30.96/9.14	YES
35.25/12.27	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
35.25/12.27	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
35.25/12.27	
35.25/12.27	
35.25/12.27	Termination w.r.t. Q of the given QTRS could be proven:
35.25/12.27	
35.25/12.27	(0) QTRS
35.25/12.27	(1) FlatCCProof [EQUIVALENT, 0 ms]
35.25/12.27	(2) QTRS
35.25/12.27	(3) RootLabelingProof [EQUIVALENT, 1 ms]
35.25/12.27	(4) QTRS
35.25/12.27	(5) DependencyPairsProof [EQUIVALENT, 62 ms]
35.25/12.27	(6) QDP
35.25/12.27	(7) DependencyGraphProof [EQUIVALENT, 4 ms]
35.25/12.27	(8) QDP
35.25/12.27	(9) QDPOrderProof [EQUIVALENT, 690 ms]
35.25/12.27	(10) QDP
35.25/12.27	(11) PisEmptyProof [EQUIVALENT, 0 ms]
35.25/12.27	(12) YES
35.25/12.27	
35.25/12.27	
35.25/12.27	----------------------------------------
35.25/12.27	
35.25/12.27	(0)
35.25/12.27	Obligation:
35.25/12.27	Q restricted rewrite system:
35.25/12.27	The TRS R consists of the following rules:
35.25/12.27	
35.25/12.27	   b(a(a(b(x1)))) -> a(a(a(b(x1))))
35.25/12.27	   b(a(a(a(x1)))) -> a(a(b(b(x1))))
35.25/12.27	   b(a(b(a(x1)))) -> b(b(a(b(x1))))
35.25/12.27	
35.25/12.27	Q is empty.
35.25/12.27	
35.25/12.27	----------------------------------------
35.25/12.27	
35.25/12.27	(1) FlatCCProof (EQUIVALENT)
35.25/12.27	We used flat context closure [ROOTLAB]
35.25/12.27	As Q is empty the flat context closure was sound AND complete.
35.25/12.27	
35.25/12.27	----------------------------------------
35.25/12.27	
35.25/12.27	(2)
35.25/12.27	Obligation:
35.25/12.27	Q restricted rewrite system:
35.25/12.28	The TRS R consists of the following rules:
35.25/12.28	
35.25/12.28	   b(a(b(a(x1)))) -> b(b(a(b(x1))))
35.25/12.28	   b(b(a(a(b(x1))))) -> b(a(a(a(b(x1)))))
35.25/12.28	   a(b(a(a(b(x1))))) -> a(a(a(a(b(x1)))))
35.25/12.28	   b(b(a(a(a(x1))))) -> b(a(a(b(b(x1)))))
35.25/12.28	   a(b(a(a(a(x1))))) -> a(a(a(b(b(x1)))))
35.25/12.28	
35.25/12.28	Q is empty.
35.25/12.28	
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(3) RootLabelingProof (EQUIVALENT)
35.25/12.28	We used plain root labeling [ROOTLAB] with the following heuristic:
35.25/12.28	LabelAll: All function symbols get labeled
35.25/12.28	
35.25/12.28	As Q is empty the root labeling was sound AND complete.
35.25/12.28	
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(4)
35.25/12.28	Obligation:
35.25/12.28	Q restricted rewrite system:
35.25/12.28	The TRS R consists of the following rules:
35.25/12.28	
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	
35.25/12.28	Q is empty.
35.25/12.28	
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(5) DependencyPairsProof (EQUIVALENT)
35.25/12.28	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(6)
35.25/12.28	Obligation:
35.25/12.28	Q DP problem:
35.25/12.28	The TRS P consists of the following rules:
35.25/12.28	
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
35.25/12.28	
35.25/12.28	The TRS R consists of the following rules:
35.25/12.28	
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	
35.25/12.28	Q is empty.
35.25/12.28	We have to consider all minimal (P,Q,R)-chains.
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(7) DependencyGraphProof (EQUIVALENT)
35.25/12.28	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(8)
35.25/12.28	Obligation:
35.25/12.28	Q DP problem:
35.25/12.28	The TRS P consists of the following rules:
35.25/12.28	
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
35.25/12.28	
35.25/12.28	The TRS R consists of the following rules:
35.25/12.28	
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	
35.25/12.28	Q is empty.
35.25/12.28	We have to consider all minimal (P,Q,R)-chains.
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(9) QDPOrderProof (EQUIVALENT)
35.25/12.28	We use the reduction pair processor [LPAR04,JAR06].
35.25/12.28	
35.25/12.28	
35.25/12.28	The following pairs can be oriented strictly and are deleted.
35.25/12.28	
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{B_1}(x1)
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
35.25/12.28	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{b_1}(x1)))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{B_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(a_{b_1}(b_{a_1}(x1)))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> A_{B_1}(b_{b_1}(x1))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{a_1}(x1)))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
35.25/12.28	   A_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> B_{B_1}(x1)
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(x1))
35.25/12.28	   B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
35.25/12.28	The remaining pairs can at least be oriented weakly.
35.25/12.28	Used ordering:  Polynomial interpretation [POLO]:
35.25/12.28	
35.25/12.28	   POL(A_{B_1}(x_1)) = 4*x_1
35.25/12.28	   POL(B_{A_1}(x_1)) = 1 + 4*x_1
35.25/12.28	   POL(B_{B_1}(x_1)) = 4 + 4*x_1
35.25/12.28	   POL(a_{a_1}(x_1)) = 1 + 2*x_1
35.25/12.28	   POL(a_{b_1}(x_1)) = 3 + 2*x_1
35.25/12.28	   POL(b_{a_1}(x_1)) = 2*x_1
35.25/12.28	   POL(b_{b_1}(x_1)) = 2 + 2*x_1
35.25/12.28	
35.25/12.28	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
35.25/12.28	
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	
35.25/12.28	
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(10)
35.25/12.28	Obligation:
35.25/12.28	Q DP problem:
35.25/12.28	P is empty.
35.25/12.28	The TRS R consists of the following rules:
35.25/12.28	
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))
35.25/12.28	   b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
35.25/12.28	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
35.25/12.28	
35.25/12.28	Q is empty.
35.25/12.28	We have to consider all minimal (P,Q,R)-chains.
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(11) PisEmptyProof (EQUIVALENT)
35.25/12.28	The TRS P is empty. Hence, there is no (P,Q,R) chain.
35.25/12.28	----------------------------------------
35.25/12.28	
35.25/12.28	(12)
35.25/12.28	YES
35.42/12.48	EOF