28.47/8.26	YES
28.73/8.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
28.73/8.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
28.73/8.28	
28.73/8.28	
28.73/8.28	Termination w.r.t. Q of the given QTRS could be proven:
28.73/8.28	
28.73/8.28	(0) QTRS
28.73/8.28	(1) QTRS Reverse [EQUIVALENT, 0 ms]
28.73/8.28	(2) QTRS
28.73/8.28	(3) RootLabelingProof [EQUIVALENT, 0 ms]
28.73/8.28	(4) QTRS
28.73/8.28	(5) DependencyPairsProof [EQUIVALENT, 34 ms]
28.73/8.28	(6) QDP
28.73/8.28	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
28.73/8.28	(8) AND
28.73/8.28	    (9) QDP
28.73/8.28	        (10) QDPOrderProof [EQUIVALENT, 40 ms]
28.73/8.28	        (11) QDP
28.73/8.28	        (12) QDPOrderProof [EQUIVALENT, 48 ms]
28.73/8.28	        (13) QDP
28.73/8.28	        (14) PisEmptyProof [EQUIVALENT, 0 ms]
28.73/8.28	        (15) YES
28.73/8.28	    (16) QDP
28.73/8.28	        (17) QDPOrderProof [EQUIVALENT, 92 ms]
28.73/8.28	        (18) QDP
28.73/8.28	        (19) PisEmptyProof [EQUIVALENT, 0 ms]
28.73/8.28	        (20) YES
28.73/8.28	
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(0)
28.73/8.28	Obligation:
28.73/8.28	Q restricted rewrite system:
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b(b(a(b(x1)))) -> b(a(b(b(b(b(x1))))))
28.73/8.28	   b(a(b(b(x1)))) -> b(b(b(x1)))
28.73/8.28	   b(b(b(x1))) -> b(b(a(a(b(a(b(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(1) QTRS Reverse (EQUIVALENT)
28.73/8.28	We applied the QTRS Reverse Processor [REVERSE].
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(2)
28.73/8.28	Obligation:
28.73/8.28	Q restricted rewrite system:
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b(a(b(b(x1)))) -> b(b(b(b(a(b(x1))))))
28.73/8.28	   b(b(a(b(x1)))) -> b(b(b(x1)))
28.73/8.28	   b(b(b(x1))) -> b(a(b(a(a(b(b(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(3) RootLabelingProof (EQUIVALENT)
28.73/8.28	We used plain root labeling [ROOTLAB] with the following heuristic:
28.73/8.28	LabelAll: All function symbols get labeled
28.73/8.28	
28.73/8.28	As Q is empty the root labeling was sound AND complete.
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(4)
28.73/8.28	Obligation:
28.73/8.28	Q restricted rewrite system:
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(5) DependencyPairsProof (EQUIVALENT)
28.73/8.28	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(6)
28.73/8.28	Obligation:
28.73/8.28	Q DP problem:
28.73/8.28	The TRS P consists of the following rules:
28.73/8.28	
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
28.73/8.28	   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   B_{B_1}(b_{b_1}(b_{b_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
28.73/8.28	   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	   B_{B_1}(b_{b_1}(b_{a_1}(x1))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
28.73/8.28	
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	We have to consider all minimal (P,Q,R)-chains.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(7) DependencyGraphProof (EQUIVALENT)
28.73/8.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 10 less nodes.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(8)
28.73/8.28	Complex Obligation (AND)
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(9)
28.73/8.28	Obligation:
28.73/8.28	Q DP problem:
28.73/8.28	The TRS P consists of the following rules:
28.73/8.28	
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
28.73/8.28	
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	We have to consider all minimal (P,Q,R)-chains.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(10) QDPOrderProof (EQUIVALENT)
28.73/8.28	We use the reduction pair processor [LPAR04,JAR06].
28.73/8.28	
28.73/8.28	
28.73/8.28	The following pairs can be oriented strictly and are deleted.
28.73/8.28	
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
28.73/8.28	The remaining pairs can at least be oriented weakly.
28.73/8.28	Used ordering:  Polynomial interpretation [POLO]:
28.73/8.28	
28.73/8.28	   POL(B_{B_1}(x_1)) = x_1
28.73/8.28	   POL(a_{a_1}(x_1)) = 0
28.73/8.28	   POL(a_{b_1}(x_1)) = 1 + x_1
28.73/8.28	   POL(b_{a_1}(x_1)) = x_1
28.73/8.28	   POL(b_{b_1}(x_1)) = 1
28.73/8.28	
28.73/8.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.73/8.28	
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(11)
28.73/8.28	Obligation:
28.73/8.28	Q DP problem:
28.73/8.28	The TRS P consists of the following rules:
28.73/8.28	
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	We have to consider all minimal (P,Q,R)-chains.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(12) QDPOrderProof (EQUIVALENT)
28.73/8.28	We use the reduction pair processor [LPAR04,JAR06].
28.73/8.28	
28.73/8.28	
28.73/8.28	The following pairs can be oriented strictly and are deleted.
28.73/8.28	
28.73/8.28	   B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	The remaining pairs can at least be oriented weakly.
28.73/8.28	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
28.73/8.28	
28.73/8.28	POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
28.73/8.28	POL( b_{b_1}_1(x_1) ) = x_1 + 1
28.73/8.28	POL( b_{a_1}_1(x_1) ) = 2x_1
28.73/8.28	POL( a_{b_1}_1(x_1) ) = 2x_1 + 1
28.73/8.28	POL( a_{a_1}_1(x_1) ) = max{0, -2}
28.73/8.28	
28.73/8.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(13)
28.73/8.28	Obligation:
28.73/8.28	Q DP problem:
28.73/8.28	P is empty.
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	We have to consider all minimal (P,Q,R)-chains.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(14) PisEmptyProof (EQUIVALENT)
28.73/8.28	The TRS P is empty. Hence, there is no (P,Q,R) chain.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(15)
28.73/8.28	YES
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(16)
28.73/8.28	Obligation:
28.73/8.28	Q DP problem:
28.73/8.28	The TRS P consists of the following rules:
28.73/8.28	
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
28.73/8.28	
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	We have to consider all minimal (P,Q,R)-chains.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(17) QDPOrderProof (EQUIVALENT)
28.73/8.28	We use the reduction pair processor [LPAR04,JAR06].
28.73/8.28	
28.73/8.28	
28.73/8.28	The following pairs can be oriented strictly and are deleted.
28.73/8.28	
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
28.73/8.28	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
28.73/8.28	The remaining pairs can at least be oriented weakly.
28.73/8.28	Used ordering:  Polynomial interpretation [POLO]:
28.73/8.28	
28.73/8.28	   POL(B_{A_1}(x_1)) = 4*x_1
28.73/8.28	   POL(a_{a_1}(x_1)) = 0
28.73/8.28	   POL(a_{b_1}(x_1)) = 4*x_1
28.73/8.28	   POL(b_{a_1}(x_1)) = 1 + x_1
28.73/8.28	   POL(b_{b_1}(x_1)) = 3 + x_1
28.73/8.28	
28.73/8.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(18)
28.73/8.28	Obligation:
28.73/8.28	Q DP problem:
28.73/8.28	P is empty.
28.73/8.28	The TRS R consists of the following rules:
28.73/8.28	
28.73/8.28	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))))
28.73/8.28	   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}(a_{b_1}(b_{a_1}(x1))))))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{b_1}(x1)))
28.73/8.28	   b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(x1)))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{b_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))))
28.73/8.28	   b_{b_1}(b_{b_1}(b_{a_1}(x1))) -> b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))))
28.73/8.28	
28.73/8.28	Q is empty.
28.73/8.28	We have to consider all minimal (P,Q,R)-chains.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(19) PisEmptyProof (EQUIVALENT)
28.73/8.28	The TRS P is empty. Hence, there is no (P,Q,R) chain.
28.73/8.28	----------------------------------------
28.73/8.28	
28.73/8.28	(20)
28.73/8.28	YES
28.88/8.40	EOF