4.72/2.07	YES
5.08/2.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.08/2.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.08/2.11	
5.08/2.11	
5.08/2.11	Termination of the given RelTRS could be proven:
5.08/2.11	
5.08/2.11	(0) RelTRS
5.08/2.11	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
5.08/2.11	(2) RelTRS
5.08/2.11	(3) FlatCCProof [EQUIVALENT, 0 ms]
5.08/2.11	(4) RelTRS
5.08/2.11	(5) RootLabelingProof [EQUIVALENT, 15 ms]
5.08/2.11	(6) RelTRS
5.08/2.11	(7) RelTRSRRRProof [EQUIVALENT, 52 ms]
5.08/2.11	(8) RelTRS
5.08/2.11	(9) RIsEmptyProof [EQUIVALENT, 4 ms]
5.08/2.11	(10) YES
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(0)
5.08/2.11	Obligation:
5.08/2.11	Relative term rewrite system:
5.08/2.11	The relative TRS consists of the following R rules:
5.08/2.11	
5.08/2.11	   a(c(a(x1))) -> a(c(b(x1)))
5.08/2.11	   b(b(c(x1))) -> b(c(c(x1)))
5.08/2.11	   c(c(b(x1))) -> c(a(b(x1)))
5.08/2.11	   b(a(a(x1))) -> a(b(a(x1)))
5.08/2.11	   b(a(c(x1))) -> a(b(a(x1)))
5.08/2.11	   a(c(a(x1))) -> b(c(c(x1)))
5.08/2.11	
5.08/2.11	The relative TRS consists of the following S rules:
5.08/2.11	
5.08/2.11	   b(c(a(x1))) -> b(a(a(x1)))
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(1) RelTRS Reverse (EQUIVALENT)
5.08/2.11	We have reversed the following relative TRS [REVERSE]:
5.08/2.11	The set of rules R is 
5.08/2.11	   a(c(a(x1))) -> a(c(b(x1)))
5.08/2.11	   b(b(c(x1))) -> b(c(c(x1)))
5.08/2.11	   c(c(b(x1))) -> c(a(b(x1)))
5.08/2.11	   b(a(a(x1))) -> a(b(a(x1)))
5.08/2.11	   b(a(c(x1))) -> a(b(a(x1)))
5.08/2.11	   a(c(a(x1))) -> b(c(c(x1)))
5.08/2.11	
5.08/2.11	The set of rules S is 
5.08/2.11	   b(c(a(x1))) -> b(a(a(x1)))
5.08/2.11	
5.08/2.11	We have obtained the following relative TRS:
5.08/2.11	The set of rules R is 
5.08/2.11	   a(c(a(x1))) -> b(c(a(x1)))
5.08/2.11	   c(b(b(x1))) -> c(c(b(x1)))
5.08/2.11	   b(c(c(x1))) -> b(a(c(x1)))
5.08/2.11	   a(a(b(x1))) -> a(b(a(x1)))
5.08/2.11	   c(a(b(x1))) -> a(b(a(x1)))
5.08/2.11	   a(c(a(x1))) -> c(c(b(x1)))
5.08/2.11	
5.08/2.11	The set of rules S is 
5.08/2.11	   a(c(b(x1))) -> a(a(b(x1)))
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(2)
5.08/2.11	Obligation:
5.08/2.11	Relative term rewrite system:
5.08/2.11	The relative TRS consists of the following R rules:
5.08/2.11	
5.08/2.11	   a(c(a(x1))) -> b(c(a(x1)))
5.08/2.11	   c(b(b(x1))) -> c(c(b(x1)))
5.08/2.11	   b(c(c(x1))) -> b(a(c(x1)))
5.08/2.11	   a(a(b(x1))) -> a(b(a(x1)))
5.08/2.11	   c(a(b(x1))) -> a(b(a(x1)))
5.08/2.11	   a(c(a(x1))) -> c(c(b(x1)))
5.08/2.11	
5.08/2.11	The relative TRS consists of the following S rules:
5.08/2.11	
5.08/2.11	   a(c(b(x1))) -> a(a(b(x1)))
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(3) FlatCCProof (EQUIVALENT)
5.08/2.11	We used flat context closure [ROOTLAB]
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(4)
5.08/2.11	Obligation:
5.08/2.11	Relative term rewrite system:
5.08/2.11	The relative TRS consists of the following R rules:
5.08/2.11	
5.08/2.11	   c(b(b(x1))) -> c(c(b(x1)))
5.08/2.11	   b(c(c(x1))) -> b(a(c(x1)))
5.08/2.11	   a(a(b(x1))) -> a(b(a(x1)))
5.08/2.11	   a(a(c(a(x1)))) -> a(b(c(a(x1))))
5.08/2.11	   c(a(c(a(x1)))) -> c(b(c(a(x1))))
5.08/2.11	   b(a(c(a(x1)))) -> b(b(c(a(x1))))
5.08/2.11	   a(c(a(b(x1)))) -> a(a(b(a(x1))))
5.08/2.11	   c(c(a(b(x1)))) -> c(a(b(a(x1))))
5.08/2.11	   b(c(a(b(x1)))) -> b(a(b(a(x1))))
5.08/2.11	   a(a(c(a(x1)))) -> a(c(c(b(x1))))
5.08/2.11	   c(a(c(a(x1)))) -> c(c(c(b(x1))))
5.08/2.11	   b(a(c(a(x1)))) -> b(c(c(b(x1))))
5.08/2.11	
5.08/2.11	The relative TRS consists of the following S rules:
5.08/2.11	
5.08/2.11	   a(c(b(x1))) -> a(a(b(x1)))
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(5) RootLabelingProof (EQUIVALENT)
5.08/2.11	We used plain root labeling [ROOTLAB] with the following heuristic:
5.08/2.11	LabelAll: All function symbols get labeled
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(6)
5.08/2.11	Obligation:
5.08/2.11	Relative term rewrite system:
5.08/2.11	The relative TRS consists of the following R rules:
5.08/2.11	
5.08/2.11	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
5.08/2.11	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
5.08/2.11	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
5.08/2.11	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
5.08/2.11	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
5.08/2.11	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
5.08/2.11	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
5.08/2.11	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.08/2.11	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.08/2.11	   a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
5.08/2.11	   a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
5.08/2.11	   a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
5.08/2.11	   c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
5.08/2.11	   c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
5.08/2.11	   c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
5.08/2.11	   b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
5.08/2.11	   b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
5.08/2.11	   b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.08/2.11	
5.08/2.11	The relative TRS consists of the following S rules:
5.08/2.11	
5.08/2.11	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
5.08/2.11	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
5.08/2.11	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(7) RelTRSRRRProof (EQUIVALENT)
5.08/2.11	We used the following monotonic ordering for rule removal:
5.08/2.11	Knuth-Bendix order [KBO] with precedence:c_{b_1}_1 > c_{c_1}_1 > a_{c_1}_1 > a_{a_1}_1 > b_{c_1}_1 > c_{a_1}_1 > b_{a_1}_1 > b_{b_1}_1 > a_{b_1}_1
5.08/2.11	
5.08/2.11	and weight map:
5.08/2.11	
5.08/2.11	   c_{b_1}_1=1
5.08/2.11	   b_{b_1}_1=6
5.08/2.11	   b_{c_1}_1=9
5.08/2.11	   c_{c_1}_1=6
5.08/2.11	   b_{a_1}_1=5
5.08/2.11	   a_{c_1}_1=10
5.08/2.11	   c_{a_1}_1=3
5.08/2.11	   a_{a_1}_1=6
5.08/2.11	   a_{b_1}_1=5
5.08/2.11	
5.08/2.11	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.08/2.11	Rules from R:
5.08/2.11	
5.08/2.11	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
5.08/2.11	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
5.08/2.11	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
5.08/2.11	   b_{c_1}(c_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
5.08/2.11	   b_{c_1}(c_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
5.08/2.11	   b_{c_1}(c_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
5.08/2.11	   a_{a_1}(a_{b_1}(b_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
5.08/2.11	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.08/2.11	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1))))
5.08/2.11	   a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
5.08/2.11	   a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
5.08/2.11	   a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
5.08/2.11	   c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
5.08/2.11	   c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
5.08/2.11	   c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
5.08/2.11	   b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{c_1}(x1))))
5.08/2.11	   b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
5.08/2.11	   b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.08/2.11	   a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.08/2.11	   c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.08/2.11	   b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.08/2.11	Rules from S:
5.08/2.11	
5.08/2.11	   a_{c_1}(c_{b_1}(b_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
5.08/2.11	   a_{c_1}(c_{b_1}(b_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
5.08/2.11	   a_{c_1}(c_{b_1}(b_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
5.08/2.11	
5.08/2.11	
5.08/2.11	
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(8)
5.08/2.11	Obligation:
5.08/2.11	Relative term rewrite system:
5.08/2.11	R is empty.
5.08/2.11	S is empty.
5.08/2.11	
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(9) RIsEmptyProof (EQUIVALENT)
5.08/2.11	The TRS R is empty. Hence, termination is trivially proven.
5.08/2.11	----------------------------------------
5.08/2.11	
5.08/2.11	(10)
5.08/2.11	YES
5.08/2.16	EOF