4.29/2.33	YES
4.59/2.36	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
4.59/2.36	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
4.59/2.36	
4.59/2.36	
4.59/2.36	Termination of the given RelTRS could be proven:
4.59/2.36	
4.59/2.36	(0) RelTRS
4.59/2.36	(1) FlatCCProof [EQUIVALENT, 0 ms]
4.59/2.36	(2) RelTRS
4.59/2.36	(3) RootLabelingProof [EQUIVALENT, 0 ms]
4.59/2.36	(4) RelTRS
4.59/2.36	(5) RelTRSRRRProof [EQUIVALENT, 15 ms]
4.59/2.36	(6) RelTRS
4.59/2.36	(7) RIsEmptyProof [EQUIVALENT, 0 ms]
4.59/2.36	(8) YES
4.59/2.36	
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(0)
4.59/2.36	Obligation:
4.59/2.36	Relative term rewrite system:
4.59/2.36	The relative TRS consists of the following R rules:
4.59/2.36	
4.59/2.36	   a(c(b(x1))) -> c(c(c(x1)))
4.59/2.36	   b(c(c(x1))) -> a(a(b(x1)))
4.59/2.36	
4.59/2.36	The relative TRS consists of the following S rules:
4.59/2.36	
4.59/2.36	   c(a(c(x1))) -> b(a(b(x1)))
4.59/2.36	   b(c(a(x1))) -> a(b(c(x1)))
4.59/2.36	   c(c(c(x1))) -> c(c(b(x1)))
4.59/2.36	
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(1) FlatCCProof (EQUIVALENT)
4.59/2.36	We used flat context closure [ROOTLAB]
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(2)
4.59/2.36	Obligation:
4.59/2.36	Relative term rewrite system:
4.59/2.36	The relative TRS consists of the following R rules:
4.59/2.36	
4.59/2.36	   a(a(c(b(x1)))) -> a(c(c(c(x1))))
4.59/2.36	   c(a(c(b(x1)))) -> c(c(c(c(x1))))
4.59/2.36	   b(a(c(b(x1)))) -> b(c(c(c(x1))))
4.59/2.36	   a(b(c(c(x1)))) -> a(a(a(b(x1))))
4.59/2.36	   c(b(c(c(x1)))) -> c(a(a(b(x1))))
4.59/2.36	   b(b(c(c(x1)))) -> b(a(a(b(x1))))
4.59/2.36	
4.59/2.36	The relative TRS consists of the following S rules:
4.59/2.36	
4.59/2.36	   c(c(c(x1))) -> c(c(b(x1)))
4.59/2.36	   a(c(a(c(x1)))) -> a(b(a(b(x1))))
4.59/2.36	   c(c(a(c(x1)))) -> c(b(a(b(x1))))
4.59/2.36	   b(c(a(c(x1)))) -> b(b(a(b(x1))))
4.59/2.36	   a(b(c(a(x1)))) -> a(a(b(c(x1))))
4.59/2.36	   c(b(c(a(x1)))) -> c(a(b(c(x1))))
4.59/2.36	   b(b(c(a(x1)))) -> b(a(b(c(x1))))
4.59/2.36	
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(3) RootLabelingProof (EQUIVALENT)
4.59/2.36	We used plain root labeling [ROOTLAB] with the following heuristic:
4.59/2.36	LabelAll: All function symbols get labeled
4.59/2.36	
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(4)
4.59/2.36	Obligation:
4.59/2.36	Relative term rewrite system:
4.59/2.36	The relative TRS consists of the following R rules:
4.59/2.36	
4.59/2.36	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.59/2.36	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.59/2.36	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.59/2.36	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.59/2.36	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.59/2.36	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.59/2.36	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.59/2.36	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.59/2.36	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	
4.59/2.36	The relative TRS consists of the following S rules:
4.59/2.36	
4.59/2.36	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
4.59/2.36	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
4.59/2.36	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
4.59/2.36	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
4.59/2.36	
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(5) RelTRSRRRProof (EQUIVALENT)
4.59/2.36	We used the following monotonic ordering for rule removal:
4.59/2.36	Knuth-Bendix order [KBO] with precedence:a_{b_1}_1 > a_{a_1}_1 > c_{c_1}_1 > a_{c_1}_1 > c_{b_1}_1 > c_{a_1}_1 > b_{c_1}_1 > b_{b_1}_1 > b_{a_1}_1
4.59/2.36	
4.59/2.36	and weight map:
4.59/2.36	
4.59/2.36	   a_{a_1}_1=5
4.59/2.36	   a_{c_1}_1=4
4.59/2.36	   c_{b_1}_1=3
4.59/2.36	   b_{a_1}_1=9
4.59/2.36	   c_{c_1}_1=5
4.59/2.36	   c_{a_1}_1=7
4.59/2.36	   b_{c_1}_1=7
4.59/2.36	   b_{b_1}_1=5
4.59/2.36	   a_{b_1}_1=1
4.59/2.36	
4.59/2.36	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
4.59/2.36	Rules from R:
4.59/2.36	
4.59/2.36	   a_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.59/2.36	   a_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.59/2.36	   a_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.59/2.36	   c_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.59/2.36	   c_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.59/2.36	   c_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.59/2.36	   b_{a_1}(a_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{a_1}(x1))))
4.59/2.36	   b_{a_1}(a_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{c_1}(x1))))
4.59/2.36	   b_{a_1}(a_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{c_1}(c_{b_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	Rules from S:
4.59/2.36	
4.59/2.36	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{c_1}(c_{b_1}(b_{a_1}(x1)))
4.59/2.36	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{c_1}(c_{b_1}(b_{c_1}(x1)))
4.59/2.36	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{c_1}(c_{b_1}(b_{b_1}(x1)))
4.59/2.36	   a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
4.59/2.36	   b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
4.59/2.36	   b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
4.59/2.36	   a_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
4.59/2.36	   c_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{a_1}(a_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1))))
4.59/2.36	   b_{b_1}(b_{c_1}(c_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1))))
4.59/2.36	
4.59/2.36	
4.59/2.36	
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(6)
4.59/2.36	Obligation:
4.59/2.36	Relative term rewrite system:
4.59/2.36	R is empty.
4.59/2.36	S is empty.
4.59/2.36	
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(7) RIsEmptyProof (EQUIVALENT)
4.59/2.36	The TRS R is empty. Hence, termination is trivially proven.
4.59/2.36	----------------------------------------
4.59/2.36	
4.59/2.36	(8)
4.59/2.36	YES
4.71/2.42	EOF