17.04/5.22	YES
17.04/5.22	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
17.04/5.22	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
17.04/5.22	
17.04/5.22	
17.04/5.22	Termination w.r.t. Q of the given QTRS could be proven:
17.04/5.22	
17.04/5.22	(0) QTRS
17.04/5.22	(1) QTRS Reverse [EQUIVALENT, 0 ms]
17.04/5.22	(2) QTRS
17.04/5.22	(3) FlatCCProof [EQUIVALENT, 0 ms]
17.04/5.22	(4) QTRS
17.04/5.22	(5) RootLabelingProof [EQUIVALENT, 0 ms]
17.04/5.22	(6) QTRS
17.04/5.22	(7) DependencyPairsProof [EQUIVALENT, 31 ms]
17.04/5.22	(8) QDP
17.04/5.22	(9) QDPOrderProof [EQUIVALENT, 117 ms]
17.04/5.22	(10) QDP
17.04/5.22	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
17.04/5.22	(12) TRUE
17.04/5.22	
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(0)
17.04/5.22	Obligation:
17.04/5.22	Q restricted rewrite system:
17.04/5.22	The TRS R consists of the following rules:
17.04/5.22	
17.04/5.22	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
17.04/5.22	   a(b(a(a(x1)))) -> b(b(b(a(x1))))
17.04/5.22	   b(b(b(a(x1)))) -> a(a(a(b(x1))))
17.04/5.22	
17.04/5.22	Q is empty.
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(1) QTRS Reverse (EQUIVALENT)
17.04/5.22	We applied the QTRS Reverse Processor [REVERSE].
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(2)
17.04/5.22	Obligation:
17.04/5.22	Q restricted rewrite system:
17.04/5.22	The TRS R consists of the following rules:
17.04/5.22	
17.04/5.22	   a(a(a(b(x1)))) -> a(a(b(b(x1))))
17.04/5.22	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
17.04/5.22	   a(b(b(b(x1)))) -> b(a(a(a(x1))))
17.04/5.22	
17.04/5.22	Q is empty.
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(3) FlatCCProof (EQUIVALENT)
17.04/5.22	We used flat context closure [ROOTLAB]
17.04/5.22	As Q is empty the flat context closure was sound AND complete.
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(4)
17.04/5.22	Obligation:
17.04/5.22	Q restricted rewrite system:
17.04/5.22	The TRS R consists of the following rules:
17.04/5.22	
17.04/5.22	   a(a(a(b(x1)))) -> a(a(b(b(x1))))
17.04/5.22	   a(a(b(a(x1)))) -> a(b(b(b(x1))))
17.04/5.22	   a(a(b(b(b(x1))))) -> a(b(a(a(a(x1)))))
17.04/5.22	   b(a(b(b(b(x1))))) -> b(b(a(a(a(x1)))))
17.04/5.22	
17.04/5.22	Q is empty.
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(5) RootLabelingProof (EQUIVALENT)
17.04/5.22	We used plain root labeling [ROOTLAB] with the following heuristic:
17.04/5.22	LabelAll: All function symbols get labeled
17.04/5.22	
17.04/5.22	As Q is empty the root labeling was sound AND complete.
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(6)
17.04/5.22	Obligation:
17.04/5.22	Q restricted rewrite system:
17.04/5.22	The TRS R consists of the following rules:
17.04/5.22	
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	
17.04/5.22	Q is empty.
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(7) DependencyPairsProof (EQUIVALENT)
17.04/5.22	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(8)
17.04/5.22	Obligation:
17.04/5.22	Q DP problem:
17.04/5.22	The TRS P consists of the following rules:
17.04/5.22	
17.04/5.22	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
17.04/5.22	
17.04/5.22	The TRS R consists of the following rules:
17.04/5.22	
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	
17.04/5.22	Q is empty.
17.04/5.22	We have to consider all minimal (P,Q,R)-chains.
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(9) QDPOrderProof (EQUIVALENT)
17.04/5.22	We use the reduction pair processor [LPAR04,JAR06].
17.04/5.22	
17.04/5.22	
17.04/5.22	The following pairs can be oriented strictly and are deleted.
17.04/5.22	
17.04/5.22	   A_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> B_{A_1}(x1)
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
17.04/5.22	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{a_1}(x1)))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(a_{a_1}(x1))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> A_{A_1}(x1)
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> B_{A_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{a_1}(a_{b_1}(x1)))
17.04/5.22	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> A_{A_1}(a_{b_1}(x1))
17.04/5.22	The remaining pairs can at least be oriented weakly.
17.04/5.22	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
17.04/5.22	
17.04/5.22	POL( A_{A_1}_1(x_1) ) = 2x_1 + 2
17.04/5.22	POL( a_{b_1}_1(x_1) ) = x_1 + 2
17.04/5.22	POL( B_{A_1}_1(x_1) ) = max{0, 2x_1 - 2}
17.04/5.22	POL( b_{a_1}_1(x_1) ) = x_1 + 2
17.04/5.22	POL( b_{b_1}_1(x_1) ) = x_1 + 2
17.04/5.22	POL( a_{a_1}_1(x_1) ) = x_1 + 2
17.04/5.22	
17.04/5.22	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
17.04/5.22	
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	
17.04/5.22	
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(10)
17.04/5.22	Obligation:
17.04/5.22	Q DP problem:
17.04/5.22	The TRS P consists of the following rules:
17.04/5.22	
17.04/5.22	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	
17.04/5.22	The TRS R consists of the following rules:
17.04/5.22	
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))))
17.04/5.22	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))))
17.04/5.22	
17.04/5.22	Q is empty.
17.04/5.22	We have to consider all minimal (P,Q,R)-chains.
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(11) DependencyGraphProof (EQUIVALENT)
17.04/5.22	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
17.04/5.22	----------------------------------------
17.04/5.22	
17.04/5.22	(12)
17.04/5.22	TRUE
17.35/5.33	EOF