23.37/6.99	YES
25.37/8.10	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
25.37/8.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
25.37/8.10	
25.37/8.10	
25.37/8.10	Termination w.r.t. Q of the given QTRS could be proven:
25.37/8.10	
25.37/8.10	(0) QTRS
25.37/8.10	(1) DependencyPairsProof [EQUIVALENT, 38 ms]
25.37/8.10	(2) QDP
25.37/8.10	(3) DependencyGraphProof [EQUIVALENT, 2 ms]
25.37/8.10	(4) QDP
25.37/8.10	(5) QDPOrderProof [EQUIVALENT, 108 ms]
25.37/8.10	(6) QDP
25.37/8.10	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
25.37/8.10	(8) TRUE
25.37/8.10	
25.37/8.10	
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(0)
25.37/8.10	Obligation:
25.37/8.10	Q restricted rewrite system:
25.37/8.10	The TRS R consists of the following rules:
25.37/8.10	
25.37/8.10	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
25.37/8.10	   b(a(a(b(x1)))) -> a(b(b(b(x1))))
25.37/8.10	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
25.37/8.10	
25.37/8.10	Q is empty.
25.37/8.10	
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(1) DependencyPairsProof (EQUIVALENT)
25.37/8.10	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(2)
25.37/8.10	Obligation:
25.37/8.10	Q DP problem:
25.37/8.10	The TRS P consists of the following rules:
25.37/8.10	
25.37/8.10	   B(a(b(b(x1)))) -> A(b(a(b(x1))))
25.37/8.10	   B(a(b(b(x1)))) -> B(a(b(x1)))
25.37/8.10	   B(a(b(b(x1)))) -> A(b(x1))
25.37/8.10	   B(a(a(b(x1)))) -> A(b(b(b(x1))))
25.37/8.10	   B(a(a(b(x1)))) -> B(b(b(x1)))
25.37/8.10	   B(a(a(b(x1)))) -> B(b(x1))
25.37/8.10	   A(b(b(a(x1)))) -> A(b(a(a(x1))))
25.37/8.10	   A(b(b(a(x1)))) -> B(a(a(x1)))
25.37/8.10	   A(b(b(a(x1)))) -> A(a(x1))
25.37/8.10	
25.37/8.10	The TRS R consists of the following rules:
25.37/8.10	
25.37/8.10	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
25.37/8.10	   b(a(a(b(x1)))) -> a(b(b(b(x1))))
25.37/8.10	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
25.37/8.10	
25.37/8.10	Q is empty.
25.37/8.10	We have to consider all minimal (P,Q,R)-chains.
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(3) DependencyGraphProof (EQUIVALENT)
25.37/8.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(4)
25.37/8.10	Obligation:
25.37/8.10	Q DP problem:
25.37/8.10	The TRS P consists of the following rules:
25.37/8.10	
25.37/8.10	   B(a(b(b(x1)))) -> A(b(x1))
25.37/8.10	   A(b(b(a(x1)))) -> B(a(a(x1)))
25.37/8.10	   B(a(b(b(x1)))) -> B(a(b(x1)))
25.37/8.10	   B(a(a(b(x1)))) -> A(b(b(b(x1))))
25.37/8.10	   B(a(a(b(x1)))) -> B(b(b(x1)))
25.37/8.10	   B(a(a(b(x1)))) -> B(b(x1))
25.37/8.10	
25.37/8.10	The TRS R consists of the following rules:
25.37/8.10	
25.37/8.10	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
25.37/8.10	   b(a(a(b(x1)))) -> a(b(b(b(x1))))
25.37/8.10	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
25.37/8.10	
25.37/8.10	Q is empty.
25.37/8.10	We have to consider all minimal (P,Q,R)-chains.
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(5) QDPOrderProof (EQUIVALENT)
25.37/8.10	We use the reduction pair processor [LPAR04,JAR06].
25.37/8.10	
25.37/8.10	
25.37/8.10	The following pairs can be oriented strictly and are deleted.
25.37/8.10	
25.37/8.10	   B(a(b(b(x1)))) -> A(b(x1))
25.37/8.10	   A(b(b(a(x1)))) -> B(a(a(x1)))
25.37/8.10	   B(a(b(b(x1)))) -> B(a(b(x1)))
25.37/8.10	   B(a(a(b(x1)))) -> B(b(b(x1)))
25.37/8.10	   B(a(a(b(x1)))) -> B(b(x1))
25.37/8.10	The remaining pairs can at least be oriented weakly.
25.37/8.10	Used ordering:  Polynomial interpretation [POLO]:
25.37/8.10	
25.37/8.10	   POL(A(x_1)) = 4*x_1
25.37/8.10	   POL(B(x_1)) = 4*x_1
25.37/8.10	   POL(a(x_1)) = 4 + 4*x_1
25.37/8.10	   POL(b(x_1)) = 4 + 4*x_1
25.37/8.10	
25.37/8.10	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
25.37/8.10	
25.37/8.10	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
25.37/8.10	   b(a(a(b(x1)))) -> a(b(b(b(x1))))
25.37/8.10	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
25.37/8.10	
25.37/8.10	
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(6)
25.37/8.10	Obligation:
25.37/8.10	Q DP problem:
25.37/8.10	The TRS P consists of the following rules:
25.37/8.10	
25.37/8.10	   B(a(a(b(x1)))) -> A(b(b(b(x1))))
25.37/8.10	
25.37/8.10	The TRS R consists of the following rules:
25.37/8.10	
25.37/8.10	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
25.37/8.10	   b(a(a(b(x1)))) -> a(b(b(b(x1))))
25.37/8.10	   a(b(b(a(x1)))) -> a(b(a(a(x1))))
25.37/8.10	
25.37/8.10	Q is empty.
25.37/8.10	We have to consider all minimal (P,Q,R)-chains.
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(7) DependencyGraphProof (EQUIVALENT)
25.37/8.10	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
25.37/8.10	----------------------------------------
25.37/8.10	
25.37/8.10	(8)
25.37/8.10	TRUE
25.37/8.14	EOF