26.99/7.83	YES
26.99/7.85	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
26.99/7.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
26.99/7.85	
26.99/7.85	
26.99/7.85	Termination w.r.t. Q of the given QTRS could be proven:
26.99/7.85	
26.99/7.85	(0) QTRS
26.99/7.85	(1) FlatCCProof [EQUIVALENT, 0 ms]
26.99/7.85	(2) QTRS
26.99/7.85	(3) RootLabelingProof [EQUIVALENT, 0 ms]
26.99/7.85	(4) QTRS
26.99/7.85	(5) QTRSRRRProof [EQUIVALENT, 48 ms]
26.99/7.85	(6) QTRS
26.99/7.85	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
26.99/7.85	(8) QDP
26.99/7.85	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
26.99/7.85	(10) QDP
26.99/7.85	(11) QDPOrderProof [EQUIVALENT, 20 ms]
26.99/7.85	(12) QDP
26.99/7.85	(13) DependencyGraphProof [EQUIVALENT, 0 ms]
26.99/7.85	(14) AND
26.99/7.85	    (15) QDP
26.99/7.85	        (16) QDPOrderProof [EQUIVALENT, 0 ms]
26.99/7.85	        (17) QDP
26.99/7.85	        (18) PisEmptyProof [EQUIVALENT, 0 ms]
26.99/7.85	        (19) YES
26.99/7.85	    (20) QDP
26.99/7.85	        (21) QDPOrderProof [EQUIVALENT, 0 ms]
26.99/7.85	        (22) QDP
26.99/7.85	        (23) PisEmptyProof [EQUIVALENT, 0 ms]
26.99/7.85	        (24) YES
26.99/7.85	
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(0)
26.99/7.85	Obligation:
26.99/7.85	Q restricted rewrite system:
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b(b(b(b(x1)))) -> b(b(b(a(x1))))
26.99/7.85	   a(b(a(a(x1)))) -> b(a(b(a(x1))))
26.99/7.85	   b(a(a(b(x1)))) -> a(b(a(b(x1))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(1) FlatCCProof (EQUIVALENT)
26.99/7.85	We used flat context closure [ROOTLAB]
26.99/7.85	As Q is empty the flat context closure was sound AND complete.
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(2)
26.99/7.85	Obligation:
26.99/7.85	Q restricted rewrite system:
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b(b(b(b(x1)))) -> b(b(b(a(x1))))
26.99/7.85	   b(a(b(a(a(x1))))) -> b(b(a(b(a(x1)))))
26.99/7.85	   a(a(b(a(a(x1))))) -> a(b(a(b(a(x1)))))
26.99/7.85	   b(b(a(a(b(x1))))) -> b(a(b(a(b(x1)))))
26.99/7.85	   a(b(a(a(b(x1))))) -> a(a(b(a(b(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(3) RootLabelingProof (EQUIVALENT)
26.99/7.85	We used plain root labeling [ROOTLAB] with the following heuristic:
26.99/7.85	LabelAll: All function symbols get labeled
26.99/7.85	
26.99/7.85	As Q is empty the root labeling was sound AND complete.
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(4)
26.99/7.85	Obligation:
26.99/7.85	Q restricted rewrite system:
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(5) QTRSRRRProof (EQUIVALENT)
26.99/7.85	Used ordering:
26.99/7.85	Polynomial interpretation [POLO]:
26.99/7.85	
26.99/7.85	   POL(a_{a_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(a_{b_1}(x_1)) = x_1
26.99/7.85	   POL(b_{a_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(b_{b_1}(x_1)) = 1 + x_1
26.99/7.85	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	
26.99/7.85	
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(6)
26.99/7.85	Obligation:
26.99/7.85	Q restricted rewrite system:
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(7) DependencyPairsProof (EQUIVALENT)
26.99/7.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(8)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	The TRS P consists of the following rules:
26.99/7.85	
26.99/7.85	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(x1)))
26.99/7.85	   B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
26.99/7.85	
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(9) DependencyGraphProof (EQUIVALENT)
26.99/7.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(10)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	The TRS P consists of the following rules:
26.99/7.85	
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
26.99/7.85	
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(11) QDPOrderProof (EQUIVALENT)
26.99/7.85	We use the reduction pair processor [LPAR04,JAR06].
26.99/7.85	
26.99/7.85	
26.99/7.85	The following pairs can be oriented strictly and are deleted.
26.99/7.85	
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
26.99/7.85	The remaining pairs can at least be oriented weakly.
26.99/7.85	Used ordering:  Polynomial interpretation [POLO]:
26.99/7.85	
26.99/7.85	   POL(A_{B_1}(x_1)) = x_1
26.99/7.85	   POL(B_{A_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(a_{a_1}(x_1)) = x_1
26.99/7.85	   POL(a_{b_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(b_{a_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(b_{b_1}(x_1)) = 0
26.99/7.85	
26.99/7.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.99/7.85	
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(12)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	The TRS P consists of the following rules:
26.99/7.85	
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(13) DependencyGraphProof (EQUIVALENT)
26.99/7.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(14)
26.99/7.85	Complex Obligation (AND)
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(15)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	The TRS P consists of the following rules:
26.99/7.85	
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(16) QDPOrderProof (EQUIVALENT)
26.99/7.85	We use the reduction pair processor [LPAR04,JAR06].
26.99/7.85	
26.99/7.85	
26.99/7.85	The following pairs can be oriented strictly and are deleted.
26.99/7.85	
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
26.99/7.85	The remaining pairs can at least be oriented weakly.
26.99/7.85	Used ordering:  Polynomial interpretation [POLO]:
26.99/7.85	
26.99/7.85	   POL(B_{A_1}(x_1)) = x_1
26.99/7.85	   POL(a_{a_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(a_{b_1}(x_1)) = x_1
26.99/7.85	   POL(b_{a_1}(x_1)) = 1 + x_1
26.99/7.85	   POL(b_{b_1}(x_1)) = 0
26.99/7.85	
26.99/7.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.99/7.85	
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(17)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	P is empty.
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(18) PisEmptyProof (EQUIVALENT)
26.99/7.85	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(19)
26.99/7.85	YES
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(20)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	The TRS P consists of the following rules:
26.99/7.85	
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
26.99/7.85	
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(21) QDPOrderProof (EQUIVALENT)
26.99/7.85	We use the reduction pair processor [LPAR04,JAR06].
26.99/7.85	
26.99/7.85	
26.99/7.85	The following pairs can be oriented strictly and are deleted.
26.99/7.85	
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
26.99/7.85	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
26.99/7.85	The remaining pairs can at least be oriented weakly.
26.99/7.85	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
26.99/7.85	
26.99/7.85	POL( A_{B_1}_1(x_1) ) = max{0, 2x_1 - 1}
26.99/7.85	POL( b_{a_1}_1(x_1) ) = x_1
26.99/7.85	POL( a_{b_1}_1(x_1) ) = 2x_1 + 1
26.99/7.85	POL( a_{a_1}_1(x_1) ) = x_1 + 1
26.99/7.85	POL( b_{b_1}_1(x_1) ) = max{0, -2}
26.99/7.85	
26.99/7.85	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
26.99/7.85	
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	
26.99/7.85	
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(22)
26.99/7.85	Obligation:
26.99/7.85	Q DP problem:
26.99/7.85	P is empty.
26.99/7.85	The TRS R consists of the following rules:
26.99/7.85	
26.99/7.85	   b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
26.99/7.85	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
26.99/7.85	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
26.99/7.85	
26.99/7.85	Q is empty.
26.99/7.85	We have to consider all minimal (P,Q,R)-chains.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(23) PisEmptyProof (EQUIVALENT)
26.99/7.85	The TRS P is empty. Hence, there is no (P,Q,R) chain.
26.99/7.85	----------------------------------------
26.99/7.85	
26.99/7.85	(24)
26.99/7.85	YES
27.54/8.12	EOF