40.36/11.26	YES
40.86/11.27	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
40.86/11.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
40.86/11.27	
40.86/11.27	
40.86/11.27	Termination w.r.t. Q of the given QTRS could be proven:
40.86/11.27	
40.86/11.27	(0) QTRS
40.86/11.27	(1) FlatCCProof [EQUIVALENT, 0 ms]
40.86/11.27	(2) QTRS
40.86/11.27	(3) RootLabelingProof [EQUIVALENT, 2 ms]
40.86/11.27	(4) QTRS
40.86/11.27	(5) DependencyPairsProof [EQUIVALENT, 19 ms]
40.86/11.27	(6) QDP
40.86/11.27	(7) DependencyGraphProof [EQUIVALENT, 1 ms]
40.86/11.27	(8) QDP
40.86/11.27	(9) QDPOrderProof [EQUIVALENT, 198 ms]
40.86/11.27	(10) QDP
40.86/11.27	(11) QDPOrderProof [EQUIVALENT, 609 ms]
40.86/11.27	(12) QDP
40.86/11.27	(13) QDPOrderProof [EQUIVALENT, 277 ms]
40.86/11.27	(14) QDP
40.86/11.27	(15) DependencyGraphProof [EQUIVALENT, 0 ms]
40.86/11.27	(16) TRUE
40.86/11.27	
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(0)
40.86/11.27	Obligation:
40.86/11.27	Q restricted rewrite system:
40.86/11.27	The TRS R consists of the following rules:
40.86/11.27	
40.86/11.27	   a(a(a(x1))) -> a(b(b(b(x1))))
40.86/11.27	   b(b(a(b(x1)))) -> a(a(a(x1)))
40.86/11.27	
40.86/11.27	Q is empty.
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(1) FlatCCProof (EQUIVALENT)
40.86/11.27	We used flat context closure [ROOTLAB]
40.86/11.27	As Q is empty the flat context closure was sound AND complete.
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(2)
40.86/11.27	Obligation:
40.86/11.27	Q restricted rewrite system:
40.86/11.27	The TRS R consists of the following rules:
40.86/11.27	
40.86/11.27	   a(a(a(x1))) -> a(b(b(b(x1))))
40.86/11.27	   a(b(b(a(b(x1))))) -> a(a(a(a(x1))))
40.86/11.27	   b(b(b(a(b(x1))))) -> b(a(a(a(x1))))
40.86/11.27	
40.86/11.27	Q is empty.
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(3) RootLabelingProof (EQUIVALENT)
40.86/11.27	We used plain root labeling [ROOTLAB] with the following heuristic:
40.86/11.27	LabelAll: All function symbols get labeled
40.86/11.27	
40.86/11.27	As Q is empty the root labeling was sound AND complete.
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(4)
40.86/11.27	Obligation:
40.86/11.27	Q restricted rewrite system:
40.86/11.27	The TRS R consists of the following rules:
40.86/11.27	
40.86/11.27	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	Q is empty.
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(5) DependencyPairsProof (EQUIVALENT)
40.86/11.27	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(6)
40.86/11.27	Obligation:
40.86/11.27	Q DP problem:
40.86/11.27	The TRS P consists of the following rules:
40.86/11.27	
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{a_1}(x1))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
40.86/11.27	
40.86/11.27	The TRS R consists of the following rules:
40.86/11.27	
40.86/11.27	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	Q is empty.
40.86/11.27	We have to consider all minimal (P,Q,R)-chains.
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(7) DependencyGraphProof (EQUIVALENT)
40.86/11.27	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(8)
40.86/11.27	Obligation:
40.86/11.27	Q DP problem:
40.86/11.27	The TRS P consists of the following rules:
40.86/11.27	
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
40.86/11.27	
40.86/11.27	The TRS R consists of the following rules:
40.86/11.27	
40.86/11.27	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	Q is empty.
40.86/11.27	We have to consider all minimal (P,Q,R)-chains.
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(9) QDPOrderProof (EQUIVALENT)
40.86/11.27	We use the reduction pair processor [LPAR04,JAR06].
40.86/11.27	
40.86/11.27	
40.86/11.27	The following pairs can be oriented strictly and are deleted.
40.86/11.27	
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(b_{b_1}(x1))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> B_{B_1}(x1)
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
40.86/11.27	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
40.86/11.27	The remaining pairs can at least be oriented weakly.
40.86/11.27	Used ordering:  Polynomial interpretation [POLO]:
40.86/11.27	
40.86/11.27	   POL(A_{A_1}(x_1)) = 2 + 2*x_1
40.86/11.27	   POL(A_{B_1}(x_1)) = 2 + 2*x_1
40.86/11.27	   POL(B_{B_1}(x_1)) = 2*x_1
40.86/11.27	   POL(a_{a_1}(x_1)) = 3 + x_1
40.86/11.27	   POL(a_{b_1}(x_1)) = 3 + x_1
40.86/11.27	   POL(b_{a_1}(x_1)) = 2 + x_1
40.86/11.27	   POL(b_{b_1}(x_1)) = 2 + x_1
40.86/11.27	
40.86/11.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.86/11.27	
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(10)
40.86/11.27	Obligation:
40.86/11.27	Q DP problem:
40.86/11.27	The TRS P consists of the following rules:
40.86/11.27	
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	The TRS R consists of the following rules:
40.86/11.27	
40.86/11.27	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	Q is empty.
40.86/11.27	We have to consider all minimal (P,Q,R)-chains.
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(11) QDPOrderProof (EQUIVALENT)
40.86/11.27	We use the reduction pair processor [LPAR04,JAR06].
40.86/11.27	
40.86/11.27	
40.86/11.27	The following pairs can be oriented strictly and are deleted.
40.86/11.27	
40.86/11.27	   A_{A_1}(a_{a_1}(a_{b_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	The remaining pairs can at least be oriented weakly.
40.86/11.27	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.86/11.27	
40.86/11.27	   <<<
40.86/11.27	 POL(A_{B_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, 0A]] 	* 	x_1
40.86/11.27	>>>
40.86/11.27	
40.86/11.27	   <<<
40.86/11.27	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [-I, -I, 0A], [0A, -I, 0A]] 	* 	x_1
40.86/11.27	>>>
40.86/11.27	
40.86/11.27	   <<<
40.86/11.27	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
40.86/11.27	>>>
40.86/11.27	
40.86/11.27	   <<<
40.86/11.27	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, -I, 0A], [1A, -I, 0A], [0A, -I, 0A]] 	* 	x_1
40.86/11.27	>>>
40.86/11.27	
40.86/11.27	   <<<
40.86/11.27	 POL(A_{A_1}(x_1)) =  	[[0A]] 	 +  	[[0A, 0A, 0A]] 	* 	x_1
40.86/11.27	>>>
40.86/11.27	
40.86/11.27	   <<<
40.86/11.27	 POL(a_{a_1}(x_1)) =  	[[-I], [0A], [0A]] 	 +  	[[0A, -I, 0A], [1A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
40.86/11.27	>>>
40.86/11.27	
40.86/11.27	
40.86/11.27	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.86/11.27	
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.27	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.27	
40.86/11.27	
40.86/11.27	----------------------------------------
40.86/11.27	
40.86/11.27	(12)
40.86/11.27	Obligation:
40.86/11.27	Q DP problem:
40.86/11.27	The TRS P consists of the following rules:
40.86/11.27	
40.86/11.27	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.27	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.28	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	
40.86/11.28	The TRS R consists of the following rules:
40.86/11.28	
40.86/11.28	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.28	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.28	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.28	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.28	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	
40.86/11.28	Q is empty.
40.86/11.28	We have to consider all minimal (P,Q,R)-chains.
40.86/11.28	----------------------------------------
40.86/11.28	
40.86/11.28	(13) QDPOrderProof (EQUIVALENT)
40.86/11.28	We use the reduction pair processor [LPAR04,JAR06].
40.86/11.28	
40.86/11.28	
40.86/11.28	The following pairs can be oriented strictly and are deleted.
40.86/11.28	
40.86/11.28	   A_{A_1}(a_{a_1}(a_{a_1}(x1))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.28	The remaining pairs can at least be oriented weakly.
40.86/11.28	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
40.86/11.28	
40.86/11.28	   <<<
40.86/11.28	 POL(A_{B_1}(x_1)) =  	[[0A]] 	 +  	[[0A, -I, -I]] 	* 	x_1
40.86/11.28	>>>
40.86/11.28	
40.86/11.28	   <<<
40.86/11.28	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[-I, -I, 0A], [0A, 0A, 0A], [-I, 0A, -I]] 	* 	x_1
40.86/11.28	>>>
40.86/11.28	
40.86/11.28	   <<<
40.86/11.28	 POL(b_{a_1}(x_1)) =  	[[0A], [-I], [1A]] 	 +  	[[-I, -I, -I], [-I, -I, -I], [-I, -I, -I]] 	* 	x_1
40.86/11.28	>>>
40.86/11.28	
40.86/11.28	   <<<
40.86/11.28	 POL(a_{b_1}(x_1)) =  	[[0A], [-I], [-I]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
40.86/11.28	>>>
40.86/11.28	
40.86/11.28	   <<<
40.86/11.28	 POL(A_{A_1}(x_1)) =  	[[1A]] 	 +  	[[-I, -I, -I]] 	* 	x_1
40.86/11.28	>>>
40.86/11.28	
40.86/11.28	   <<<
40.86/11.28	 POL(a_{a_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
40.86/11.28	>>>
40.86/11.28	
40.86/11.28	
40.86/11.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
40.86/11.28	
40.86/11.28	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.28	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	
40.86/11.28	
40.86/11.28	----------------------------------------
40.86/11.28	
40.86/11.28	(14)
40.86/11.28	Obligation:
40.86/11.28	Q DP problem:
40.86/11.28	The TRS P consists of the following rules:
40.86/11.28	
40.86/11.28	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.28	   A_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	
40.86/11.28	The TRS R consists of the following rules:
40.86/11.28	
40.86/11.28	   a_{a_1}(a_{a_1}(a_{a_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
40.86/11.28	   a_{a_1}(a_{a_1}(a_{b_1}(x1))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
40.86/11.28	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.28	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
40.86/11.28	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
40.86/11.28	
40.86/11.28	Q is empty.
40.86/11.28	We have to consider all minimal (P,Q,R)-chains.
40.86/11.28	----------------------------------------
40.86/11.28	
40.86/11.28	(15) DependencyGraphProof (EQUIVALENT)
40.86/11.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
40.86/11.28	----------------------------------------
40.86/11.28	
40.86/11.28	(16)
40.86/11.28	TRUE
40.93/11.32	EOF