10.24/3.64	YES
10.39/3.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
10.39/3.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.39/3.71	
10.39/3.71	
10.39/3.71	Termination w.r.t. Q of the given QTRS could be proven:
10.39/3.71	
10.39/3.71	(0) QTRS
10.39/3.71	(1) FlatCCProof [EQUIVALENT, 0 ms]
10.39/3.71	(2) QTRS
10.39/3.71	(3) RootLabelingProof [EQUIVALENT, 0 ms]
10.39/3.71	(4) QTRS
10.39/3.71	(5) QTRSRRRProof [EQUIVALENT, 25 ms]
10.39/3.71	(6) QTRS
10.39/3.71	(7) DependencyPairsProof [EQUIVALENT, 0 ms]
10.39/3.71	(8) QDP
10.39/3.71	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
10.39/3.71	(10) QDP
10.39/3.71	(11) QDPOrderProof [EQUIVALENT, 5 ms]
10.39/3.71	(12) QDP
10.39/3.71	(13) PisEmptyProof [EQUIVALENT, 0 ms]
10.39/3.71	(14) YES
10.39/3.71	
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(0)
10.39/3.71	Obligation:
10.39/3.71	Q restricted rewrite system:
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   b(a(a(a(x1)))) -> a(b(b(b(x1))))
10.39/3.71	   b(a(a(a(x1)))) -> b(b(b(a(x1))))
10.39/3.71	   a(b(b(a(x1)))) -> a(a(a(a(x1))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(1) FlatCCProof (EQUIVALENT)
10.39/3.71	We used flat context closure [ROOTLAB]
10.39/3.71	As Q is empty the flat context closure was sound AND complete.
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(2)
10.39/3.71	Obligation:
10.39/3.71	Q restricted rewrite system:
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   b(a(a(a(x1)))) -> b(b(b(a(x1))))
10.39/3.71	   a(b(b(a(x1)))) -> a(a(a(a(x1))))
10.39/3.71	   b(b(a(a(a(x1))))) -> b(a(b(b(b(x1)))))
10.39/3.71	   a(b(a(a(a(x1))))) -> a(a(b(b(b(x1)))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(3) RootLabelingProof (EQUIVALENT)
10.39/3.71	We used plain root labeling [ROOTLAB] with the following heuristic:
10.39/3.71	LabelAll: All function symbols get labeled
10.39/3.71	
10.39/3.71	As Q is empty the root labeling was sound AND complete.
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(4)
10.39/3.71	Obligation:
10.39/3.71	Q restricted rewrite system:
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
10.39/3.71	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
10.39/3.71	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(5) QTRSRRRProof (EQUIVALENT)
10.39/3.71	Used ordering:
10.39/3.71	Polynomial interpretation [POLO]:
10.39/3.71	
10.39/3.71	   POL(a_{a_1}(x_1)) = 1 + x_1
10.39/3.71	   POL(a_{b_1}(x_1)) = 2 + x_1
10.39/3.71	   POL(b_{a_1}(x_1)) = 1 + x_1
10.39/3.71	   POL(b_{b_1}(x_1)) = x_1
10.39/3.71	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
10.39/3.71	
10.39/3.71	   b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
10.39/3.71	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
10.39/3.71	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
10.39/3.71	   a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	
10.39/3.71	
10.39/3.71	
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(6)
10.39/3.71	Obligation:
10.39/3.71	Q restricted rewrite system:
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(7) DependencyPairsProof (EQUIVALENT)
10.39/3.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(8)
10.39/3.71	Obligation:
10.39/3.71	Q DP problem:
10.39/3.71	The TRS P consists of the following rules:
10.39/3.71	
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> A_{B_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1))))
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
10.39/3.71	
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	We have to consider all minimal (P,Q,R)-chains.
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(9) DependencyGraphProof (EQUIVALENT)
10.39/3.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(10)
10.39/3.71	Obligation:
10.39/3.71	Q DP problem:
10.39/3.71	The TRS P consists of the following rules:
10.39/3.71	
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
10.39/3.71	
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	We have to consider all minimal (P,Q,R)-chains.
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(11) QDPOrderProof (EQUIVALENT)
10.39/3.71	We use the reduction pair processor [LPAR04,JAR06].
10.39/3.71	
10.39/3.71	
10.39/3.71	The following pairs can be oriented strictly and are deleted.
10.39/3.71	
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
10.39/3.71	   B_{B_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
10.39/3.71	The remaining pairs can at least be oriented weakly.
10.39/3.71	Used ordering:  Polynomial interpretation [POLO]:
10.39/3.71	
10.39/3.71	   POL(B_{B_1}(x_1)) = x_1
10.39/3.71	   POL(a_{a_1}(x_1)) = 1 + x_1
10.39/3.71	   POL(a_{b_1}(x_1)) = 1 + x_1
10.39/3.71	   POL(b_{a_1}(x_1)) = 1 + x_1
10.39/3.71	   POL(b_{b_1}(x_1)) = 1 + x_1
10.39/3.71	
10.39/3.71	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
10.39/3.71	
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
10.39/3.71	
10.39/3.71	
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(12)
10.39/3.71	Obligation:
10.39/3.71	Q DP problem:
10.39/3.71	P is empty.
10.39/3.71	The TRS R consists of the following rules:
10.39/3.71	
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
10.39/3.71	   a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
10.39/3.71	   b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
10.39/3.71	
10.39/3.71	Q is empty.
10.39/3.71	We have to consider all minimal (P,Q,R)-chains.
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(13) PisEmptyProof (EQUIVALENT)
10.39/3.71	The TRS P is empty. Hence, there is no (P,Q,R) chain.
10.39/3.71	----------------------------------------
10.39/3.71	
10.39/3.71	(14)
10.39/3.71	YES
10.66/3.76	EOF