40.44/11.15	YES
40.44/11.16	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.44/11.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.44/11.16	
40.44/11.16	
40.44/11.16	Termination of the given RelTRS could be proven:
40.44/11.16	
40.44/11.16	(0) RelTRS
40.44/11.16	(1) FlatCCProof [EQUIVALENT, 0 ms]
40.44/11.16	(2) RelTRS
40.44/11.16	(3) RootLabelingProof [EQUIVALENT, 0 ms]
40.44/11.16	(4) RelTRS
40.44/11.16	(5) RelTRSRRRProof [EQUIVALENT, 105 ms]
40.44/11.16	(6) RelTRS
40.44/11.16	(7) RelTRSRRRProof [EQUIVALENT, 25 ms]
40.44/11.16	(8) RelTRS
40.44/11.16	(9) RelTRSRRRProof [EQUIVALENT, 378 ms]
40.44/11.16	(10) RelTRS
40.44/11.16	(11) RelTRSRRRProof [EQUIVALENT, 341 ms]
40.44/11.16	(12) RelTRS
40.44/11.16	(13) RelTRSRRRProof [EQUIVALENT, 16 ms]
40.44/11.16	(14) RelTRS
40.44/11.16	(15) RIsEmptyProof [EQUIVALENT, 0 ms]
40.44/11.16	(16) YES
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(0)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a(a(c(x1))) -> a(c(c(x1)))
40.44/11.16	   b(b(c(x1))) -> a(b(b(x1)))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   c(c(b(x1))) -> a(c(a(x1)))
40.44/11.16	   b(a(c(x1))) -> c(a(c(x1)))
40.44/11.16	   c(b(c(x1))) -> b(b(b(x1)))
40.44/11.16	   b(b(a(x1))) -> b(a(c(x1)))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(1) FlatCCProof (EQUIVALENT)
40.44/11.16	We used flat context closure [ROOTLAB]
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(2)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a(a(c(x1))) -> a(c(c(x1)))
40.44/11.16	   a(b(b(c(x1)))) -> a(a(b(b(x1))))
40.44/11.16	   c(b(b(c(x1)))) -> c(a(b(b(x1))))
40.44/11.16	   b(b(b(c(x1)))) -> b(a(b(b(x1))))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b(b(a(x1))) -> b(a(c(x1)))
40.44/11.16	   a(c(c(b(x1)))) -> a(a(c(a(x1))))
40.44/11.16	   c(c(c(b(x1)))) -> c(a(c(a(x1))))
40.44/11.16	   b(c(c(b(x1)))) -> b(a(c(a(x1))))
40.44/11.16	   a(b(a(c(x1)))) -> a(c(a(c(x1))))
40.44/11.16	   c(b(a(c(x1)))) -> c(c(a(c(x1))))
40.44/11.16	   b(b(a(c(x1)))) -> b(c(a(c(x1))))
40.44/11.16	   a(c(b(c(x1)))) -> a(b(b(b(x1))))
40.44/11.16	   c(c(b(c(x1)))) -> c(b(b(b(x1))))
40.44/11.16	   b(c(b(c(x1)))) -> b(b(b(b(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(3) RootLabelingProof (EQUIVALENT)
40.44/11.16	We used plain root labeling [ROOTLAB] with the following heuristic:
40.44/11.16	LabelAll: All function symbols get labeled
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(4)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
40.44/11.16	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
40.44/11.16	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
40.44/11.16	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(5) RelTRSRRRProof (EQUIVALENT)
40.44/11.16	We used the following monotonic ordering for rule removal:
40.44/11.16	Polynomial interpretation [POLO]:
40.44/11.16	
40.44/11.16	   POL(a_{a_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(a_{b_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(a_{c_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(b_{a_1}(x_1)) = x_1
40.44/11.16	   POL(b_{b_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(b_{c_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(c_{a_1}(x_1)) = x_1
40.44/11.16	   POL(c_{b_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(c_{c_1}(x_1)) = 1 + x_1
40.44/11.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
40.44/11.16	Rules from R:
40.44/11.16	
40.44/11.16	   c_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   b_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	Rules from S:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{c_1}(c_{a_1}(x1)))
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
40.44/11.16	   c_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> c_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
40.44/11.16	   b_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   b_{c_1}(c_{c_1}(c_{b_1}(b_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{c_1}(x1))))
40.44/11.16	   b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(x1)))) -> b_{a_1}(a_{c_1}(c_{a_1}(a_{b_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(6)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(7) RelTRSRRRProof (EQUIVALENT)
40.44/11.16	We used the following monotonic ordering for rule removal:
40.44/11.16	Polynomial interpretation [POLO]:
40.44/11.16	
40.44/11.16	   POL(a_{a_1}(x_1)) = x_1
40.44/11.16	   POL(a_{b_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(a_{c_1}(x_1)) = x_1
40.44/11.16	   POL(b_{a_1}(x_1)) = x_1
40.44/11.16	   POL(b_{b_1}(x_1)) = x_1
40.44/11.16	   POL(b_{c_1}(x_1)) = x_1
40.44/11.16	   POL(c_{a_1}(x_1)) = x_1
40.44/11.16	   POL(c_{b_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(c_{c_1}(x_1)) = x_1
40.44/11.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
40.44/11.16	Rules from R:
40.44/11.16	
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	Rules from S:
40.44/11.16	
40.44/11.16	   a_{c_1}(c_{c_1}(c_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{c_1}(c_{a_1}(a_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   a_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   c_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(8)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(9) RelTRSRRRProof (EQUIVALENT)
40.44/11.16	We used the following monotonic ordering for rule removal:
40.44/11.16	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 2]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(a_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 2], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(c_{a_1}(x_1)) =  	[[0], [2]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(c_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
40.44/11.16	Rules from R:
40.44/11.16	
40.44/11.16	   a_{a_1}(a_{c_1}(c_{a_1}(x1))) -> a_{c_1}(c_{c_1}(c_{a_1}(x1)))
40.44/11.16	Rules from S:
40.44/11.16	none
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(10)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(11) RelTRSRRRProof (EQUIVALENT)
40.44/11.16	We used the following monotonic ordering for rule removal:
40.44/11.16	Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(a_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[1, 2], [0, 2]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(a_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [1, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(c_{c_1}(x_1)) =  	[[0], [2]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(c_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(a_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 1], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(b_{b_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(b_{c_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(c_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	   <<<
40.44/11.16	 POL(b_{a_1}(x_1)) =  	[[0], [0]] 	 +  	[[2, 0], [0, 0]] 	* 	x_1
40.44/11.16	>>>
40.44/11.16	
40.44/11.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
40.44/11.16	Rules from R:
40.44/11.16	
40.44/11.16	   a_{a_1}(a_{c_1}(c_{c_1}(x1))) -> a_{c_1}(c_{c_1}(c_{c_1}(x1)))
40.44/11.16	   a_{a_1}(a_{c_1}(c_{b_1}(x1))) -> a_{c_1}(c_{c_1}(c_{b_1}(x1)))
40.44/11.16	Rules from S:
40.44/11.16	none
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(12)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	The relative TRS consists of the following R rules:
40.44/11.16	
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(13) RelTRSRRRProof (EQUIVALENT)
40.44/11.16	We used the following monotonic ordering for rule removal:
40.44/11.16	Polynomial interpretation [POLO]:
40.44/11.16	
40.44/11.16	   POL(a_{a_1}(x_1)) = x_1
40.44/11.16	   POL(a_{b_1}(x_1)) = x_1
40.44/11.16	   POL(a_{c_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(b_{a_1}(x_1)) = x_1
40.44/11.16	   POL(b_{b_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(b_{c_1}(x_1)) = x_1
40.44/11.16	   POL(c_{a_1}(x_1)) = 1 + x_1
40.44/11.16	   POL(c_{b_1}(x_1)) = x_1
40.44/11.16	   POL(c_{c_1}(x_1)) = 1 + x_1
40.44/11.16	With this ordering the following rules can be removed [MATRO] because they are oriented strictly:
40.44/11.16	Rules from R:
40.44/11.16	
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{b_1}(b_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	Rules from S:
40.44/11.16	none
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(14)
40.44/11.16	Obligation:
40.44/11.16	Relative term rewrite system:
40.44/11.16	R is empty.
40.44/11.16	The relative TRS consists of the following S rules:
40.44/11.16	
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{a_1}(a_{c_1}(c_{c_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{c_1}(c_{b_1}(x1)))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{a_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{c_1}(x1))))
40.44/11.16	   b_{b_1}(b_{a_1}(a_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{a_1}(a_{c_1}(c_{b_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.44/11.16	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{b_1}(b_{b_1}(b_{b_1}(b_{c_1}(x1))))
40.44/11.16	
40.44/11.16	
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(15) RIsEmptyProof (EQUIVALENT)
40.44/11.16	The TRS R is empty. Hence, termination is trivially proven.
40.44/11.16	----------------------------------------
40.44/11.16	
40.44/11.16	(16)
40.44/11.16	YES
40.51/11.20	EOF