven:
4.22/1.75	
4.22/1.75	(0) RelTRS
4.22/1.75	(1) FlatCCProof [EQUIVALENT, 0 ms]
4.22/1.75	(2) RelTRS
4.22/1.75	(3) RootLabelingProof [EQUIVALENT, 0 ms]
4.22/1.75	(4) RelTRS
4.22/1.75	(5) RelTRSRRRProof [EQUIVALENT, 12 ms]
4.22/1.75	(6) RelTRS
4.22/1.75	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
4.22/1.75	(8) YES
4.22/1.75	
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(0)
4.22/1.75	Obligation:
4.22/1.75	Relative term rewrite system:
4.22/1.75	The relative TRS consists of the following R rules:
4.22/1.75	
4.22/1.75	   b(c(a(x1))) -> a(b(a(x1)))
4.22/1.75	   a(a(a(x1))) -> c(c(c(x1)))
4.22/1.75	   a(a(a(x1))) -> a(b(a(x1)))
4.22/1.75	
4.22/1.75	The relative TRS consists of the following S rules:
4.22/1.75	
4.22/1.75	   c(c(a(x1))) -> c(a(c(x1)))
4.22/1.75	   b(a(b(x1))) -> b(c(c(x1)))
4.22/1.75	   c(a(b(x1))) -> a(c(b(x1)))
4.22/1.75	
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(1) FlatCCProof (EQUIVALENT)
4.22/1.75	We used flat context closure [ROOTLAB]
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(2)
4.22/1.75	Obligation:
4.22/1.75	Relative term rewrite system:
4.22/1.75	The relative TRS consists of the following R rules:
4.22/1.75	
4.22/1.75	   a(a(a(x1))) -> a(b(a(x1)))
4.22/1.75	   b(b(c(a(x1)))) -> b(a(b(a(x1))))
4.22/1.75	   c(b(c(a(x1)))) -> c(a(b(a(x1))))
4.22/1.75	   a(b(c(a(x1)))) -> a(a(b(a(x1))))
4.22/1.75	   b(a(a(a(x1)))) -> b(c(c(c(x1))))
4.22/1.75	   c(a(a(a(x1)))) -> c(c(c(c(x1))))
4.22/1.75	   a(a(a(a(x1)))) -> a(c(c(c(x1))))
4.22/1.75	
4.22/1.75	The relative TRS consists of the following S rules:
4.22/1.75	
4.22/1.75	   c(c(a(x1))) -> c(a(c(x1)))
4.22/1.75	   b(a(b(x1))) -> b(c(c(x1)))
4.22/1.75	   b(c(a(b(x1)))) -> b(a(c(b(x1))))
4.22/1.75	   c(c(a(b(x1)))) -> c(a(c(b(x1))))
4.22/1.75	   a(c(a(b(x1)))) -> a(a(c(b(x1))))
4.22/1.75	
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(3) RootLabelingProof (EQUIVALENT)
4.22/1.75	We used plain root labeling [ROOTLAB] with the following heuristic:
4.22/1.75	LabelAll: All function symbols get labeled
4.22/1.75	
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(4)
4.22/1.75	Obligation:
4.22/1.75	Relative term rewrite system:
4.22/1.75	The relative TRS consists of the following R rules:
4.22/1.75	
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
4.22/1.75	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.22/1.75	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.22/1.75	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
4.22/1.75	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.22/1.75	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.22/1.75	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
4.22/1.75	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.22/1.75	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.22/1.75	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
4.22/1.75	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.22/1.75	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.22/1.75	   b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.22/1.75	   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.22/1.75	   c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.22/1.75	   c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.22/1.75	
4.22/1.75	The relative TRS consists of the following S rules:
4.22/1.75	
4.22/1.75	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
4.22/1.75	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{a_1}(x1)))
4.22/1.75	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(x1)))
4.22/1.75	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{c_1}(x1)))
4.22/1.75	   b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
4.22/1.75	   b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
4.22/1.75	   b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
4.22/1.75	   a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
4.22/1.75	   a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
4.22/1.75	   a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
4.22/1.75	
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(5) RelTRSRRRProof (EQUIVALENT)
4.22/1.75	We used the following monotonic ordering for rule removal:
4.22/1.75	Knuth-Bendix order [KBO] with precedence:c_{c_1}_1 > a_{b_1}_1 > b_{a_1}_1 > b_{c_1}_1 > b_{b_1}_1 > a_{c_1}_1 > a_{a_1}_1 > c_{a_1}_1 > c_{b_1}_1
4.22/1.75	
4.22/1.75	and weight map:
4.22/1.75	
4.22/1.75	   a_{a_1}_1=15
4.22/1.75	   a_{b_1}_1=15
4.22/1.75	   b_{a_1}_1=13
4.22/1.75	   a_{c_1}_1=4
4.22/1.75	   b_{b_1}_1=14
4.22/1.75	   b_{c_1}_1=16
4.22/1.75	   c_{a_1}_1=14
4.22/1.75	   c_{b_1}_1=13
4.22/1.75	   c_{c_1}_1=10
4.22/1.75	
4.22/1.75	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.22/1.75	Rules from R:
4.22/1.75	
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
4.22/1.75	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.22/1.75	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.22/1.75	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
4.22/1.75	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.22/1.75	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.22/1.75	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
4.22/1.75	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
4.22/1.75	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
4.22/1.75	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
4.22/1.75	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.22/1.75	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.22/1.75	   b_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.22/1.75	   c_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.22/1.75	   c_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.22/1.75	   c_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.22/1.75	   a_{a_1}(a_{a_1}(a_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.22/1.75	Rules from S:
4.22/1.75	
4.22/1.75	   c_{c_1}(c_{a_1}(a_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
4.22/1.75	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{c_1}(c_{a_1}(x1)))
4.22/1.75	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{c_1}(c_{b_1}(x1)))
4.22/1.75	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{c_1}(c_{c_1}(x1)))
4.22/1.75	   b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
4.22/1.75	   b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
4.22/1.75	   b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
4.22/1.75	   c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
4.22/1.75	   a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1))))
4.22/1.75	   a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1))))
4.22/1.75	   a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1))))
4.22/1.75	
4.22/1.75	
4.22/1.75	
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(6)
4.22/1.75	Obligation:
4.22/1.75	Relative term rewrite system:
4.22/1.75	R is empty.
4.22/1.75	S is empty.
4.22/1.75	
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(7) RIsEmptyProof (EQUIVALENT)
4.22/1.75	The TRS R is empty. Hence, termination is trivially proven.
4.22/1.75	----------------------------------------
4.22/1.75	
4.22/1.75	(8)
4.22/1.75	YES
4.22/1.78	EOF