5.32/2.27	YES
5.32/2.29	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
5.32/2.29	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.32/2.29	
5.32/2.29	
5.32/2.29	Termination of the given RelTRS could be proven:
5.32/2.29	
5.32/2.29	(0) RelTRS
5.32/2.29	(1) FlatCCProof [EQUIVALENT, 0 ms]
5.32/2.29	(2) RelTRS
5.32/2.29	(3) RootLabelingProof [EQUIVALENT, 0 ms]
5.32/2.29	(4) RelTRS
5.32/2.29	(5) RelTRSRRRProof [EQUIVALENT, 16 ms]
5.32/2.29	(6) RelTRS
5.32/2.29	(7) RIsEmptyProof [EQUIVALENT, 1 ms]
5.32/2.29	(8) YES
5.32/2.29	
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(0)
5.32/2.29	Obligation:
5.32/2.29	Relative term rewrite system:
5.32/2.29	The relative TRS consists of the following R rules:
5.32/2.29	
5.32/2.29	   b(b(c(x1))) -> b(a(a(x1)))
5.32/2.29	   a(c(c(x1))) -> a(a(b(x1)))
5.32/2.29	
5.32/2.29	The relative TRS consists of the following S rules:
5.32/2.29	
5.32/2.29	   a(c(b(x1))) -> c(a(b(x1)))
5.32/2.29	   a(c(c(x1))) -> c(c(b(x1)))
5.32/2.29	   a(a(a(x1))) -> a(b(a(x1)))
5.32/2.29	   a(b(a(x1))) -> a(b(c(x1)))
5.32/2.29	
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(1) FlatCCProof (EQUIVALENT)
5.32/2.29	We used flat context closure [ROOTLAB]
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(2)
5.32/2.29	Obligation:
5.32/2.29	Relative term rewrite system:
5.32/2.29	The relative TRS consists of the following R rules:
5.32/2.29	
5.32/2.29	   b(b(c(x1))) -> b(a(a(x1)))
5.32/2.29	   a(c(c(x1))) -> a(a(b(x1)))
5.32/2.29	
5.32/2.29	The relative TRS consists of the following S rules:
5.32/2.29	
5.32/2.29	   a(a(a(x1))) -> a(b(a(x1)))
5.32/2.29	   a(b(a(x1))) -> a(b(c(x1)))
5.32/2.29	   b(a(c(b(x1)))) -> b(c(a(b(x1))))
5.32/2.29	   c(a(c(b(x1)))) -> c(c(a(b(x1))))
5.32/2.29	   a(a(c(b(x1)))) -> a(c(a(b(x1))))
5.32/2.29	   b(a(c(c(x1)))) -> b(c(c(b(x1))))
5.32/2.29	   c(a(c(c(x1)))) -> c(c(c(b(x1))))
5.32/2.29	   a(a(c(c(x1)))) -> a(c(c(b(x1))))
5.32/2.29	
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(3) RootLabelingProof (EQUIVALENT)
5.32/2.29	We used plain root labeling [ROOTLAB] with the following heuristic:
5.32/2.29	LabelAll: All function symbols get labeled
5.32/2.29	
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(4)
5.32/2.29	Obligation:
5.32/2.29	Relative term rewrite system:
5.32/2.29	The relative TRS consists of the following R rules:
5.32/2.29	
5.32/2.29	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.32/2.29	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1)))
5.32/2.29	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.32/2.29	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
5.32/2.29	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
5.32/2.29	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
5.32/2.29	
5.32/2.29	The relative TRS consists of the following S rules:
5.32/2.29	
5.32/2.29	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.32/2.29	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
5.32/2.29	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.32/2.29	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
5.32/2.29	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
5.32/2.29	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.32/2.29	
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(5) RelTRSRRRProof (EQUIVALENT)
5.32/2.29	We used the following monotonic ordering for rule removal:
5.32/2.29	Knuth-Bendix order [KBO] with precedence:a_{b_1}_1 > a_{a_1}_1 > c_{b_1}_1 > a_{c_1}_1 > c_{a_1}_1 > c_{c_1}_1 > b_{b_1}_1 > b_{a_1}_1 > b_{c_1}_1
5.32/2.29	
5.32/2.29	and weight map:
5.32/2.29	
5.32/2.29	   b_{b_1}_1=8
5.32/2.29	   b_{c_1}_1=10
5.32/2.29	   c_{b_1}_1=5
5.32/2.29	   b_{a_1}_1=14
5.32/2.29	   a_{a_1}_1=8
5.32/2.29	   a_{b_1}_1=1
5.32/2.29	   c_{c_1}_1=9
5.32/2.29	   a_{c_1}_1=5
5.32/2.29	   c_{a_1}_1=12
5.32/2.29	
5.32/2.29	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.32/2.29	Rules from R:
5.32/2.29	
5.32/2.29	   b_{b_1}(b_{c_1}(c_{b_1}(x1))) -> b_{a_1}(a_{a_1}(a_{b_1}(x1)))
5.32/2.29	   b_{b_1}(b_{c_1}(c_{c_1}(x1))) -> b_{a_1}(a_{a_1}(a_{c_1}(x1)))
5.32/2.29	   b_{b_1}(b_{c_1}(c_{a_1}(x1))) -> b_{a_1}(a_{a_1}(a_{a_1}(x1)))
5.32/2.29	   a_{c_1}(c_{c_1}(c_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{b_1}(x1)))
5.32/2.29	   a_{c_1}(c_{c_1}(c_{c_1}(x1))) -> a_{a_1}(a_{b_1}(b_{c_1}(x1)))
5.32/2.29	   a_{c_1}(c_{c_1}(c_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(x1)))
5.32/2.29	Rules from S:
5.32/2.29	
5.32/2.29	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
5.32/2.29	   a_{a_1}(a_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{a_1}(a_{c_1}(x1)))
5.32/2.29	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
5.32/2.29	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{c_1}(c_{b_1}(x1)))
5.32/2.29	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{b_1}(b_{c_1}(c_{c_1}(x1)))
5.32/2.29	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{c_1}(c_{a_1}(x1)))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{c_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{a_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.32/2.29	   b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.32/2.29	   c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1))))
5.32/2.29	   a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1))))
5.32/2.29	
5.32/2.29	
5.32/2.29	
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(6)
5.32/2.29	Obligation:
5.32/2.29	Relative term rewrite system:
5.32/2.29	R is empty.
5.32/2.29	S is empty.
5.32/2.29	
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(7) RIsEmptyProof (EQUIVALENT)
5.32/2.29	The TRS R is empty. Hence, termination is trivially proven.
5.32/2.29	----------------------------------------
5.32/2.29	
5.32/2.29	(8)
5.32/2.29	YES
5.67/2.37	EOF