27.64/8.00	YES
27.64/8.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
27.64/8.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
27.64/8.01	
27.64/8.01	
27.64/8.01	Termination w.r.t. Q of the given QTRS could be proven:
27.64/8.01	
27.64/8.01	(0) QTRS
27.64/8.01	(1) FlatCCProof [EQUIVALENT, 0 ms]
27.64/8.01	(2) QTRS
27.64/8.01	(3) RootLabelingProof [EQUIVALENT, 1 ms]
27.64/8.01	(4) QTRS
27.64/8.01	(5) DependencyPairsProof [EQUIVALENT, 21 ms]
27.64/8.01	(6) QDP
27.64/8.01	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
27.64/8.01	(8) AND
27.64/8.01	    (9) QDP
27.64/8.01	        (10) UsableRulesProof [EQUIVALENT, 0 ms]
27.64/8.01	        (11) QDP
27.64/8.01	        (12) QDPSizeChangeProof [EQUIVALENT, 0 ms]
27.64/8.01	        (13) YES
27.64/8.01	    (14) QDP
27.64/8.01	        (15) QDPOrderProof [EQUIVALENT, 42 ms]
27.64/8.01	        (16) QDP
27.64/8.01	        (17) DependencyGraphProof [EQUIVALENT, 0 ms]
27.64/8.01	        (18) QDP
27.64/8.01	        (19) QDPOrderProof [EQUIVALENT, 493 ms]
27.64/8.01	        (20) QDP
27.64/8.01	        (21) PisEmptyProof [EQUIVALENT, 0 ms]
27.64/8.01	        (22) YES
27.64/8.01	
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(0)
27.64/8.01	Obligation:
27.64/8.01	Q restricted rewrite system:
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
27.64/8.01	   b(a(b(b(x1)))) -> a(a(a(a(x1))))
27.64/8.01	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(1) FlatCCProof (EQUIVALENT)
27.64/8.01	We used flat context closure [ROOTLAB]
27.64/8.01	As Q is empty the flat context closure was sound AND complete.
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(2)
27.64/8.01	Obligation:
27.64/8.01	Q restricted rewrite system:
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b(b(a(a(x1)))) -> b(b(a(b(x1))))
27.64/8.01	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
27.64/8.01	   b(b(a(b(b(x1))))) -> b(a(a(a(a(x1)))))
27.64/8.01	   a(b(a(b(b(x1))))) -> a(a(a(a(a(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(3) RootLabelingProof (EQUIVALENT)
27.64/8.01	We used plain root labeling [ROOTLAB] with the following heuristic:
27.64/8.01	LabelAll: All function symbols get labeled
27.64/8.01	
27.64/8.01	As Q is empty the root labeling was sound AND complete.
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(4)
27.64/8.01	Obligation:
27.64/8.01	Q restricted rewrite system:
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(5) DependencyPairsProof (EQUIVALENT)
27.64/8.01	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(6)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	The TRS P consists of the following rules:
27.64/8.01	
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
27.64/8.01	   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))))
27.64/8.01	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
27.64/8.01	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{B_1}(x1)
27.64/8.01	   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))))
27.64/8.01	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
27.64/8.01	   B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
27.64/8.01	
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(7) DependencyGraphProof (EQUIVALENT)
27.64/8.01	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(8)
27.64/8.01	Complex Obligation (AND)
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(9)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	The TRS P consists of the following rules:
27.64/8.01	
27.64/8.01	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
27.64/8.01	
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(10) UsableRulesProof (EQUIVALENT)
27.64/8.01	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.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(11)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	The TRS P consists of the following rules:
27.64/8.01	
27.64/8.01	   A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
27.64/8.01	
27.64/8.01	R is empty.
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(12) QDPSizeChangeProof (EQUIVALENT)
27.64/8.01	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. 
27.64/8.01	
27.64/8.01	From the DPs we obtained the following set of size-change graphs:
27.64/8.01	*A_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{B_1}(x1)
27.64/8.01	The graph contains the following edges 1 > 1
27.64/8.01	
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(13)
27.64/8.01	YES
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(14)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	The TRS P consists of the following rules:
27.64/8.01	
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
27.64/8.01	   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))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
27.64/8.01	   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))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
27.64/8.01	
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(15) QDPOrderProof (EQUIVALENT)
27.64/8.01	We use the reduction pair processor [LPAR04,JAR06].
27.64/8.01	
27.64/8.01	
27.64/8.01	The following pairs can be oriented strictly and are deleted.
27.64/8.01	
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
27.64/8.01	The remaining pairs can at least be oriented weakly.
27.64/8.01	Used ordering:  Polynomial interpretation [POLO]:
27.64/8.01	
27.64/8.01	   POL(B_{A_1}(x_1)) = x_1
27.64/8.01	   POL(B_{B_1}(x_1)) = x_1
27.64/8.01	   POL(a_{a_1}(x_1)) = 1 + x_1
27.64/8.01	   POL(a_{b_1}(x_1)) = 1 + x_1
27.64/8.01	   POL(b_{a_1}(x_1)) = 1 + x_1
27.64/8.01	   POL(b_{b_1}(x_1)) = 1 + x_1
27.64/8.01	
27.64/8.01	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.64/8.01	
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(16)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	The TRS P consists of the following rules:
27.64/8.01	
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(17) DependencyGraphProof (EQUIVALENT)
27.64/8.01	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(18)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	The TRS P consists of the following rules:
27.64/8.01	
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(19) QDPOrderProof (EQUIVALENT)
27.64/8.01	We use the reduction pair processor [LPAR04,JAR06].
27.64/8.01	
27.64/8.01	
27.64/8.01	The following pairs can be oriented strictly and are deleted.
27.64/8.01	
27.64/8.01	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	The remaining pairs can at least be oriented weakly.
27.64/8.01	Used ordering:  Matrix interpretation [MATRO] with arctic natural numbers [ARCTIC]:
27.64/8.01	
27.64/8.01	   <<<
27.64/8.01	 POL(B_{B_1}(x_1)) =  	[[0A]] 	 +  	[[-I, 0A, -I]] 	* 	x_1
27.64/8.01	>>>
27.64/8.01	
27.64/8.01	   <<<
27.64/8.01	 POL(b_{a_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[0A, 0A, 0A], [-I, 0A, -I], [0A, 0A, 1A]] 	* 	x_1
27.64/8.01	>>>
27.64/8.01	
27.64/8.01	   <<<
27.64/8.01	 POL(a_{a_1}(x_1)) =  	[[-I], [-I], [0A]] 	 +  	[[0A, 0A, 0A], [0A, 0A, 1A], [0A, 0A, 0A]] 	* 	x_1
27.64/8.01	>>>
27.64/8.01	
27.64/8.01	   <<<
27.64/8.01	 POL(a_{b_1}(x_1)) =  	[[0A], [0A], [-I]] 	 +  	[[1A, 0A, 1A], [0A, 0A, -I], [-I, -I, 0A]] 	* 	x_1
27.64/8.01	>>>
27.64/8.01	
27.64/8.01	   <<<
27.64/8.01	 POL(b_{b_1}(x_1)) =  	[[0A], [0A], [0A]] 	 +  	[[1A, 0A, 1A], [0A, 0A, 0A], [0A, 0A, 0A]] 	* 	x_1
27.64/8.01	>>>
27.64/8.01	
27.64/8.01	
27.64/8.01	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
27.64/8.01	
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(20)
27.64/8.01	Obligation:
27.64/8.01	Q DP problem:
27.64/8.01	P is empty.
27.64/8.01	The TRS R consists of the following rules:
27.64/8.01	
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
27.64/8.01	   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))))
27.64/8.01	   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))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
27.64/8.01	   a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
27.64/8.01	
27.64/8.01	Q is empty.
27.64/8.01	We have to consider all minimal (P,Q,R)-chains.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(21) PisEmptyProof (EQUIVALENT)
27.64/8.01	The TRS P is empty. Hence, there is no (P,Q,R) chain.
27.64/8.01	----------------------------------------
27.64/8.01	
27.64/8.01	(22)
27.64/8.01	YES
27.93/8.07	EOF