34.59/9.79	YES
34.87/9.82	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
34.87/9.82	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
34.87/9.82	
34.87/9.82	
34.87/9.82	Termination w.r.t. Q of the given QTRS could be proven:
34.87/9.82	
34.87/9.82	(0) QTRS
34.87/9.82	(1) QTRS Reverse [EQUIVALENT, 3 ms]
34.87/9.82	(2) QTRS
34.87/9.82	(3) RootLabelingProof [EQUIVALENT, 0 ms]
34.87/9.82	(4) QTRS
34.87/9.82	(5) DependencyPairsProof [EQUIVALENT, 7 ms]
34.87/9.82	(6) QDP
34.87/9.82	(7) DependencyGraphProof [EQUIVALENT, 2 ms]
34.87/9.82	(8) QDP
34.87/9.82	(9) QDPOrderProof [EQUIVALENT, 38 ms]
34.87/9.82	(10) QDP
34.87/9.82	(11) QDPOrderProof [EQUIVALENT, 11 ms]
34.87/9.82	(12) QDP
34.87/9.82	(13) QDPOrderProof [EQUIVALENT, 511 ms]
34.87/9.82	(14) QDP
34.87/9.82	(15) PisEmptyProof [EQUIVALENT, 0 ms]
34.87/9.82	(16) YES
34.87/9.82	
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(0)
34.87/9.82	Obligation:
34.87/9.82	Q restricted rewrite system:
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   a(b(a(b(x1)))) -> b(b(a(b(x1))))
34.87/9.82	   b(b(a(a(x1)))) -> a(b(a(a(x1))))
34.87/9.82	   a(a(b(b(x1)))) -> b(a(a(b(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(1) QTRS Reverse (EQUIVALENT)
34.87/9.82	We applied the QTRS Reverse Processor [REVERSE].
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(2)
34.87/9.82	Obligation:
34.87/9.82	Q restricted rewrite system:
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b(a(b(a(x1)))) -> b(a(b(b(x1))))
34.87/9.82	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
34.87/9.82	   b(b(a(a(x1)))) -> b(a(a(b(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(3) RootLabelingProof (EQUIVALENT)
34.87/9.82	We used plain root labeling [ROOTLAB] with the following heuristic:
34.87/9.82	LabelAll: All function symbols get labeled
34.87/9.82	
34.87/9.82	As Q is empty the root labeling was sound AND complete.
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(4)
34.87/9.82	Obligation:
34.87/9.82	Q restricted rewrite system:
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(5) DependencyPairsProof (EQUIVALENT)
34.87/9.82	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(6)
34.87/9.82	Obligation:
34.87/9.82	Q DP problem:
34.87/9.82	The TRS P consists of the following rules:
34.87/9.82	
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
34.87/9.82	   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}(x1))))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{a_1}(x1)))
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
34.87/9.82	
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	We have to consider all minimal (P,Q,R)-chains.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(7) DependencyGraphProof (EQUIVALENT)
34.87/9.82	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(8)
34.87/9.82	Obligation:
34.87/9.82	Q DP problem:
34.87/9.82	The TRS P consists of the following rules:
34.87/9.82	
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
34.87/9.82	   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}(x1))))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
34.87/9.82	
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	We have to consider all minimal (P,Q,R)-chains.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(9) QDPOrderProof (EQUIVALENT)
34.87/9.82	We use the reduction pair processor [LPAR04,JAR06].
34.87/9.82	
34.87/9.82	
34.87/9.82	The following pairs can be oriented strictly and are deleted.
34.87/9.82	
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{b_1}(x1))
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(x1)))
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
34.87/9.82	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(x1))
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
34.87/9.82	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
34.87/9.82	The remaining pairs can at least be oriented weakly.
34.87/9.82	Used ordering:  Polynomial interpretation [POLO]:
34.87/9.82	
34.87/9.82	   POL(A_{A_1}(x_1)) = x_1
34.87/9.82	   POL(B_{A_1}(x_1)) = x_1
34.87/9.82	   POL(B_{B_1}(x_1)) = 1 + x_1
34.87/9.82	   POL(a_{a_1}(x_1)) = 1 + x_1
34.87/9.82	   POL(a_{b_1}(x_1)) = 1 + x_1
34.87/9.82	   POL(b_{a_1}(x_1)) = 1 + x_1
34.87/9.82	   POL(b_{b_1}(x_1)) = 1 + x_1
34.87/9.82	
34.87/9.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.87/9.82	
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(10)
34.87/9.82	Obligation:
34.87/9.82	Q DP problem:
34.87/9.82	The TRS P consists of the following rules:
34.87/9.82	
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	We have to consider all minimal (P,Q,R)-chains.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(11) QDPOrderProof (EQUIVALENT)
34.87/9.82	We use the reduction pair processor [LPAR04,JAR06].
34.87/9.82	
34.87/9.82	
34.87/9.82	The following pairs can be oriented strictly and are deleted.
34.87/9.82	
34.87/9.82	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	The remaining pairs can at least be oriented weakly.
34.87/9.82	Used ordering:  Polynomial interpretation [POLO]:
34.87/9.82	
34.87/9.82	   POL(B_{A_1}(x_1)) = x_1
34.87/9.82	   POL(a_{a_1}(x_1)) = 0
34.87/9.82	   POL(a_{b_1}(x_1)) = 1 + x_1
34.87/9.82	   POL(b_{a_1}(x_1)) = x_1
34.87/9.82	   POL(b_{b_1}(x_1)) = 0
34.87/9.82	
34.87/9.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.87/9.82	
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(12)
34.87/9.82	Obligation:
34.87/9.82	Q DP problem:
34.87/9.82	The TRS P consists of the following rules:
34.87/9.82	
34.87/9.82	   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}(x1))))
34.87/9.82	
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	We have to consider all minimal (P,Q,R)-chains.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(13) QDPOrderProof (EQUIVALENT)
34.87/9.82	We use the reduction pair processor [LPAR04,JAR06].
34.87/9.82	
34.87/9.82	
34.87/9.82	The following pairs can be oriented strictly and are deleted.
34.87/9.82	
34.87/9.82	   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}(x1))))
34.87/9.82	The remaining pairs can at least be oriented weakly.
34.87/9.82	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
34.87/9.82	
34.87/9.82	   <<<
34.87/9.82	 POL(B_{A_1}(x_1)) =  	[[-I]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
34.87/9.82	>>>
34.87/9.82	
34.87/9.82	   <<<
34.87/9.82	 POL(a_{b_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [-I, 0A, -I]] 	* 	x_1
34.87/9.82	>>>
34.87/9.82	
34.87/9.82	   <<<
34.87/9.82	 POL(b_{a_1}(x_1)) =  	[[0A], [-I], [0A]] 	 +  	[[-I, -I, 0A], [-I, -I, -I], [0A, 0A, 0A]] 	* 	x_1
34.87/9.82	>>>
34.87/9.82	
34.87/9.82	   <<<
34.87/9.82	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [1A]] 	 +  	[[-I, 0A, 0A], [-I, 0A, 1A], [-I, 0A, 0A]] 	* 	x_1
34.87/9.82	>>>
34.87/9.82	
34.87/9.82	   <<<
34.87/9.82	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, -I], [-I, -I, -I], [0A, 0A, -I]] 	* 	x_1
34.87/9.82	>>>
34.87/9.82	
34.87/9.82	
34.87/9.82	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
34.87/9.82	
34.87/9.82	   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}(x1))))
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	
34.87/9.82	
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(14)
34.87/9.82	Obligation:
34.87/9.82	Q DP problem:
34.87/9.82	P is empty.
34.87/9.82	The TRS R consists of the following rules:
34.87/9.82	
34.87/9.82	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
34.87/9.82	   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}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
34.87/9.82	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))
34.87/9.82	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))
34.87/9.82	
34.87/9.82	Q is empty.
34.87/9.82	We have to consider all minimal (P,Q,R)-chains.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(15) PisEmptyProof (EQUIVALENT)
34.87/9.82	The TRS P is empty. Hence, there is no (P,Q,R) chain.
34.87/9.82	----------------------------------------
34.87/9.82	
34.87/9.82	(16)
34.87/9.82	YES
35.20/10.05	EOF