5.03/2.09	YES
5.03/2.09	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.03/2.09	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.03/2.09	
5.03/2.09	
5.03/2.09	Termination of the given RelTRS could be proven:
5.03/2.09	
5.03/2.09	(0) RelTRS
5.03/2.09	(1) FlatCCProof [EQUIVALENT, 0 ms]
5.03/2.09	(2) RelTRS
5.03/2.09	(3) RootLabelingProof [EQUIVALENT, 11 ms]
5.03/2.09	(4) RelTRS
5.03/2.09	(5) RelTRSRRRProof [EQUIVALENT, 11 ms]
5.03/2.09	(6) RelTRS
5.03/2.09	(7) RIsEmptyProof [EQUIVALENT, 1 ms]
5.03/2.09	(8) YES
5.03/2.09	
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(0)
5.03/2.09	Obligation:
5.03/2.09	Relative term rewrite system:
5.03/2.09	The relative TRS consists of the following R rules:
5.03/2.09	
5.03/2.09	   b(c(b(x1))) -> a(c(c(x1)))
5.03/2.09	   c(b(b(x1))) -> c(c(b(x1)))
5.03/2.09	
5.03/2.09	The relative TRS consists of the following S rules:
5.03/2.09	
5.03/2.09	   c(a(a(x1))) -> c(b(b(x1)))
5.03/2.09	   a(c(a(x1))) -> b(a(a(x1)))
5.03/2.09	   a(c(b(x1))) -> b(a(c(x1)))
5.03/2.09	
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(1) FlatCCProof (EQUIVALENT)
5.03/2.09	We used flat context closure [ROOTLAB]
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(2)
5.03/2.09	Obligation:
5.03/2.09	Relative term rewrite system:
5.03/2.09	The relative TRS consists of the following R rules:
5.03/2.09	
5.03/2.09	   c(b(b(x1))) -> c(c(b(x1)))
5.03/2.09	   b(b(c(b(x1)))) -> b(a(c(c(x1))))
5.03/2.09	   c(b(c(b(x1)))) -> c(a(c(c(x1))))
5.03/2.09	   a(b(c(b(x1)))) -> a(a(c(c(x1))))
5.03/2.09	
5.03/2.09	The relative TRS consists of the following S rules:
5.03/2.09	
5.03/2.09	   c(a(a(x1))) -> c(b(b(x1)))
5.03/2.09	   b(a(c(a(x1)))) -> b(b(a(a(x1))))
5.03/2.09	   c(a(c(a(x1)))) -> c(b(a(a(x1))))
5.03/2.09	   a(a(c(a(x1)))) -> a(b(a(a(x1))))
5.03/2.09	   b(a(c(b(x1)))) -> b(b(a(c(x1))))
5.03/2.09	   c(a(c(b(x1)))) -> c(b(a(c(x1))))
5.03/2.09	   a(a(c(b(x1)))) -> a(b(a(c(x1))))
5.03/2.09	
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(3) RootLabelingProof (EQUIVALENT)
5.03/2.09	We used plain root labeling [ROOTLAB] with the following heuristic:
5.03/2.09	LabelAll: All function symbols get labeled
5.03/2.09	
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(4)
5.03/2.09	Obligation:
5.03/2.09	Relative term rewrite system:
5.03/2.09	The relative TRS consists of the following R rules:
5.03/2.09	
5.03/2.09	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
5.03/2.09	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
5.03/2.09	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
5.03/2.09	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
5.03/2.09	   b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
5.03/2.09	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
5.03/2.09	   c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
5.03/2.09	   c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
5.03/2.09	   c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
5.03/2.09	   a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
5.03/2.09	   a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
5.03/2.09	   a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
5.03/2.09	
5.03/2.09	The relative TRS consists of the following S rules:
5.03/2.09	
5.03/2.09	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
5.03/2.09	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
5.03/2.09	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.03/2.09	
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(5) RelTRSRRRProof (EQUIVALENT)
5.03/2.09	We used the following monotonic ordering for rule removal:
5.03/2.09	Knuth-Bendix order [KBO] with precedence:c_{a_1}_1 > b_{a_1}_1 > c_{b_1}_1 > b_{c_1}_1 > a_{c_1}_1 > c_{c_1}_1 > a_{a_1}_1 > a_{b_1}_1 > b_{b_1}_1
5.03/2.09	
5.03/2.09	and weight map:
5.03/2.09	
5.03/2.09	   c_{b_1}_1=1
5.03/2.09	   b_{b_1}_1=8
5.03/2.09	   b_{c_1}_1=24
5.03/2.09	   c_{c_1}_1=7
5.03/2.09	   b_{a_1}_1=10
5.03/2.09	   a_{c_1}_1=18
5.03/2.09	   c_{a_1}_1=3
5.03/2.09	   a_{b_1}_1=9
5.03/2.09	   a_{a_1}_1=12
5.03/2.09	
5.03/2.09	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
5.03/2.09	Rules from R:
5.03/2.09	
5.03/2.09	   c_{b_1}(b_{b_1}(b_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
5.03/2.09	   c_{b_1}(b_{b_1}(b_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
5.03/2.09	   c_{b_1}(b_{b_1}(b_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
5.03/2.09	   b_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
5.03/2.09	   b_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
5.03/2.09	   b_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
5.03/2.09	   c_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
5.03/2.09	   c_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
5.03/2.09	   c_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
5.03/2.09	   a_{b_1}(b_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1))))
5.03/2.09	   a_{b_1}(b_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1))))
5.03/2.09	   a_{b_1}(b_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1))))
5.03/2.09	Rules from S:
5.03/2.09	
5.03/2.09	   c_{a_1}(a_{a_1}(a_{c_1}(x1))) -> c_{b_1}(b_{b_1}(b_{c_1}(x1)))
5.03/2.09	   c_{a_1}(a_{a_1}(a_{b_1}(x1))) -> c_{b_1}(b_{b_1}(b_{b_1}(x1)))
5.03/2.09	   c_{a_1}(a_{a_1}(a_{a_1}(x1))) -> c_{b_1}(b_{b_1}(b_{a_1}(x1)))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.03/2.09	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.03/2.09	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1))))
5.03/2.09	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1))))
5.03/2.09	
5.03/2.09	
5.03/2.09	
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(6)
5.03/2.09	Obligation:
5.03/2.09	Relative term rewrite system:
5.03/2.09	R is empty.
5.03/2.09	S is empty.
5.03/2.09	
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(7) RIsEmptyProof (EQUIVALENT)
5.03/2.09	The TRS R is empty. Hence, termination is trivially proven.
5.03/2.09	----------------------------------------
5.03/2.09	
5.03/2.09	(8)
5.03/2.09	YES
5.35/2.12	EOF