13.05/4.27	YES
13.30/4.35	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.30/4.35	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.30/4.35	
13.30/4.35	
13.30/4.35	Termination w.r.t. Q of the given QTRS could be proven:
13.30/4.35	
13.30/4.35	(0) QTRS
13.30/4.35	(1) FlatCCProof [EQUIVALENT, 0 ms]
13.30/4.35	(2) QTRS
13.30/4.35	(3) RootLabelingProof [EQUIVALENT, 0 ms]
13.30/4.35	(4) QTRS
13.30/4.35	(5) QTRSRRRProof [EQUIVALENT, 30 ms]
13.30/4.35	(6) QTRS
13.30/4.35	(7) QTRSRRRProof [EQUIVALENT, 0 ms]
13.30/4.35	(8) QTRS
13.30/4.35	(9) DependencyPairsProof [EQUIVALENT, 0 ms]
13.30/4.35	(10) QDP
13.30/4.35	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
13.30/4.35	(12) QDP
13.30/4.35	(13) QDPOrderProof [EQUIVALENT, 78 ms]
13.30/4.35	(14) QDP
13.30/4.35	(15) PisEmptyProof [EQUIVALENT, 0 ms]
13.30/4.35	(16) YES
13.30/4.35	
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(0)
13.30/4.35	Obligation:
13.30/4.35	Q restricted rewrite system:
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   a(a(a(a(x1)))) -> b(a(b(b(x1))))
13.30/4.35	   b(b(a(b(x1)))) -> a(b(b(b(x1))))
13.30/4.35	   a(a(b(a(x1)))) -> a(a(a(a(x1))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(1) FlatCCProof (EQUIVALENT)
13.30/4.35	We used flat context closure [ROOTLAB]
13.30/4.35	As Q is empty the flat context closure was sound AND complete.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(2)
13.30/4.35	Obligation:
13.30/4.35	Q restricted rewrite system:
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   a(a(b(a(x1)))) -> a(a(a(a(x1))))
13.30/4.35	   a(a(a(a(a(x1))))) -> a(b(a(b(b(x1)))))
13.30/4.35	   b(a(a(a(a(x1))))) -> b(b(a(b(b(x1)))))
13.30/4.35	   a(b(b(a(b(x1))))) -> a(a(b(b(b(x1)))))
13.30/4.35	   b(b(b(a(b(x1))))) -> b(a(b(b(b(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(3) RootLabelingProof (EQUIVALENT)
13.30/4.35	We used plain root labeling [ROOTLAB] with the following heuristic:
13.30/4.35	LabelAll: All function symbols get labeled
13.30/4.35	
13.30/4.35	As Q is empty the root labeling was sound AND complete.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(4)
13.30/4.35	Obligation:
13.30/4.35	Q restricted rewrite system:
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.30/4.35	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.30/4.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(5) QTRSRRRProof (EQUIVALENT)
13.30/4.35	Used ordering:
13.30/4.35	Polynomial interpretation [POLO]:
13.30/4.35	
13.30/4.35	   POL(a_{a_1}(x_1)) = 1 + x_1
13.30/4.35	   POL(a_{b_1}(x_1)) = x_1
13.30/4.35	   POL(b_{a_1}(x_1)) = 2 + x_1
13.30/4.35	   POL(b_{b_1}(x_1)) = x_1
13.30/4.35	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
13.30/4.35	
13.30/4.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> a_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))) -> b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	
13.30/4.35	
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(6)
13.30/4.35	Obligation:
13.30/4.35	Q restricted rewrite system:
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.30/4.35	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(7) QTRSRRRProof (EQUIVALENT)
13.30/4.35	Used ordering:
13.30/4.35	Polynomial interpretation [POLO]:
13.30/4.35	
13.30/4.35	   POL(a_{a_1}(x_1)) = x_1
13.30/4.35	   POL(a_{b_1}(x_1)) = x_1
13.30/4.35	   POL(b_{a_1}(x_1)) = 1 + x_1
13.30/4.35	   POL(b_{b_1}(x_1)) = x_1
13.30/4.35	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
13.30/4.35	
13.30/4.35	   a_{a_1}(a_{b_1}(b_{a_1}(a_{a_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.30/4.35	   a_{a_1}(a_{b_1}(b_{a_1}(a_{b_1}(x1)))) -> a_{a_1}(a_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.30/4.35	
13.30/4.35	
13.30/4.35	
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(8)
13.30/4.35	Obligation:
13.30/4.35	Q restricted rewrite system:
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(9) DependencyPairsProof (EQUIVALENT)
13.30/4.35	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(10)
13.30/4.35	Obligation:
13.30/4.35	Q DP problem:
13.30/4.35	The TRS P consists of the following rules:
13.30/4.35	
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{a_1}(x1))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
13.30/4.35	
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	We have to consider all minimal (P,Q,R)-chains.
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(11) DependencyGraphProof (EQUIVALENT)
13.30/4.35	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(12)
13.30/4.35	Obligation:
13.30/4.35	Q DP problem:
13.30/4.35	The TRS P consists of the following rules:
13.30/4.35	
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
13.30/4.35	
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	We have to consider all minimal (P,Q,R)-chains.
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(13) QDPOrderProof (EQUIVALENT)
13.30/4.35	We use the reduction pair processor [LPAR04,JAR06].
13.30/4.35	
13.30/4.35	
13.30/4.35	The following pairs can be oriented strictly and are deleted.
13.30/4.35	
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{b_1}(x1)))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> B_{B_1}(b_{b_1}(b_{a_1}(x1)))
13.30/4.35	   B_{B_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> B_{B_1}(b_{b_1}(x1))
13.30/4.35	The remaining pairs can at least be oriented weakly.
13.30/4.35	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
13.30/4.35	
13.30/4.35	POL( B_{B_1}_1(x_1) ) = 2x_1 + 2
13.30/4.35	POL( b_{b_1}_1(x_1) ) = 2x_1
13.30/4.35	POL( b_{a_1}_1(x_1) ) = x_1 + 1
13.30/4.35	POL( a_{b_1}_1(x_1) ) = 2x_1 + 1
13.30/4.35	
13.30/4.35	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
13.30/4.35	
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(14)
13.30/4.35	Obligation:
13.30/4.35	Q DP problem:
13.30/4.35	P is empty.
13.30/4.35	The TRS R consists of the following rules:
13.30/4.35	
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{a_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{a_1}(x1)))))
13.30/4.35	   b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(b_{b_1}(x1))))) -> b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(b_{b_1}(x1)))))
13.30/4.35	
13.30/4.35	Q is empty.
13.30/4.35	We have to consider all minimal (P,Q,R)-chains.
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(15) PisEmptyProof (EQUIVALENT)
13.30/4.35	The TRS P is empty. Hence, there is no (P,Q,R) chain.
13.30/4.35	----------------------------------------
13.30/4.35	
13.30/4.35	(16)
13.30/4.35	YES
13.58/4.41	EOF