9.93/3.40	YES
10.24/3.46	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.24/3.46	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.24/3.46	
10.24/3.46	
10.24/3.46	Termination w.r.t. Q of the given QTRS could be proven:
10.24/3.46	
10.24/3.46	(0) QTRS
10.24/3.46	(1) FlatCCProof [EQUIVALENT, 0 ms]
10.24/3.46	(2) QTRS
10.24/3.46	(3) RootLabelingProof [EQUIVALENT, 0 ms]
10.24/3.46	(4) QTRS
10.24/3.46	(5) QTRSRRRProof [EQUIVALENT, 64 ms]
10.24/3.46	(6) QTRS
10.24/3.46	(7) DependencyPairsProof [EQUIVALENT, 8 ms]
10.24/3.46	(8) QDP
10.24/3.46	(9) QDPOrderProof [EQUIVALENT, 17 ms]
10.24/3.46	(10) QDP
10.24/3.46	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
10.24/3.46	(12) TRUE
10.24/3.46	
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(0)
10.24/3.46	Obligation:
10.24/3.46	Q restricted rewrite system:
10.24/3.46	The TRS R consists of the following rules:
10.24/3.46	
10.24/3.46	   a(b(b(a(x1)))) -> b(a(b(a(x1))))
10.24/3.46	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
10.24/3.46	   a(a(a(a(x1)))) -> b(b(b(a(x1))))
10.24/3.46	
10.24/3.46	Q is empty.
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(1) FlatCCProof (EQUIVALENT)
10.24/3.46	We used flat context closure [ROOTLAB]
10.24/3.46	As Q is empty the flat context closure was sound AND complete.
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(2)
10.24/3.46	Obligation:
10.24/3.46	Q restricted rewrite system:
10.24/3.46	The TRS R consists of the following rules:
10.24/3.46	
10.24/3.46	   a(b(b(b(x1)))) -> a(a(b(a(x1))))
10.24/3.46	   a(a(b(b(a(x1))))) -> a(b(a(b(a(x1)))))
10.24/3.46	   b(a(b(b(a(x1))))) -> b(b(a(b(a(x1)))))
10.24/3.46	   a(a(a(a(a(x1))))) -> a(b(b(b(a(x1)))))
10.24/3.46	   b(a(a(a(a(x1))))) -> b(b(b(b(a(x1)))))
10.24/3.46	
10.24/3.46	Q is empty.
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(3) RootLabelingProof (EQUIVALENT)
10.24/3.46	We used plain root labeling [ROOTLAB] with the following heuristic:
10.24/3.46	LabelAll: All function symbols get labeled
10.24/3.46	
10.24/3.46	As Q is empty the root labeling was sound AND complete.
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(4)
10.24/3.46	Obligation:
10.24/3.46	Q restricted rewrite system:
10.24/3.46	The TRS R consists of the following rules:
10.24/3.46	
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
10.24/3.46	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	
10.24/3.46	Q is empty.
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(5) QTRSRRRProof (EQUIVALENT)
10.24/3.46	Used ordering:
10.24/3.46	Polynomial interpretation [POLO]:
10.24/3.46	
10.24/3.46	   POL(a_{a_1}(x_1)) = 1 + x_1
10.24/3.46	   POL(a_{b_1}(x_1)) = 1 + x_1
10.24/3.46	   POL(b_{a_1}(x_1)) = x_1
10.24/3.46	   POL(b_{b_1}(x_1)) = 1 + x_1
10.24/3.46	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
10.24/3.46	
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
10.24/3.46	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	
10.24/3.46	
10.24/3.46	
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(6)
10.24/3.46	Obligation:
10.24/3.46	Q restricted rewrite system:
10.24/3.46	The TRS R consists of the following rules:
10.24/3.46	
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	
10.24/3.46	Q is empty.
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(7) DependencyPairsProof (EQUIVALENT)
10.24/3.46	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(8)
10.24/3.46	Obligation:
10.24/3.46	Q DP problem:
10.24/3.46	The TRS P consists of the following rules:
10.24/3.46	
10.24/3.46	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
10.24/3.46	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
10.24/3.46	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
10.24/3.46	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
10.24/3.46	
10.24/3.46	The TRS R consists of the following rules:
10.24/3.46	
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	
10.24/3.46	Q is empty.
10.24/3.46	We have to consider all minimal (P,Q,R)-chains.
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(9) QDPOrderProof (EQUIVALENT)
10.24/3.46	We use the reduction pair processor [LPAR04,JAR06].
10.24/3.46	
10.24/3.46	
10.24/3.46	The following pairs can be oriented strictly and are deleted.
10.24/3.46	
10.24/3.46	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
10.24/3.46	   A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1))))
10.24/3.46	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{A_1}(a_{a_1}(x1))
10.24/3.46	   B_{A_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> B_{A_1}(a_{b_1}(x1))
10.24/3.46	The remaining pairs can at least be oriented weakly.
10.24/3.46	Used ordering:  Polynomial interpretation [POLO]:
10.24/3.46	
10.24/3.46	   POL(A_{B_1}(x_1)) = 1 + x_1
10.24/3.46	   POL(B_{A_1}(x_1)) = x_1
10.24/3.46	   POL(a_{a_1}(x_1)) = 1 + x_1
10.24/3.46	   POL(a_{b_1}(x_1)) = x_1
10.24/3.46	   POL(b_{a_1}(x_1)) = 1 + x_1
10.24/3.46	   POL(b_{b_1}(x_1)) = 1 + x_1
10.24/3.46	
10.24/3.46	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
10.24/3.46	
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	
10.24/3.46	
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(10)
10.24/3.46	Obligation:
10.24/3.46	Q DP problem:
10.24/3.46	The TRS P consists of the following rules:
10.24/3.46	
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{a_1}(x1)))
10.24/3.46	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> A_{B_1}(b_{a_1}(a_{b_1}(x1)))
10.24/3.46	
10.24/3.46	The TRS R consists of the following rules:
10.24/3.46	
10.24/3.46	   a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))))
10.24/3.46	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))))
10.24/3.46	
10.24/3.46	Q is empty.
10.24/3.46	We have to consider all minimal (P,Q,R)-chains.
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(11) DependencyGraphProof (EQUIVALENT)
10.24/3.46	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
10.24/3.46	----------------------------------------
10.24/3.46	
10.24/3.46	(12)
10.24/3.46	TRUE
10.36/4.50	EOF