11.08/3.67	YES
11.97/3.85	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
11.97/3.85	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.97/3.85	
11.97/3.85	
11.97/3.85	Termination w.r.t. Q of the given QTRS could be proven:
11.97/3.85	
11.97/3.85	(0) QTRS
11.97/3.85	(1) DependencyPairsProof [EQUIVALENT, 0 ms]
11.97/3.85	(2) QDP
11.97/3.85	(3) DependencyGraphProof [EQUIVALENT, 0 ms]
11.97/3.85	(4) QDP
11.97/3.85	(5) QDPOrderProof [EQUIVALENT, 35 ms]
11.97/3.85	(6) QDP
11.97/3.85	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
11.97/3.85	(8) TRUE
11.97/3.85	
11.97/3.85	
11.97/3.85	----------------------------------------
11.97/3.85	
11.97/3.85	(0)
11.97/3.85	Obligation:
11.97/3.85	Q restricted rewrite system:
11.97/3.85	The TRS R consists of the following rules:
11.97/3.85	
11.97/3.85	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
11.97/3.85	   a(b(b(b(x1)))) -> a(a(a(a(x1))))
11.97/3.85	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
11.97/3.85	
11.97/3.85	Q is empty.
11.97/3.85	
11.97/3.85	----------------------------------------
11.97/3.85	
11.97/3.85	(1) DependencyPairsProof (EQUIVALENT)
11.97/3.85	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.97/3.85	----------------------------------------
11.97/3.85	
11.97/3.85	(2)
11.97/3.85	Obligation:
11.97/3.85	Q DP problem:
11.97/3.85	The TRS P consists of the following rules:
11.97/3.85	
11.97/3.85	   A(b(a(a(x1)))) -> A(a(b(b(x1))))
11.97/3.85	   A(b(a(a(x1)))) -> A(b(b(x1)))
11.97/3.85	   A(b(a(a(x1)))) -> B(b(x1))
11.97/3.85	   A(b(a(a(x1)))) -> B(x1)
11.97/3.85	   A(b(b(b(x1)))) -> A(a(a(a(x1))))
11.97/3.85	   A(b(b(b(x1)))) -> A(a(a(x1)))
11.97/3.85	   A(b(b(b(x1)))) -> A(a(x1))
11.97/3.85	   A(b(b(b(x1)))) -> A(x1)
11.97/3.85	   B(b(a(b(x1)))) -> A(b(a(a(x1))))
11.97/3.85	   B(b(a(b(x1)))) -> B(a(a(x1)))
11.97/3.85	   B(b(a(b(x1)))) -> A(a(x1))
11.97/3.85	   B(b(a(b(x1)))) -> A(x1)
11.97/3.85	
11.97/3.85	The TRS R consists of the following rules:
11.97/3.85	
11.97/3.85	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
11.97/3.85	   a(b(b(b(x1)))) -> a(a(a(a(x1))))
11.97/3.85	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
11.97/3.85	
11.97/3.85	Q is empty.
11.97/3.85	We have to consider all minimal (P,Q,R)-chains.
11.97/3.85	----------------------------------------
11.97/3.85	
11.97/3.85	(3) DependencyGraphProof (EQUIVALENT)
11.97/3.85	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 6 less nodes.
11.97/3.85	----------------------------------------
11.97/3.85	
11.97/3.85	(4)
11.97/3.85	Obligation:
11.97/3.85	Q DP problem:
11.97/3.85	The TRS P consists of the following rules:
11.97/3.85	
11.97/3.85	   A(b(a(a(x1)))) -> B(b(x1))
11.97/3.85	   B(b(a(b(x1)))) -> A(b(a(a(x1))))
11.97/3.85	   A(b(a(a(x1)))) -> A(b(b(x1)))
11.97/3.85	   A(b(a(a(x1)))) -> B(x1)
11.97/3.85	   B(b(a(b(x1)))) -> A(x1)
11.97/3.85	   A(b(b(b(x1)))) -> A(x1)
11.97/3.85	
11.97/3.85	The TRS R consists of the following rules:
11.97/3.86	
11.97/3.86	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
11.97/3.86	   a(b(b(b(x1)))) -> a(a(a(a(x1))))
11.97/3.86	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
11.97/3.86	
11.97/3.86	Q is empty.
11.97/3.86	We have to consider all minimal (P,Q,R)-chains.
11.97/3.86	----------------------------------------
11.97/3.86	
11.97/3.86	(5) QDPOrderProof (EQUIVALENT)
11.97/3.86	We use the reduction pair processor [LPAR04,JAR06].
11.97/3.86	
11.97/3.86	
11.97/3.86	The following pairs can be oriented strictly and are deleted.
11.97/3.86	
11.97/3.86	   A(b(a(a(x1)))) -> B(b(x1))
11.97/3.86	   A(b(a(a(x1)))) -> A(b(b(x1)))
11.97/3.86	   A(b(a(a(x1)))) -> B(x1)
11.97/3.86	   B(b(a(b(x1)))) -> A(x1)
11.97/3.86	   A(b(b(b(x1)))) -> A(x1)
11.97/3.86	The remaining pairs can at least be oriented weakly.
11.97/3.86	Used ordering:  Polynomial interpretation [POLO]:
11.97/3.86	
11.97/3.86	   POL(A(x_1)) = 1 + x_1
11.97/3.86	   POL(B(x_1)) = 1 + x_1
11.97/3.86	   POL(a(x_1)) = 1 + x_1
11.97/3.86	   POL(b(x_1)) = 1 + x_1
11.97/3.86	
11.97/3.86	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.97/3.86	
11.97/3.86	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
11.97/3.86	   a(b(b(b(x1)))) -> a(a(a(a(x1))))
11.97/3.86	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
11.97/3.86	
11.97/3.86	
11.97/3.86	----------------------------------------
11.97/3.86	
11.97/3.86	(6)
11.97/3.86	Obligation:
11.97/3.86	Q DP problem:
11.97/3.86	The TRS P consists of the following rules:
11.97/3.86	
11.97/3.86	   B(b(a(b(x1)))) -> A(b(a(a(x1))))
11.97/3.86	
11.97/3.86	The TRS R consists of the following rules:
11.97/3.86	
11.97/3.86	   a(b(a(a(x1)))) -> a(a(b(b(x1))))
11.97/3.86	   a(b(b(b(x1)))) -> a(a(a(a(x1))))
11.97/3.86	   b(b(a(b(x1)))) -> a(b(a(a(x1))))
11.97/3.86	
11.97/3.86	Q is empty.
11.97/3.86	We have to consider all minimal (P,Q,R)-chains.
11.97/3.86	----------------------------------------
11.97/3.86	
11.97/3.86	(7) DependencyGraphProof (EQUIVALENT)
11.97/3.86	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
11.97/3.86	----------------------------------------
11.97/3.86	
11.97/3.86	(8)
11.97/3.86	TRUE
12.23/6.05	EOF