21.15/6.31	YES
21.15/6.34	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
21.15/6.34	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
21.15/6.34	
21.15/6.34	
21.15/6.34	Termination w.r.t. Q of the given QTRS could be proven:
21.15/6.34	
21.15/6.34	(0) QTRS
21.15/6.34	(1) QTRS Reverse [EQUIVALENT, 0 ms]
21.15/6.34	(2) QTRS
21.15/6.34	(3) FlatCCProof [EQUIVALENT, 0 ms]
21.15/6.34	(4) QTRS
21.15/6.34	(5) RootLabelingProof [EQUIVALENT, 0 ms]
21.15/6.34	(6) QTRS
21.15/6.34	(7) QTRSRRRProof [EQUIVALENT, 54 ms]
21.15/6.34	(8) QTRS
21.15/6.34	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
21.15/6.34	(10) QDP
21.15/6.34	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
21.15/6.34	(12) AND
21.15/6.34	    (13) QDP
21.15/6.34	        (14) UsableRulesProof [EQUIVALENT, 2 ms]
21.15/6.34	        (15) QDP
21.15/6.34	        (16) QDPSizeChangeProof [EQUIVALENT, 1 ms]
21.15/6.34	        (17) YES
21.15/6.34	    (18) QDP
21.15/6.34	        (19) QDPOrderProof [EQUIVALENT, 48 ms]
21.15/6.34	        (20) QDP
21.15/6.34	        (21) UsableRulesProof [EQUIVALENT, 0 ms]
21.15/6.34	        (22) QDP
21.15/6.34	        (23) QDPSizeChangeProof [EQUIVALENT, 0 ms]
21.15/6.34	        (24) YES
21.15/6.34	
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(0)
21.15/6.34	Obligation:
21.15/6.34	Q restricted rewrite system:
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   a(x1) -> x1
21.15/6.34	   a(b(x1)) -> b(b(a(c(x1))))
21.15/6.34	   b(x1) -> x1
21.15/6.34	   c(b(c(x1))) -> a(x1)
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(1) QTRS Reverse (EQUIVALENT)
21.15/6.34	We applied the QTRS Reverse Processor [REVERSE].
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(2)
21.15/6.34	Obligation:
21.15/6.34	Q restricted rewrite system:
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   a(x1) -> x1
21.15/6.34	   b(a(x1)) -> c(a(b(b(x1))))
21.15/6.34	   b(x1) -> x1
21.15/6.34	   c(b(c(x1))) -> a(x1)
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(3) FlatCCProof (EQUIVALENT)
21.15/6.34	We used flat context closure [ROOTLAB]
21.15/6.34	As Q is empty the flat context closure was sound AND complete.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(4)
21.15/6.34	Obligation:
21.15/6.34	Q restricted rewrite system:
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   a(a(x1)) -> a(x1)
21.15/6.34	   b(a(x1)) -> b(x1)
21.15/6.34	   c(a(x1)) -> c(x1)
21.15/6.34	   a(b(a(x1))) -> a(c(a(b(b(x1)))))
21.15/6.34	   b(b(a(x1))) -> b(c(a(b(b(x1)))))
21.15/6.34	   c(b(a(x1))) -> c(c(a(b(b(x1)))))
21.15/6.34	   a(b(x1)) -> a(x1)
21.15/6.34	   b(b(x1)) -> b(x1)
21.15/6.34	   c(b(x1)) -> c(x1)
21.15/6.34	   a(c(b(c(x1)))) -> a(a(x1))
21.15/6.34	   b(c(b(c(x1)))) -> b(a(x1))
21.15/6.34	   c(c(b(c(x1)))) -> c(a(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(5) RootLabelingProof (EQUIVALENT)
21.15/6.34	We used plain root labeling [ROOTLAB] with the following heuristic:
21.15/6.34	LabelAll: All function symbols get labeled
21.15/6.34	
21.15/6.34	As Q is empty the root labeling was sound AND complete.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(6)
21.15/6.34	Obligation:
21.15/6.34	Q restricted rewrite system:
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(x1))
21.15/6.34	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(x1))
21.15/6.34	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(x1))
21.15/6.34	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(7) QTRSRRRProof (EQUIVALENT)
21.15/6.34	Used ordering:
21.15/6.34	Polynomial interpretation [POLO]:
21.15/6.34	
21.15/6.34	   POL(a_{a_1}(x_1)) = 2 + x_1
21.15/6.34	   POL(a_{b_1}(x_1)) = 2 + x_1
21.15/6.34	   POL(a_{c_1}(x_1)) = 2 + x_1
21.15/6.34	   POL(b_{a_1}(x_1)) = x_1
21.15/6.34	   POL(b_{b_1}(x_1)) = x_1
21.15/6.34	   POL(b_{c_1}(x_1)) = x_1
21.15/6.34	   POL(c_{a_1}(x_1)) = x_1
21.15/6.34	   POL(c_{b_1}(x_1)) = 2 + x_1
21.15/6.34	   POL(c_{c_1}(x_1)) = 1 + x_1
21.15/6.34	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
21.15/6.34	
21.15/6.34	   a_{a_1}(a_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{a_1}(a_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{a_1}(a_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{a_1}(a_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{a_1}(a_{a_1}(x1)) -> c_{a_1}(x1)
21.15/6.34	   c_{a_1}(a_{c_1}(x1)) -> c_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{a_1}(a_{a_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   c_{b_1}(b_{a_1}(a_{b_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   c_{b_1}(b_{a_1}(a_{c_1}(x1))) -> c_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   c_{b_1}(b_{a_1}(x1)) -> c_{a_1}(x1)
21.15/6.34	   c_{b_1}(b_{c_1}(x1)) -> c_{c_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(x1))
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> a_{a_1}(a_{c_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> b_{a_1}(a_{c_1}(x1))
21.15/6.34	   c_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> c_{a_1}(a_{a_1}(x1))
21.15/6.34	   c_{c_1}(c_{b_1}(b_{c_1}(c_{b_1}(x1)))) -> c_{a_1}(a_{b_1}(x1))
21.15/6.34	   c_{c_1}(c_{b_1}(b_{c_1}(c_{c_1}(x1)))) -> c_{a_1}(a_{c_1}(x1))
21.15/6.34	
21.15/6.34	
21.15/6.34	
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(8)
21.15/6.34	Obligation:
21.15/6.34	Q restricted rewrite system:
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(9) DependencyPairsProof (EQUIVALENT)
21.15/6.34	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(10)
21.15/6.34	Obligation:
21.15/6.34	Q DP problem:
21.15/6.34	The TRS P consists of the following rules:
21.15/6.34	
21.15/6.34	   C_{A_1}(a_{b_1}(x1)) -> C_{B_1}(x1)
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> C_{A_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1))))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{C_1}(x1)
21.15/6.34	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
21.15/6.34	   A_{B_1}(b_{c_1}(x1)) -> A_{C_1}(x1)
21.15/6.34	   C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
21.15/6.34	
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	We have to consider all minimal (P,Q,R)-chains.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(11) DependencyGraphProof (EQUIVALENT)
21.15/6.34	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 16 less nodes.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(12)
21.15/6.34	Complex Obligation (AND)
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(13)
21.15/6.34	Obligation:
21.15/6.34	Q DP problem:
21.15/6.34	The TRS P consists of the following rules:
21.15/6.34	
21.15/6.34	   C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
21.15/6.34	
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	We have to consider all minimal (P,Q,R)-chains.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(14) UsableRulesProof (EQUIVALENT)
21.15/6.34	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.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(15)
21.15/6.34	Obligation:
21.15/6.34	Q DP problem:
21.15/6.34	The TRS P consists of the following rules:
21.15/6.34	
21.15/6.34	   C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
21.15/6.34	
21.15/6.34	R is empty.
21.15/6.34	Q is empty.
21.15/6.34	We have to consider all minimal (P,Q,R)-chains.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(16) QDPSizeChangeProof (EQUIVALENT)
21.15/6.34	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. 
21.15/6.34	
21.15/6.34	From the DPs we obtained the following set of size-change graphs:
21.15/6.34	*C_{B_1}(b_{b_1}(x1)) -> C_{B_1}(x1)
21.15/6.34	The graph contains the following edges 1 > 1
21.15/6.34	
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(17)
21.15/6.34	YES
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(18)
21.15/6.34	Obligation:
21.15/6.34	Q DP problem:
21.15/6.34	The TRS P consists of the following rules:
21.15/6.34	
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.15/6.34	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
21.15/6.34	
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	We have to consider all minimal (P,Q,R)-chains.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(19) QDPOrderProof (EQUIVALENT)
21.15/6.34	We use the reduction pair processor [LPAR04,JAR06].
21.15/6.34	
21.15/6.34	
21.15/6.34	The following pairs can be oriented strictly and are deleted.
21.15/6.34	
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> A_{B_1}(b_{b_1}(b_{c_1}(x1)))
21.15/6.34	   A_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.15/6.34	   B_{B_1}(b_{a_1}(a_{c_1}(x1))) -> B_{B_1}(b_{c_1}(x1))
21.15/6.34	The remaining pairs can at least be oriented weakly.
21.15/6.34	Used ordering:  Polynomial interpretation [POLO]:
21.15/6.34	
21.15/6.34	   POL(A_{B_1}(x_1)) = x_1
21.15/6.34	   POL(B_{B_1}(x_1)) = x_1
21.15/6.34	   POL(a_{a_1}(x_1)) = 1 + x_1
21.15/6.34	   POL(a_{b_1}(x_1)) = 1 + x_1
21.15/6.34	   POL(a_{c_1}(x_1)) = 1 + x_1
21.15/6.34	   POL(b_{a_1}(x_1)) = x_1
21.15/6.34	   POL(b_{b_1}(x_1)) = x_1
21.15/6.34	   POL(b_{c_1}(x_1)) = x_1
21.15/6.34	   POL(c_{a_1}(x_1)) = x_1
21.15/6.34	   POL(c_{b_1}(x_1)) = 1 + x_1
21.15/6.34	
21.15/6.34	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
21.15/6.34	
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(20)
21.15/6.34	Obligation:
21.15/6.34	Q DP problem:
21.15/6.34	The TRS P consists of the following rules:
21.15/6.34	
21.15/6.34	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
21.15/6.34	
21.15/6.34	The TRS R consists of the following rules:
21.15/6.34	
21.15/6.34	   c_{a_1}(a_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(a_{c_1}(x1))) -> a_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
21.15/6.34	   b_{b_1}(b_{a_1}(a_{c_1}(x1))) -> b_{c_1}(c_{a_1}(a_{b_1}(b_{b_1}(b_{c_1}(x1)))))
21.15/6.34	   a_{b_1}(b_{a_1}(x1)) -> a_{a_1}(x1)
21.15/6.34	   a_{b_1}(b_{b_1}(x1)) -> a_{b_1}(x1)
21.15/6.34	   a_{b_1}(b_{c_1}(x1)) -> a_{c_1}(x1)
21.15/6.34	   b_{b_1}(b_{a_1}(x1)) -> b_{a_1}(x1)
21.15/6.34	   b_{b_1}(b_{b_1}(x1)) -> b_{b_1}(x1)
21.15/6.34	   b_{b_1}(b_{c_1}(x1)) -> b_{c_1}(x1)
21.15/6.34	   c_{b_1}(b_{b_1}(x1)) -> c_{b_1}(x1)
21.15/6.34	   a_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(x1))
21.15/6.34	   b_{c_1}(c_{b_1}(b_{c_1}(c_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(x1))
21.15/6.34	
21.15/6.34	Q is empty.
21.15/6.34	We have to consider all minimal (P,Q,R)-chains.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(21) UsableRulesProof (EQUIVALENT)
21.15/6.34	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.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(22)
21.15/6.34	Obligation:
21.15/6.34	Q DP problem:
21.15/6.34	The TRS P consists of the following rules:
21.15/6.34	
21.15/6.34	   A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
21.15/6.34	
21.15/6.34	R is empty.
21.15/6.34	Q is empty.
21.15/6.34	We have to consider all minimal (P,Q,R)-chains.
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(23) QDPSizeChangeProof (EQUIVALENT)
21.15/6.34	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. 
21.15/6.34	
21.15/6.34	From the DPs we obtained the following set of size-change graphs:
21.15/6.34	*A_{B_1}(b_{b_1}(x1)) -> A_{B_1}(x1)
21.15/6.34	The graph contains the following edges 1 > 1
21.15/6.34	
21.15/6.34	
21.15/6.34	----------------------------------------
21.15/6.34	
21.15/6.34	(24)
21.15/6.34	YES
21.40/6.42	EOF