42.71/11.88	YES
45.26/12.53	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
45.26/12.53	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
45.26/12.53	
45.26/12.53	
45.26/12.53	Termination w.r.t. Q of the given QTRS could be proven:
45.26/12.53	
45.26/12.53	(0) QTRS
45.26/12.53	(1) FlatCCProof [EQUIVALENT, 0 ms]
45.26/12.53	(2) QTRS
45.26/12.53	(3) RootLabelingProof [EQUIVALENT, 0 ms]
45.26/12.53	(4) QTRS
45.26/12.53	(5) QTRSRRRProof [EQUIVALENT, 47 ms]
45.26/12.53	(6) QTRS
45.26/12.53	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
45.26/12.53	(8) QDP
45.26/12.53	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
45.26/12.53	(10) AND
45.26/12.53	    (11) QDP
45.26/12.53	        (12) UsableRulesProof [EQUIVALENT, 0 ms]
45.26/12.53	        (13) QDP
45.26/12.53	        (14) QDPSizeChangeProof [EQUIVALENT, 0 ms]
45.26/12.53	        (15) YES
45.26/12.53	    (16) QDP
45.26/12.53	        (17) UsableRulesProof [EQUIVALENT, 0 ms]
45.26/12.53	        (18) QDP
45.26/12.53	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
45.26/12.53	        (20) YES
45.26/12.53	    (21) QDP
45.26/12.53	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
45.26/12.53	        (23) QDP
45.26/12.53	        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
45.26/12.53	        (25) YES
45.26/12.53	    (26) QDP
45.26/12.53	        (27) UsableRulesProof [EQUIVALENT, 0 ms]
45.26/12.53	        (28) QDP
45.26/12.53	        (29) QDPSizeChangeProof [EQUIVALENT, 0 ms]
45.26/12.53	        (30) YES
45.26/12.53	    (31) QDP
45.26/12.53	        (32) UsableRulesProof [EQUIVALENT, 0 ms]
45.26/12.53	        (33) QDP
45.26/12.53	        (34) QDPSizeChangeProof [EQUIVALENT, 0 ms]
45.26/12.53	        (35) YES
45.26/12.53	    (36) QDP
45.26/12.53	        (37) QDPOrderProof [EQUIVALENT, 17 ms]
45.26/12.53	        (38) QDP
45.26/12.53	        (39) UsableRulesProof [EQUIVALENT, 0 ms]
45.26/12.53	        (40) QDP
45.26/12.53	        (41) QDPSizeChangeProof [EQUIVALENT, 0 ms]
45.26/12.53	        (42) YES
45.26/12.53	
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(0)
45.26/12.53	Obligation:
45.26/12.53	Q restricted rewrite system:
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a(b(c(x1))) -> c(c(b(b(a(a(x1))))))
45.26/12.53	   a(x1) -> x1
45.26/12.53	   b(x1) -> x1
45.26/12.53	   c(x1) -> x1
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(1) FlatCCProof (EQUIVALENT)
45.26/12.53	We used flat context closure [ROOTLAB]
45.26/12.53	As Q is empty the flat context closure was sound AND complete.
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(2)
45.26/12.53	Obligation:
45.26/12.53	Q restricted rewrite system:
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a(a(b(c(x1)))) -> a(c(c(b(b(a(a(x1)))))))
45.26/12.53	   b(a(b(c(x1)))) -> b(c(c(b(b(a(a(x1)))))))
45.26/12.53	   c(a(b(c(x1)))) -> c(c(c(b(b(a(a(x1)))))))
45.26/12.53	   a(a(x1)) -> a(x1)
45.26/12.53	   b(a(x1)) -> b(x1)
45.26/12.53	   c(a(x1)) -> c(x1)
45.26/12.53	   a(b(x1)) -> a(x1)
45.26/12.53	   b(b(x1)) -> b(x1)
45.26/12.53	   c(b(x1)) -> c(x1)
45.26/12.53	   a(c(x1)) -> a(x1)
45.26/12.53	   b(c(x1)) -> b(x1)
45.26/12.53	   c(c(x1)) -> c(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(3) RootLabelingProof (EQUIVALENT)
45.26/12.53	We used plain root labeling [ROOTLAB] with the following heuristic:
45.26/12.53	LabelAll: All function symbols get labeled
45.26/12.53	
45.26/12.53	As Q is empty the root labeling was sound AND complete.
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(4)
45.26/12.53	Obligation:
45.26/12.53	Q restricted rewrite system:
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))
45.26/12.53	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	   a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(5) QTRSRRRProof (EQUIVALENT)
45.26/12.53	Used ordering:
45.26/12.53	Polynomial interpretation [POLO]:
45.26/12.53	
45.26/12.53	   POL(a_{a_1}(x_1)) = x_1
45.26/12.53	   POL(a_{b_1}(x_1)) = 1 + x_1
45.26/12.53	   POL(a_{c_1}(x_1)) = x_1
45.26/12.53	   POL(b_{a_1}(x_1)) = 1 + x_1
45.26/12.53	   POL(b_{b_1}(x_1)) = x_1
45.26/12.53	   POL(b_{c_1}(x_1)) = 1 + x_1
45.26/12.53	   POL(c_{a_1}(x_1)) = 1 + x_1
45.26/12.53	   POL(c_{b_1}(x_1)) = 1 + x_1
45.26/12.53	   POL(c_{c_1}(x_1)) = x_1
45.26/12.53	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))
45.26/12.53	   c_{a_1}(a_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))))
45.26/12.53	   c_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   c_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	   a_{c_1}(c_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   b_{c_1}(c_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{c_1}(c_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	
45.26/12.53	
45.26/12.53	
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(6)
45.26/12.53	Obligation:
45.26/12.53	Q restricted rewrite system:
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(7) DependencyPairsProof (EQUIVALENT)
45.26/12.53	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(8)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(x1)))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
45.26/12.53	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(x1)
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{B_1}(x1)
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{C_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{C_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> C_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1))))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(x1)))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
45.26/12.53	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{C_1}(x1)
45.26/12.53	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
45.26/12.53	   B_{A_1}(a_{c_1}(x1)) -> B_{C_1}(x1)
45.26/12.53	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
45.26/12.53	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
45.26/12.53	   C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
45.26/12.53	   A_{C_1}(c_{b_1}(x1)) -> A_{B_1}(x1)
45.26/12.53	   A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1)
45.26/12.53	   B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
45.26/12.53	
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(9) DependencyGraphProof (EQUIVALENT)
45.26/12.53	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 6 SCCs with 22 less nodes.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(10)
45.26/12.53	Complex Obligation (AND)
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(11)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
45.26/12.53	
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(12) UsableRulesProof (EQUIVALENT)
45.26/12.53	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(13)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
45.26/12.53	
45.26/12.53	R is empty.
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(14) QDPSizeChangeProof (EQUIVALENT)
45.26/12.53	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
45.26/12.53	
45.26/12.53	From the DPs we obtained the following set of size-change graphs:
45.26/12.53	*B_{C_1}(c_{c_1}(x1)) -> B_{C_1}(x1)
45.26/12.53	The graph contains the following edges 1 > 1
45.26/12.53	
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(15)
45.26/12.53	YES
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(16)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
45.26/12.53	
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(17) UsableRulesProof (EQUIVALENT)
45.26/12.53	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(18)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
45.26/12.53	
45.26/12.53	R is empty.
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(19) QDPSizeChangeProof (EQUIVALENT)
45.26/12.53	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
45.26/12.53	
45.26/12.53	From the DPs we obtained the following set of size-change graphs:
45.26/12.53	*C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
45.26/12.53	The graph contains the following edges 1 > 1
45.26/12.53	
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(20)
45.26/12.53	YES
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(21)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
45.26/12.53	
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(22) UsableRulesProof (EQUIVALENT)
45.26/12.53	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(23)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
45.26/12.53	
45.26/12.53	R is empty.
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(24) QDPSizeChangeProof (EQUIVALENT)
45.26/12.53	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
45.26/12.53	
45.26/12.53	From the DPs we obtained the following set of size-change graphs:
45.26/12.53	*A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
45.26/12.53	The graph contains the following edges 1 > 1
45.26/12.53	
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(25)
45.26/12.53	YES
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(26)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1)
45.26/12.53	
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.53	
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(27) UsableRulesProof (EQUIVALENT)
45.26/12.53	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(28)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1)
45.26/12.53	
45.26/12.53	R is empty.
45.26/12.53	Q is empty.
45.26/12.53	We have to consider all minimal (P,Q,R)-chains.
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(29) QDPSizeChangeProof (EQUIVALENT)
45.26/12.53	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
45.26/12.53	
45.26/12.53	From the DPs we obtained the following set of size-change graphs:
45.26/12.53	*A_{C_1}(c_{c_1}(x1)) -> A_{C_1}(x1)
45.26/12.53	The graph contains the following edges 1 > 1
45.26/12.53	
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(30)
45.26/12.53	YES
45.26/12.53	
45.26/12.53	----------------------------------------
45.26/12.53	
45.26/12.53	(31)
45.26/12.53	Obligation:
45.26/12.53	Q DP problem:
45.26/12.53	The TRS P consists of the following rules:
45.26/12.53	
45.26/12.53	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
45.26/12.53	
45.26/12.53	The TRS R consists of the following rules:
45.26/12.53	
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.53	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.53	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.53	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.53	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.53	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.53	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.53	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.53	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.53	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.54	
45.26/12.54	Q is empty.
45.26/12.54	We have to consider all minimal (P,Q,R)-chains.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(32) UsableRulesProof (EQUIVALENT)
45.26/12.54	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(33)
45.26/12.54	Obligation:
45.26/12.54	Q DP problem:
45.26/12.54	The TRS P consists of the following rules:
45.26/12.54	
45.26/12.54	   C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
45.26/12.54	
45.26/12.54	R is empty.
45.26/12.54	Q is empty.
45.26/12.54	We have to consider all minimal (P,Q,R)-chains.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(34) QDPSizeChangeProof (EQUIVALENT)
45.26/12.54	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
45.26/12.54	
45.26/12.54	From the DPs we obtained the following set of size-change graphs:
45.26/12.54	*C_{A_1}(a_{a_1}(x1)) -> C_{A_1}(x1)
45.26/12.54	The graph contains the following edges 1 > 1
45.26/12.54	
45.26/12.54	
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(35)
45.26/12.54	YES
45.26/12.54	
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(36)
45.26/12.54	Obligation:
45.26/12.54	Q DP problem:
45.26/12.54	The TRS P consists of the following rules:
45.26/12.54	
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
45.26/12.54	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
45.26/12.54	
45.26/12.54	The TRS R consists of the following rules:
45.26/12.54	
45.26/12.54	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.54	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.54	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.54	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.54	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.54	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.54	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.54	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.54	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.54	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.54	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.54	
45.26/12.54	Q is empty.
45.26/12.54	We have to consider all minimal (P,Q,R)-chains.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(37) QDPOrderProof (EQUIVALENT)
45.26/12.54	We use the reduction pair processor [LPAR04,JAR06].
45.26/12.54	
45.26/12.54	
45.26/12.54	The following pairs can be oriented strictly and are deleted.
45.26/12.54	
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{c_1}(x1)))
45.26/12.54	   B_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
45.26/12.54	   A_{A_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> A_{A_1}(a_{c_1}(x1))
45.26/12.54	The remaining pairs can at least be oriented weakly.
45.26/12.54	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
45.26/12.54	
45.26/12.54	POL( A_{A_1}_1(x_1) ) = max{0, 2x_1 - 2}
45.26/12.54	POL( B_{A_1}_1(x_1) ) = 2x_1 + 2
45.26/12.54	POL( a_{b_1}_1(x_1) ) = 2x_1
45.26/12.54	POL( b_{b_1}_1(x_1) ) = x_1
45.26/12.54	POL( a_{a_1}_1(x_1) ) = x_1
45.26/12.54	POL( b_{c_1}_1(x_1) ) = x_1
45.26/12.54	POL( c_{b_1}_1(x_1) ) = 2x_1 + 2
45.26/12.54	POL( a_{c_1}_1(x_1) ) = x_1
45.26/12.54	POL( c_{c_1}_1(x_1) ) = x_1 + 2
45.26/12.54	POL( b_{a_1}_1(x_1) ) = x_1
45.26/12.54	POL( c_{a_1}_1(x_1) ) = 0
45.26/12.54	
45.26/12.54	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
45.26/12.54	
45.26/12.54	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.54	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.54	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.54	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.54	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.54	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.54	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.54	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.54	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.54	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.54	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.54	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	
45.26/12.54	
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(38)
45.26/12.54	Obligation:
45.26/12.54	Q DP problem:
45.26/12.54	The TRS P consists of the following rules:
45.26/12.54	
45.26/12.54	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
45.26/12.54	
45.26/12.54	The TRS R consists of the following rules:
45.26/12.54	
45.26/12.54	   a_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.54	   a_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.54	   b_{a_1}(a_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))))
45.26/12.54	   b_{a_1}(a_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{c_1}(c_{c_1}(c_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(a_{c_1}(x1)))))))
45.26/12.54	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
45.26/12.54	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.54	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.54	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
45.26/12.54	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
45.26/12.54	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   a_{c_1}(c_{b_1}(x1)) -> a_{b_1}(x1)
45.26/12.54	   a_{c_1}(c_{c_1}(x1)) -> a_{c_1}(x1)
45.26/12.54	   b_{c_1}(c_{c_1}(x1)) -> b_{c_1}(x1)
45.26/12.54	   c_{c_1}(c_{a_1}(x1)) -> c_{a_1}(x1)
45.26/12.54	   c_{c_1}(c_{b_1}(x1)) -> c_{b_1}(x1)
45.26/12.54	   c_{c_1}(c_{c_1}(x1)) -> c_{c_1}(x1)
45.26/12.54	
45.26/12.54	Q is empty.
45.26/12.54	We have to consider all minimal (P,Q,R)-chains.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(39) UsableRulesProof (EQUIVALENT)
45.26/12.54	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(40)
45.26/12.54	Obligation:
45.26/12.54	Q DP problem:
45.26/12.54	The TRS P consists of the following rules:
45.26/12.54	
45.26/12.54	   B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
45.26/12.54	
45.26/12.54	R is empty.
45.26/12.54	Q is empty.
45.26/12.54	We have to consider all minimal (P,Q,R)-chains.
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(41) QDPSizeChangeProof (EQUIVALENT)
45.26/12.54	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
45.26/12.54	
45.26/12.54	From the DPs we obtained the following set of size-change graphs:
45.26/12.54	*B_{A_1}(a_{a_1}(x1)) -> B_{A_1}(x1)
45.26/12.54	The graph contains the following edges 1 > 1
45.26/12.54	
45.26/12.54	
45.26/12.54	----------------------------------------
45.26/12.54	
45.26/12.54	(42)
45.26/12.54	YES
45.43/12.63	EOF