20.39/6.16	YES
20.66/6.22	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
20.66/6.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.66/6.22	
20.66/6.22	
20.66/6.22	Termination w.r.t. Q of the given QTRS could be proven:
20.66/6.22	
20.66/6.22	(0) QTRS
20.66/6.22	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.66/6.22	(2) QTRS
20.66/6.22	(3) FlatCCProof [EQUIVALENT, 0 ms]
20.66/6.22	(4) QTRS
20.66/6.22	(5) RootLabelingProof [EQUIVALENT, 0 ms]
20.66/6.22	(6) QTRS
20.66/6.22	(7) QTRSRRRProof [EQUIVALENT, 91 ms]
20.66/6.22	(8) QTRS
20.66/6.22	(9) DependencyPairsProof [EQUIVALENT, 49 ms]
20.66/6.22	(10) QDP
20.66/6.22	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
20.66/6.22	(12) AND
20.66/6.22	    (13) QDP
20.66/6.22	        (14) UsableRulesProof [EQUIVALENT, 1 ms]
20.66/6.22	        (15) QDP
20.66/6.22	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
20.66/6.22	        (17) YES
20.66/6.22	    (18) QDP
20.66/6.22	        (19) QDPOrderProof [EQUIVALENT, 62 ms]
20.66/6.22	        (20) QDP
20.66/6.22	        (21) PisEmptyProof [EQUIVALENT, 0 ms]
20.66/6.22	        (22) YES
20.66/6.22	
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(0)
20.66/6.22	Obligation:
20.66/6.22	Q restricted rewrite system:
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   a(x1) -> b(x1)
20.66/6.22	   a(a(x1)) -> a(b(a(x1)))
20.66/6.22	   a(b(x1)) -> b(b(b(x1)))
20.66/6.22	   a(a(a(x1))) -> a(a(b(a(a(x1)))))
20.66/6.22	   a(a(b(x1))) -> a(b(b(a(b(x1)))))
20.66/6.22	   a(b(a(x1))) -> b(a(b(b(a(x1)))))
20.66/6.22	   a(b(b(x1))) -> b(b(b(b(b(x1)))))
20.66/6.22	   b(a(x1)) -> b(b(b(x1)))
20.66/6.22	   a(b(a(x1))) -> a(b(b(a(b(x1)))))
20.66/6.22	   b(a(a(x1))) -> b(a(b(b(a(x1)))))
20.66/6.22	   b(b(a(x1))) -> b(b(b(b(b(x1)))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(1) QTRS Reverse (EQUIVALENT)
20.66/6.22	We applied the QTRS Reverse Processor [REVERSE].
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(2)
20.66/6.22	Obligation:
20.66/6.22	Q restricted rewrite system:
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   a(x1) -> b(x1)
20.66/6.22	   a(a(x1)) -> a(b(a(x1)))
20.66/6.22	   b(a(x1)) -> b(b(b(x1)))
20.66/6.22	   a(a(a(x1))) -> a(a(b(a(a(x1)))))
20.66/6.22	   b(a(a(x1))) -> b(a(b(b(a(x1)))))
20.66/6.22	   a(b(a(x1))) -> a(b(b(a(b(x1)))))
20.66/6.22	   b(b(a(x1))) -> b(b(b(b(b(x1)))))
20.66/6.22	   a(b(x1)) -> b(b(b(x1)))
20.66/6.22	   a(b(a(x1))) -> b(a(b(b(a(x1)))))
20.66/6.22	   a(a(b(x1))) -> a(b(b(a(b(x1)))))
20.66/6.22	   a(b(b(x1))) -> b(b(b(b(b(x1)))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(3) FlatCCProof (EQUIVALENT)
20.66/6.22	We used flat context closure [ROOTLAB]
20.66/6.22	As Q is empty the flat context closure was sound AND complete.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(4)
20.66/6.22	Obligation:
20.66/6.22	Q restricted rewrite system:
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   a(a(x1)) -> a(b(a(x1)))
20.66/6.22	   b(a(x1)) -> b(b(b(x1)))
20.66/6.22	   a(a(a(x1))) -> a(a(b(a(a(x1)))))
20.66/6.22	   b(a(a(x1))) -> b(a(b(b(a(x1)))))
20.66/6.22	   a(b(a(x1))) -> a(b(b(a(b(x1)))))
20.66/6.22	   b(b(a(x1))) -> b(b(b(b(b(x1)))))
20.66/6.22	   a(a(b(x1))) -> a(b(b(a(b(x1)))))
20.66/6.22	   a(a(x1)) -> a(b(x1))
20.66/6.22	   b(a(x1)) -> b(b(x1))
20.66/6.22	   a(a(b(x1))) -> a(b(b(b(x1))))
20.66/6.22	   b(a(b(x1))) -> b(b(b(b(x1))))
20.66/6.22	   a(a(b(a(x1)))) -> a(b(a(b(b(a(x1))))))
20.66/6.22	   b(a(b(a(x1)))) -> b(b(a(b(b(a(x1))))))
20.66/6.22	   a(a(b(b(x1)))) -> a(b(b(b(b(b(x1))))))
20.66/6.22	   b(a(b(b(x1)))) -> b(b(b(b(b(b(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(5) RootLabelingProof (EQUIVALENT)
20.66/6.22	We used plain root labeling [ROOTLAB] with the following heuristic:
20.66/6.22	LabelAll: All function symbols get labeled
20.66/6.22	
20.66/6.22	As Q is empty the root labeling was sound AND complete.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(6)
20.66/6.22	Obligation:
20.66/6.22	Q restricted rewrite system:
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
20.66/6.22	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
20.66/6.22	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
20.66/6.22	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(7) QTRSRRRProof (EQUIVALENT)
20.66/6.22	Used ordering:
20.66/6.22	Polynomial interpretation [POLO]:
20.66/6.22	
20.66/6.22	   POL(a_{a_1}(x_1)) = 1 + x_1
20.66/6.22	   POL(a_{b_1}(x_1)) = x_1
20.66/6.22	   POL(b_{a_1}(x_1)) = x_1
20.66/6.22	   POL(b_{b_1}(x_1)) = x_1
20.66/6.22	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
20.66/6.22	
20.66/6.22	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(a_{a_1}(x1)))
20.66/6.22	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{a_1}(a_{b_1}(x1)))
20.66/6.22	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
20.66/6.22	   b_{a_1}(a_{a_1}(a_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{a_1}(a_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(x1)) -> a_{b_1}(b_{a_1}(x1))
20.66/6.22	   a_{a_1}(a_{b_1}(x1)) -> a_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{a_1}(x1)) -> b_{b_1}(b_{a_1}(x1))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	
20.66/6.22	
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(8)
20.66/6.22	Obligation:
20.66/6.22	Q restricted rewrite system:
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(9) DependencyPairsProof (EQUIVALENT)
20.66/6.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(10)
20.66/6.22	Obligation:
20.66/6.22	Q DP problem:
20.66/6.22	The TRS P consists of the following rules:
20.66/6.22	
20.66/6.22	   B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(b_{b_1}(x1))
20.66/6.22	   B_{A_1}(a_{b_1}(x1)) -> B_{B_1}(x1)
20.66/6.22	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
20.66/6.22	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.66/6.22	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
20.66/6.22	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(x1))
20.66/6.22	   A_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1)))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(x1)))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	We have to consider all minimal (P,Q,R)-chains.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(11) DependencyGraphProof (EQUIVALENT)
20.66/6.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 38 less nodes.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(12)
20.66/6.22	Complex Obligation (AND)
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(13)
20.66/6.22	Obligation:
20.66/6.22	Q DP problem:
20.66/6.22	The TRS P consists of the following rules:
20.66/6.22	
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
20.66/6.22	
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	We have to consider all minimal (P,Q,R)-chains.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(14) UsableRulesProof (EQUIVALENT)
20.66/6.22	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.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(15)
20.66/6.22	Obligation:
20.66/6.22	Q DP problem:
20.66/6.22	The TRS P consists of the following rules:
20.66/6.22	
20.66/6.22	   B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
20.66/6.22	
20.66/6.22	R is empty.
20.66/6.22	Q is empty.
20.66/6.22	We have to consider all minimal (P,Q,R)-chains.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(16) QDPSizeChangeProof (EQUIVALENT)
20.66/6.22	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. 
20.66/6.22	
20.66/6.22	From the DPs we obtained the following set of size-change graphs:
20.66/6.22	*B_{B_1}(b_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
20.66/6.22	The graph contains the following edges 1 > 1
20.66/6.22	
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(17)
20.66/6.22	YES
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(18)
20.66/6.22	Obligation:
20.66/6.22	Q DP problem:
20.66/6.22	The TRS P consists of the following rules:
20.66/6.22	
20.66/6.22	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	We have to consider all minimal (P,Q,R)-chains.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(19) QDPOrderProof (EQUIVALENT)
20.66/6.22	We use the reduction pair processor [LPAR04,JAR06].
20.66/6.22	
20.66/6.22	
20.66/6.22	The following pairs can be oriented strictly and are deleted.
20.66/6.22	
20.66/6.22	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	The remaining pairs can at least be oriented weakly.
20.66/6.22	Used ordering:  Polynomial interpretation [POLO]:
20.66/6.22	
20.66/6.22	   POL(A_{A_1}(x_1)) = x_1
20.66/6.22	   POL(a_{a_1}(x_1)) = 1 + x_1
20.66/6.22	   POL(a_{b_1}(x_1)) = 1
20.66/6.22	   POL(b_{a_1}(x_1)) = 0
20.66/6.22	   POL(b_{b_1}(x_1)) = 0
20.66/6.22	
20.66/6.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.66/6.22	
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	
20.66/6.22	
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(20)
20.66/6.22	Obligation:
20.66/6.22	Q DP problem:
20.66/6.22	P is empty.
20.66/6.22	The TRS R consists of the following rules:
20.66/6.22	
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))))
20.66/6.22	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))))
20.66/6.22	   a_{b_1}(b_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{b_1}(b_{a_1}(a_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
20.66/6.22	   b_{a_1}(a_{b_1}(x1)) -> b_{b_1}(b_{b_1}(x1))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(x1))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))))
20.66/6.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))))
20.66/6.22	
20.66/6.22	Q is empty.
20.66/6.22	We have to consider all minimal (P,Q,R)-chains.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(21) PisEmptyProof (EQUIVALENT)
20.66/6.22	The TRS P is empty. Hence, there is no (P,Q,R) chain.
20.66/6.22	----------------------------------------
20.66/6.22	
20.66/6.22	(22)
20.66/6.22	YES
20.86/8.07	EOF