85.19/22.66	YES
85.49/22.70	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
85.49/22.70	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
85.49/22.70	
85.49/22.70	
85.49/22.70	Termination w.r.t. Q of the given QTRS could be proven:
85.49/22.70	
85.49/22.70	(0) QTRS
85.49/22.70	(1) QTRS Reverse [EQUIVALENT, 0 ms]
85.49/22.70	(2) QTRS
85.49/22.70	(3) FlatCCProof [EQUIVALENT, 0 ms]
85.49/22.70	(4) QTRS
85.49/22.70	(5) RootLabelingProof [EQUIVALENT, 0 ms]
85.49/22.70	(6) QTRS
85.49/22.70	(7) DependencyPairsProof [EQUIVALENT, 35 ms]
85.49/22.70	(8) QDP
85.49/22.70	(9) DependencyGraphProof [EQUIVALENT, 10 ms]
85.49/22.70	(10) QDP
85.49/22.70	(11) QDPOrderProof [EQUIVALENT, 269 ms]
85.49/22.70	(12) QDP
85.49/22.70	(13) QDPOrderProof [EQUIVALENT, 639 ms]
85.49/22.70	(14) QDP
85.49/22.70	(15) QDPOrderProof [EQUIVALENT, 553 ms]
85.49/22.70	(16) QDP
85.49/22.70	(17) PisEmptyProof [EQUIVALENT, 0 ms]
85.49/22.70	(18) YES
85.49/22.70	
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(0)
85.49/22.70	Obligation:
85.49/22.70	Q restricted rewrite system:
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   b(b(b(b(x1)))) -> b(a(a(a(x1))))
85.49/22.70	   a(a(b(a(x1)))) -> b(b(b(a(x1))))
85.49/22.70	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(1) QTRS Reverse (EQUIVALENT)
85.49/22.70	We applied the QTRS Reverse Processor [REVERSE].
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(2)
85.49/22.70	Obligation:
85.49/22.70	Q restricted rewrite system:
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   b(b(b(b(x1)))) -> a(a(a(b(x1))))
85.49/22.70	   a(b(a(a(x1)))) -> a(b(b(b(x1))))
85.49/22.70	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(3) FlatCCProof (EQUIVALENT)
85.49/22.70	We used flat context closure [ROOTLAB]
85.49/22.70	As Q is empty the flat context closure was sound AND complete.
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(4)
85.49/22.70	Obligation:
85.49/22.70	Q restricted rewrite system:
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a(b(a(a(x1)))) -> a(b(b(b(x1))))
85.49/22.70	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
85.49/22.70	   b(b(b(b(b(x1))))) -> b(a(a(a(b(x1)))))
85.49/22.70	   a(b(b(b(b(x1))))) -> a(a(a(a(b(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(5) RootLabelingProof (EQUIVALENT)
85.49/22.70	We used plain root labeling [ROOTLAB] with the following heuristic:
85.49/22.70	LabelAll: All function symbols get labeled
85.49/22.70	
85.49/22.70	As Q is empty the root labeling was sound AND complete.
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(6)
85.49/22.70	Obligation:
85.49/22.70	Q restricted rewrite system:
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(7) DependencyPairsProof (EQUIVALENT)
85.49/22.70	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(8)
85.49/22.70	Obligation:
85.49/22.70	Q DP problem:
85.49/22.70	The TRS P consists of the following rules:
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
85.49/22.70	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
85.49/22.70	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
85.49/22.70	
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	We have to consider all minimal (P,Q,R)-chains.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(9) DependencyGraphProof (EQUIVALENT)
85.49/22.70	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(10)
85.49/22.70	Obligation:
85.49/22.70	Q DP problem:
85.49/22.70	The TRS P consists of the following rules:
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
85.49/22.70	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
85.49/22.70	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
85.49/22.70	
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	We have to consider all minimal (P,Q,R)-chains.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(11) QDPOrderProof (EQUIVALENT)
85.49/22.70	We use the reduction pair processor [LPAR04,JAR06].
85.49/22.70	
85.49/22.70	
85.49/22.70	The following pairs can be oriented strictly and are deleted.
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
85.49/22.70	   B_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
85.49/22.70	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
85.49/22.70	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(x1))
85.49/22.70	   A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(x1))
85.49/22.70	The remaining pairs can at least be oriented weakly.
85.49/22.70	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
85.49/22.70	
85.49/22.70	POL( A_{B_1}_1(x_1) ) = 2x_1 + 2
85.49/22.70	POL( B_{B_1}_1(x_1) ) = max{0, 2x_1 - 1}
85.49/22.70	POL( b_{a_1}_1(x_1) ) = 2x_1 + 2
85.49/22.70	POL( a_{b_1}_1(x_1) ) = 2x_1 + 2
85.49/22.70	POL( b_{b_1}_1(x_1) ) = 2x_1 + 2
85.49/22.70	POL( a_{a_1}_1(x_1) ) = 2x_1 + 2
85.49/22.70	POL( B_{A_1}_1(x_1) ) = 2x_1 + 1
85.49/22.70	
85.49/22.70	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
85.49/22.70	
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(12)
85.49/22.70	Obligation:
85.49/22.70	Q DP problem:
85.49/22.70	The TRS P consists of the following rules:
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	We have to consider all minimal (P,Q,R)-chains.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(13) QDPOrderProof (EQUIVALENT)
85.49/22.70	We use the reduction pair processor [LPAR04,JAR06].
85.49/22.70	
85.49/22.70	
85.49/22.70	The following pairs can be oriented strictly and are deleted.
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	The remaining pairs can at least be oriented weakly.
85.49/22.70	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(A_{B_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 1A]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, -I], [0A, 0A, -I]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[-I, -I, -I], [0A, -I, -I], [0A, 0A, 0A]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(b_{b_1}(x_1)) =  	[[1A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [-I, -I, -I], [-I, 0A, -I]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(a_{b_1}(x_1)) =  	[[1A], [1A], [1A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	
85.49/22.70	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
85.49/22.70	
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(14)
85.49/22.70	Obligation:
85.49/22.70	Q DP problem:
85.49/22.70	The TRS P consists of the following rules:
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	We have to consider all minimal (P,Q,R)-chains.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(15) QDPOrderProof (EQUIVALENT)
85.49/22.70	We use the reduction pair processor [LPAR04,JAR06].
85.49/22.70	
85.49/22.70	
85.49/22.70	The following pairs can be oriented strictly and are deleted.
85.49/22.70	
85.49/22.70	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	The remaining pairs can at least be oriented weakly.
85.49/22.70	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(A_{B_1}(x_1)) =  	[[-I]] 	 +  	[[0A, -I, -I]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(a_{a_1}(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [1A, 0A, 0A]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [-I, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	   <<<
85.49/22.70	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
85.49/22.70	>>>
85.49/22.70	
85.49/22.70	
85.49/22.70	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
85.49/22.70	
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(16)
85.49/22.70	Obligation:
85.49/22.70	Q DP problem:
85.49/22.70	P is empty.
85.49/22.70	The TRS R consists of the following rules:
85.49/22.70	
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
85.49/22.70	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
85.49/22.70	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
85.49/22.70	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
85.49/22.70	
85.49/22.70	Q is empty.
85.49/22.70	We have to consider all minimal (P,Q,R)-chains.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(17) PisEmptyProof (EQUIVALENT)
85.49/22.70	The TRS P is empty. Hence, there is no (P,Q,R) chain.
85.49/22.70	----------------------------------------
85.49/22.70	
85.49/22.70	(18)
85.49/22.70	YES
85.81/22.81	EOF