19.73/6.04	YES
20.55/6.31	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
20.55/6.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
20.55/6.31	
20.55/6.31	
20.55/6.31	Termination w.r.t. Q of the given QTRS could be proven:
20.55/6.31	
20.55/6.31	(0) QTRS
20.55/6.31	(1) QTRS Reverse [EQUIVALENT, 0 ms]
20.55/6.31	(2) QTRS
20.55/6.31	(3) FlatCCProof [EQUIVALENT, 0 ms]
20.55/6.31	(4) QTRS
20.55/6.31	(5) RootLabelingProof [EQUIVALENT, 0 ms]
20.55/6.31	(6) QTRS
20.55/6.31	(7) QTRSRRRProof [EQUIVALENT, 29 ms]
20.55/6.31	(8) QTRS
20.55/6.31	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
20.55/6.31	(10) QDP
20.55/6.31	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
20.55/6.31	(12) AND
20.55/6.31	    (13) QDP
20.55/6.31	        (14) QDPOrderProof [EQUIVALENT, 21 ms]
20.55/6.31	        (15) QDP
20.55/6.31	        (16) PisEmptyProof [EQUIVALENT, 0 ms]
20.55/6.31	        (17) YES
20.55/6.31	    (18) QDP
20.55/6.31	        (19) UsableRulesProof [EQUIVALENT, 0 ms]
20.55/6.31	        (20) QDP
20.55/6.31	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
20.55/6.31	        (22) YES
20.55/6.31	    (23) QDP
20.55/6.31	        (24) UsableRulesProof [EQUIVALENT, 0 ms]
20.55/6.31	        (25) QDP
20.55/6.31	        (26) MNOCProof [EQUIVALENT, 2 ms]
20.55/6.31	        (27) QDP
20.55/6.31	        (28) QDPOrderProof [EQUIVALENT, 8 ms]
20.55/6.31	        (29) QDP
20.55/6.31	        (30) PisEmptyProof [EQUIVALENT, 0 ms]
20.55/6.31	        (31) YES
20.55/6.31	
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(0)
20.55/6.31	Obligation:
20.55/6.31	Q restricted rewrite system:
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   a(b(b(b(x1)))) -> b(b(b(b(x1))))
20.55/6.31	   b(b(b(a(x1)))) -> b(a(b(a(x1))))
20.55/6.31	   b(b(a(a(x1)))) -> b(a(b(a(x1))))
20.55/6.31	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(1) QTRS Reverse (EQUIVALENT)
20.55/6.31	We applied the QTRS Reverse Processor [REVERSE].
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(2)
20.55/6.31	Obligation:
20.55/6.31	Q restricted rewrite system:
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b(b(b(a(x1)))) -> b(b(b(b(x1))))
20.55/6.31	   a(b(b(b(x1)))) -> a(b(a(b(x1))))
20.55/6.31	   a(a(b(b(x1)))) -> a(b(a(b(x1))))
20.55/6.31	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(3) FlatCCProof (EQUIVALENT)
20.55/6.31	We used flat context closure [ROOTLAB]
20.55/6.31	As Q is empty the flat context closure was sound AND complete.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(4)
20.55/6.31	Obligation:
20.55/6.31	Q restricted rewrite system:
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b(b(b(a(x1)))) -> b(b(b(b(x1))))
20.55/6.31	   a(b(b(b(x1)))) -> a(b(a(b(x1))))
20.55/6.31	   a(a(b(b(x1)))) -> a(b(a(b(x1))))
20.55/6.31	   b(b(a(a(a(x1))))) -> b(a(a(b(a(x1)))))
20.55/6.31	   a(b(a(a(a(x1))))) -> a(a(a(b(a(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(5) RootLabelingProof (EQUIVALENT)
20.55/6.31	We used plain root labeling [ROOTLAB] with the following heuristic:
20.55/6.31	LabelAll: All function symbols get labeled
20.55/6.31	
20.55/6.31	As Q is empty the root labeling was sound AND complete.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(6)
20.55/6.31	Obligation:
20.55/6.31	Q restricted rewrite system:
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(7) QTRSRRRProof (EQUIVALENT)
20.55/6.31	Used ordering:
20.55/6.31	Polynomial interpretation [POLO]:
20.55/6.31	
20.55/6.31	   POL(a_{a_1}(x_1)) = 1 + x_1
20.55/6.31	   POL(a_{b_1}(x_1)) = x_1
20.55/6.31	   POL(b_{a_1}(x_1)) = x_1
20.55/6.31	   POL(b_{b_1}(x_1)) = x_1
20.55/6.31	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	
20.55/6.31	
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(8)
20.55/6.31	Obligation:
20.55/6.31	Q restricted rewrite system:
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(9) DependencyPairsProof (EQUIVALENT)
20.55/6.31	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(10)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
20.55/6.31	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
20.55/6.31	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
20.55/6.31	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
20.55/6.31	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(11) DependencyGraphProof (EQUIVALENT)
20.55/6.31	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 5 less nodes.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(12)
20.55/6.31	Complex Obligation (AND)
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(13)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
20.55/6.31	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(14) QDPOrderProof (EQUIVALENT)
20.55/6.31	We use the reduction pair processor [LPAR04,JAR06].
20.55/6.31	
20.55/6.31	
20.55/6.31	The following pairs can be oriented strictly and are deleted.
20.55/6.31	
20.55/6.31	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
20.55/6.31	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
20.55/6.31	The remaining pairs can at least be oriented weakly.
20.55/6.31	Used ordering:  Polynomial interpretation [POLO]:
20.55/6.31	
20.55/6.31	   POL(A_{B_1}(x_1)) = x_1
20.55/6.31	   POL(a_{a_1}(x_1)) = 1 + x_1
20.55/6.31	   POL(a_{b_1}(x_1)) = 1 + x_1
20.55/6.31	   POL(b_{a_1}(x_1)) = 1 + x_1
20.55/6.31	   POL(b_{b_1}(x_1)) = 1 + x_1
20.55/6.31	
20.55/6.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.55/6.31	
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(15)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	P is empty.
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(16) PisEmptyProof (EQUIVALENT)
20.55/6.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(17)
20.55/6.31	YES
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(18)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(19) UsableRulesProof (EQUIVALENT)
20.55/6.31	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.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(20)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(21) QDPSizeChangeProof (EQUIVALENT)
20.55/6.31	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. 
20.55/6.31	
20.55/6.31	From the DPs we obtained the following set of size-change graphs:
20.55/6.31	*A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
20.55/6.31	The graph contains the following edges 1 > 1
20.55/6.31	
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(22)
20.55/6.31	YES
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(23)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
20.55/6.31	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
20.55/6.31	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(24) UsableRulesProof (EQUIVALENT)
20.55/6.31	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.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(25)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	
20.55/6.31	Q is empty.
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(26) MNOCProof (EQUIVALENT)
20.55/6.31	We use the modular non-overlap check [LPAR04] to enlarge Q to all left-hand sides of R.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(27)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	The TRS P consists of the following rules:
20.55/6.31	
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
20.55/6.31	
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	
20.55/6.31	The set Q consists of the following terms:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x0))))
20.55/6.31	
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(28) QDPOrderProof (EQUIVALENT)
20.55/6.31	We use the reduction pair processor [LPAR04,JAR06].
20.55/6.31	
20.55/6.31	
20.55/6.31	The following pairs can be oriented strictly and are deleted.
20.55/6.31	
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
20.55/6.31	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
20.55/6.31	The remaining pairs can at least be oriented weakly.
20.55/6.31	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
20.55/6.31	
20.55/6.31	POL( B_{B_1}_1(x_1) ) = max{0, 2x_1 - 1}
20.55/6.31	POL( b_{b_1}_1(x_1) ) = 2x_1
20.55/6.31	POL( b_{a_1}_1(x_1) ) = 2x_1 + 1
20.55/6.31	POL( a_{b_1}_1(x_1) ) = 2x_1 + 1
20.55/6.31	
20.55/6.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	
20.55/6.31	
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(29)
20.55/6.31	Obligation:
20.55/6.31	Q DP problem:
20.55/6.31	P is empty.
20.55/6.31	The TRS R consists of the following rules:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
20.55/6.31	
20.55/6.31	The set Q consists of the following terms:
20.55/6.31	
20.55/6.31	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x0))))
20.55/6.31	
20.55/6.31	We have to consider all minimal (P,Q,R)-chains.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(30) PisEmptyProof (EQUIVALENT)
20.55/6.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
20.55/6.31	----------------------------------------
20.55/6.31	
20.55/6.31	(31)
20.55/6.31	YES
20.87/8.95	EOF