6.33/2.37	YES
6.53/2.44	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
6.53/2.44	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.53/2.44	
6.53/2.44	
6.53/2.44	Termination of the given RelTRS could be proven:
6.53/2.44	
6.53/2.44	(0) RelTRS
6.53/2.44	(1) RelTRS Reverse [EQUIVALENT, 0 ms]
6.53/2.44	(2) RelTRS
6.53/2.44	(3) FlatCCProof [EQUIVALENT, 0 ms]
6.53/2.44	(4) RelTRS
6.53/2.44	(5) RootLabelingProof [EQUIVALENT, 0 ms]
6.53/2.44	(6) RelTRS
6.53/2.44	(7) RelTRSRRRProof [EQUIVALENT, 81 ms]
6.53/2.44	(8) RelTRS
6.53/2.44	(9) RelTRSRRRProof [EQUIVALENT, 7 ms]
6.53/2.44	(10) RelTRS
6.53/2.44	(11) RIsEmptyProof [EQUIVALENT, 0 ms]
6.53/2.44	(12) YES
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(0)
6.53/2.44	Obligation:
6.53/2.44	Relative term rewrite system:
6.53/2.44	The relative TRS consists of the following R rules:
6.53/2.44	
6.53/2.44	   c(c(a(x1))) -> c(c(b(x1)))
6.53/2.44	   b(b(c(x1))) -> a(a(b(x1)))
6.53/2.44	
6.53/2.44	The relative TRS consists of the following S rules:
6.53/2.44	
6.53/2.44	   b(a(b(x1))) -> a(c(b(x1)))
6.53/2.44	   c(a(a(x1))) -> b(a(b(x1)))
6.53/2.44	   c(c(c(x1))) -> c(a(c(x1)))
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(1) RelTRS Reverse (EQUIVALENT)
6.53/2.44	We have reversed the following relative TRS [REVERSE]:
6.53/2.44	The set of rules R is 
6.53/2.44	   c(c(a(x1))) -> c(c(b(x1)))
6.53/2.44	   b(b(c(x1))) -> a(a(b(x1)))
6.53/2.44	
6.53/2.44	The set of rules S is 
6.53/2.44	   b(a(b(x1))) -> a(c(b(x1)))
6.53/2.44	   c(a(a(x1))) -> b(a(b(x1)))
6.53/2.44	   c(c(c(x1))) -> c(a(c(x1)))
6.53/2.44	
6.53/2.44	We have obtained the following relative TRS:
6.53/2.44	The set of rules R is 
6.53/2.44	   a(c(c(x1))) -> b(c(c(x1)))
6.53/2.44	   c(b(b(x1))) -> b(a(a(x1)))
6.53/2.44	
6.53/2.44	The set of rules S is 
6.53/2.44	   b(a(b(x1))) -> b(c(a(x1)))
6.53/2.44	   a(a(c(x1))) -> b(a(b(x1)))
6.53/2.44	   c(c(c(x1))) -> c(a(c(x1)))
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(2)
6.53/2.44	Obligation:
6.53/2.44	Relative term rewrite system:
6.53/2.44	The relative TRS consists of the following R rules:
6.53/2.44	
6.53/2.44	   a(c(c(x1))) -> b(c(c(x1)))
6.53/2.44	   c(b(b(x1))) -> b(a(a(x1)))
6.53/2.44	
6.53/2.44	The relative TRS consists of the following S rules:
6.53/2.44	
6.53/2.44	   b(a(b(x1))) -> b(c(a(x1)))
6.53/2.44	   a(a(c(x1))) -> b(a(b(x1)))
6.53/2.44	   c(c(c(x1))) -> c(a(c(x1)))
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(3) FlatCCProof (EQUIVALENT)
6.53/2.44	We used flat context closure [ROOTLAB]
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(4)
6.53/2.44	Obligation:
6.53/2.44	Relative term rewrite system:
6.53/2.44	The relative TRS consists of the following R rules:
6.53/2.44	
6.53/2.44	   a(a(c(c(x1)))) -> a(b(c(c(x1))))
6.53/2.44	   c(a(c(c(x1)))) -> c(b(c(c(x1))))
6.53/2.44	   b(a(c(c(x1)))) -> b(b(c(c(x1))))
6.53/2.44	   a(c(b(b(x1)))) -> a(b(a(a(x1))))
6.53/2.44	   c(c(b(b(x1)))) -> c(b(a(a(x1))))
6.53/2.44	   b(c(b(b(x1)))) -> b(b(a(a(x1))))
6.53/2.44	
6.53/2.44	The relative TRS consists of the following S rules:
6.53/2.44	
6.53/2.44	   b(a(b(x1))) -> b(c(a(x1)))
6.53/2.44	   c(c(c(x1))) -> c(a(c(x1)))
6.53/2.44	   a(a(a(c(x1)))) -> a(b(a(b(x1))))
6.53/2.44	   c(a(a(c(x1)))) -> c(b(a(b(x1))))
6.53/2.44	   b(a(a(c(x1)))) -> b(b(a(b(x1))))
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(5) RootLabelingProof (EQUIVALENT)
6.53/2.44	We used plain root labeling [ROOTLAB] with the following heuristic:
6.53/2.44	LabelAll: All function symbols get labeled
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(6)
6.53/2.44	Obligation:
6.53/2.44	Relative term rewrite system:
6.53/2.44	The relative TRS consists of the following R rules:
6.53/2.44	
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	   c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	
6.53/2.44	The relative TRS consists of the following S rules:
6.53/2.44	
6.53/2.44	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1)))
6.53/2.44	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1)))
6.53/2.44	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1)))
6.53/2.44	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
6.53/2.44	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
6.53/2.44	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(7) RelTRSRRRProof (EQUIVALENT)
6.53/2.44	We used the following monotonic ordering for rule removal:
6.53/2.44	Polynomial interpretation [POLO]:
6.53/2.44	
6.53/2.44	   POL(a_{a_1}(x_1)) = x_1
6.53/2.44	   POL(a_{b_1}(x_1)) = x_1
6.53/2.44	   POL(a_{c_1}(x_1)) = x_1
6.53/2.44	   POL(b_{a_1}(x_1)) = x_1
6.53/2.44	   POL(b_{b_1}(x_1)) = x_1
6.53/2.44	   POL(b_{c_1}(x_1)) = x_1
6.53/2.44	   POL(c_{a_1}(x_1)) = x_1
6.53/2.44	   POL(c_{b_1}(x_1)) = x_1
6.53/2.44	   POL(c_{c_1}(x_1)) = 1 + x_1
6.53/2.44	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.53/2.44	Rules from R:
6.53/2.44	
6.53/2.44	   c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   c_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   c_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	Rules from S:
6.53/2.44	
6.53/2.44	   c_{c_1}(c_{c_1}(c_{a_1}(x1))) -> c_{a_1}(a_{c_1}(c_{a_1}(x1)))
6.53/2.44	   c_{c_1}(c_{c_1}(c_{c_1}(x1))) -> c_{a_1}(a_{c_1}(c_{c_1}(x1)))
6.53/2.44	   c_{c_1}(c_{c_1}(c_{b_1}(x1))) -> c_{a_1}(a_{c_1}(c_{b_1}(x1)))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{c_1}(x1))))
6.53/2.44	
6.53/2.44	
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(8)
6.53/2.44	Obligation:
6.53/2.44	Relative term rewrite system:
6.53/2.44	The relative TRS consists of the following R rules:
6.53/2.44	
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	
6.53/2.44	The relative TRS consists of the following S rules:
6.53/2.44	
6.53/2.44	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1)))
6.53/2.44	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1)))
6.53/2.44	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1)))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(9) RelTRSRRRProof (EQUIVALENT)
6.53/2.44	We used the following monotonic ordering for rule removal:
6.53/2.44	Knuth-Bendix order [KBO] with precedence:a_{c_1}_1 > c_{c_1}_1 > b_{a_1}_1 > a_{a_1}_1 > a_{b_1}_1 > b_{c_1}_1 > b_{b_1}_1 > c_{a_1}_1 > c_{b_1}_1
6.53/2.44	
6.53/2.44	and weight map:
6.53/2.44	
6.53/2.44	   a_{a_1}_1=3
6.53/2.44	   a_{c_1}_1=5
6.53/2.44	   c_{c_1}_1=1
6.53/2.44	   c_{a_1}_1=1
6.53/2.44	   a_{b_1}_1=4
6.53/2.44	   b_{c_1}_1=4
6.53/2.44	   c_{b_1}_1=2
6.53/2.44	   b_{a_1}_1=2
6.53/2.44	   b_{b_1}_1=3
6.53/2.44	
6.53/2.44	The variable weight is 1With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
6.53/2.44	Rules from R:
6.53/2.44	
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   a_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   c_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{a_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{c_1}(x1))))
6.53/2.44	   b_{a_1}(a_{c_1}(c_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{c_1}(c_{c_1}(c_{b_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   a_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{c_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
6.53/2.44	   b_{c_1}(c_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
6.53/2.44	Rules from S:
6.53/2.44	
6.53/2.44	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{a_1}(x1)))
6.53/2.44	   b_{a_1}(a_{b_1}(b_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{c_1}(x1)))
6.53/2.44	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(x1)))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   a_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   c_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
6.53/2.44	   b_{a_1}(a_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
6.53/2.44	
6.53/2.44	
6.53/2.44	
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(10)
6.53/2.44	Obligation:
6.53/2.44	Relative term rewrite system:
6.53/2.44	R is empty.
6.53/2.44	S is empty.
6.53/2.44	
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(11) RIsEmptyProof (EQUIVALENT)
6.53/2.44	The TRS R is empty. Hence, termination is trivially proven.
6.53/2.44	----------------------------------------
6.53/2.44	
6.53/2.44	(12)
6.53/2.44	YES
6.69/2.49	EOF