13.06/4.18	YES
13.52/4.31	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
13.52/4.31	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
13.52/4.31	
13.52/4.31	
13.52/4.31	Termination w.r.t. Q of the given QTRS could be proven:
13.52/4.31	
13.52/4.31	(0) QTRS
13.52/4.31	(1) FlatCCProof [EQUIVALENT, 0 ms]
13.52/4.31	(2) QTRS
13.52/4.31	(3) RootLabelingProof [EQUIVALENT, 0 ms]
13.52/4.31	(4) QTRS
13.52/4.31	(5) DependencyPairsProof [EQUIVALENT, 29 ms]
13.52/4.31	(6) QDP
13.52/4.31	(7) QDPOrderProof [EQUIVALENT, 212 ms]
13.52/4.31	(8) QDP
13.52/4.31	(9) PisEmptyProof [EQUIVALENT, 0 ms]
13.52/4.31	(10) YES
13.52/4.31	
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(0)
13.52/4.31	Obligation:
13.52/4.31	Q restricted rewrite system:
13.52/4.31	The TRS R consists of the following rules:
13.52/4.31	
13.52/4.31	   a(b(b(x1))) -> b(a(a(x1)))
13.52/4.31	   a(a(b(x1))) -> b(b(a(x1)))
13.52/4.31	
13.52/4.31	Q is empty.
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(1) FlatCCProof (EQUIVALENT)
13.52/4.31	We used flat context closure [ROOTLAB]
13.52/4.31	As Q is empty the flat context closure was sound AND complete.
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(2)
13.52/4.31	Obligation:
13.52/4.31	Q restricted rewrite system:
13.52/4.31	The TRS R consists of the following rules:
13.52/4.31	
13.52/4.31	   a(a(b(b(x1)))) -> a(b(a(a(x1))))
13.52/4.31	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
13.52/4.31	   a(a(a(b(x1)))) -> a(b(b(a(x1))))
13.52/4.31	   b(a(a(b(x1)))) -> b(b(b(a(x1))))
13.52/4.31	
13.52/4.31	Q is empty.
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(3) RootLabelingProof (EQUIVALENT)
13.52/4.31	We used plain root labeling [ROOTLAB] with the following heuristic:
13.52/4.31	LabelAll: All function symbols get labeled
13.52/4.31	
13.52/4.31	As Q is empty the root labeling was sound AND complete.
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(4)
13.52/4.31	Obligation:
13.52/4.31	Q restricted rewrite system:
13.52/4.31	The TRS R consists of the following rules:
13.52/4.31	
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	
13.52/4.31	Q is empty.
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(5) DependencyPairsProof (EQUIVALENT)
13.52/4.31	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(6)
13.52/4.31	Obligation:
13.52/4.31	Q DP problem:
13.52/4.31	The TRS P consists of the following rules:
13.52/4.31	
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
13.52/4.31	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
13.52/4.31	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
13.52/4.31	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
13.52/4.31	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
13.52/4.31	
13.52/4.31	The TRS R consists of the following rules:
13.52/4.31	
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	
13.52/4.31	Q is empty.
13.52/4.31	We have to consider all minimal (P,Q,R)-chains.
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(7) QDPOrderProof (EQUIVALENT)
13.52/4.31	We use the reduction pair processor [LPAR04,JAR06].
13.52/4.31	
13.52/4.31	
13.52/4.31	The following pairs can be oriented strictly and are deleted.
13.52/4.31	
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
13.52/4.31	   A_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{a_1}(x1)))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(a_{a_1}(x1))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{a_1}(a_{b_1}(x1)))
13.52/4.31	   B_{A_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> A_{A_1}(a_{b_1}(x1))
13.52/4.31	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
13.52/4.31	   A_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   A_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
13.52/4.31	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> B_{A_1}(a_{a_1}(x1))
13.52/4.31	   B_{A_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> A_{A_1}(x1)
13.52/4.31	   B_{A_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> B_{A_1}(a_{b_1}(x1))
13.52/4.31	The remaining pairs can at least be oriented weakly.
13.52/4.31	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
13.52/4.31	
13.52/4.31	POL( A_{A_1}_1(x_1) ) = 2x_1
13.52/4.31	POL( B_{A_1}_1(x_1) ) = 2x_1 + 2
13.52/4.31	POL( a_{a_1}_1(x_1) ) = 2x_1
13.52/4.31	POL( a_{b_1}_1(x_1) ) = 2x_1 + 2
13.52/4.31	POL( b_{b_1}_1(x_1) ) = 2x_1 + 2
13.52/4.31	POL( b_{a_1}_1(x_1) ) = 2x_1
13.52/4.31	
13.52/4.31	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
13.52/4.31	
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	
13.52/4.31	
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(8)
13.52/4.31	Obligation:
13.52/4.31	Q DP problem:
13.52/4.31	P is empty.
13.52/4.31	The TRS R consists of the following rules:
13.52/4.31	
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{b_1}(b_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{a_1}(a_{a_1}(a_{b_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   a_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> a_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{a_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{a_1}(x1))))
13.52/4.31	   b_{a_1}(a_{a_1}(a_{b_1}(b_{b_1}(x1)))) -> b_{b_1}(b_{b_1}(b_{a_1}(a_{b_1}(x1))))
13.52/4.31	
13.52/4.31	Q is empty.
13.52/4.31	We have to consider all minimal (P,Q,R)-chains.
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(9) PisEmptyProof (EQUIVALENT)
13.52/4.31	The TRS P is empty. Hence, there is no (P,Q,R) chain.
13.52/4.31	----------------------------------------
13.52/4.31	
13.52/4.31	(10)
13.52/4.31	YES
13.69/4.37	EOF