10.72/3.63	YES
11.09/3.71	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
11.09/3.71	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.09/3.71	
11.09/3.71	
11.09/3.71	Termination w.r.t. Q of the given QTRS could be proven:
11.09/3.71	
11.09/3.71	(0) QTRS
11.09/3.71	(1) QTRS Reverse [EQUIVALENT, 0 ms]
11.09/3.71	(2) QTRS
11.09/3.71	(3) DependencyPairsProof [EQUIVALENT, 23 ms]
11.09/3.71	(4) QDP
11.09/3.71	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
11.09/3.71	(6) QDP
11.09/3.71	(7) QDPOrderProof [EQUIVALENT, 6 ms]
11.09/3.71	(8) QDP
11.09/3.71	(9) PisEmptyProof [EQUIVALENT, 0 ms]
11.09/3.71	(10) YES
11.09/3.71	
11.09/3.71	
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(0)
11.09/3.71	Obligation:
11.09/3.71	Q restricted rewrite system:
11.09/3.71	The TRS R consists of the following rules:
11.09/3.71	
11.09/3.71	   b(b(a(a(x1)))) -> a(b(a(a(x1))))
11.09/3.71	   b(a(a(a(x1)))) -> a(a(b(a(x1))))
11.09/3.71	   a(a(a(a(x1)))) -> b(a(b(a(x1))))
11.09/3.71	
11.09/3.71	Q is empty.
11.09/3.71	
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(1) QTRS Reverse (EQUIVALENT)
11.09/3.71	We applied the QTRS Reverse Processor [REVERSE].
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(2)
11.09/3.71	Obligation:
11.09/3.71	Q restricted rewrite system:
11.09/3.71	The TRS R consists of the following rules:
11.09/3.71	
11.09/3.71	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
11.09/3.71	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
11.09/3.71	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
11.09/3.71	
11.09/3.71	Q is empty.
11.09/3.71	
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(3) DependencyPairsProof (EQUIVALENT)
11.09/3.71	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(4)
11.09/3.71	Obligation:
11.09/3.71	Q DP problem:
11.09/3.71	The TRS P consists of the following rules:
11.09/3.71	
11.09/3.71	   A(a(b(b(x1)))) -> A(a(b(a(x1))))
11.09/3.71	   A(a(b(b(x1)))) -> A(b(a(x1)))
11.09/3.71	   A(a(b(b(x1)))) -> A(x1)
11.09/3.71	   A(a(a(b(x1)))) -> A(b(a(a(x1))))
11.09/3.71	   A(a(a(b(x1)))) -> A(a(x1))
11.09/3.71	   A(a(a(b(x1)))) -> A(x1)
11.09/3.71	   A(a(a(a(x1)))) -> A(b(a(b(x1))))
11.09/3.71	   A(a(a(a(x1)))) -> A(b(x1))
11.09/3.71	
11.09/3.71	The TRS R consists of the following rules:
11.09/3.71	
11.09/3.71	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
11.09/3.71	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
11.09/3.71	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
11.09/3.71	
11.09/3.71	Q is empty.
11.09/3.71	We have to consider all minimal (P,Q,R)-chains.
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(5) DependencyGraphProof (EQUIVALENT)
11.09/3.71	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(6)
11.09/3.71	Obligation:
11.09/3.71	Q DP problem:
11.09/3.71	The TRS P consists of the following rules:
11.09/3.71	
11.09/3.71	   A(a(a(b(x1)))) -> A(a(x1))
11.09/3.71	   A(a(b(b(x1)))) -> A(x1)
11.09/3.71	   A(a(a(b(x1)))) -> A(x1)
11.09/3.71	
11.09/3.71	The TRS R consists of the following rules:
11.09/3.71	
11.09/3.71	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
11.09/3.71	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
11.09/3.71	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
11.09/3.71	
11.09/3.71	Q is empty.
11.09/3.71	We have to consider all minimal (P,Q,R)-chains.
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(7) QDPOrderProof (EQUIVALENT)
11.09/3.71	We use the reduction pair processor [LPAR04,JAR06].
11.09/3.71	
11.09/3.71	
11.09/3.71	The following pairs can be oriented strictly and are deleted.
11.09/3.71	
11.09/3.71	   A(a(a(b(x1)))) -> A(a(x1))
11.09/3.71	   A(a(b(b(x1)))) -> A(x1)
11.09/3.71	   A(a(a(b(x1)))) -> A(x1)
11.09/3.71	The remaining pairs can at least be oriented weakly.
11.09/3.71	Used ordering:  Polynomial interpretation [POLO]:
11.09/3.71	
11.09/3.71	   POL(A(x_1)) = x_1
11.09/3.71	   POL(a(x_1)) = 1 + x_1
11.09/3.71	   POL(b(x_1)) = 1 + x_1
11.09/3.71	
11.09/3.71	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.09/3.71	
11.09/3.71	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
11.09/3.71	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
11.09/3.71	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
11.09/3.71	
11.09/3.71	
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(8)
11.09/3.71	Obligation:
11.09/3.71	Q DP problem:
11.09/3.71	P is empty.
11.09/3.71	The TRS R consists of the following rules:
11.09/3.71	
11.09/3.71	   a(a(b(b(x1)))) -> a(a(b(a(x1))))
11.09/3.71	   a(a(a(b(x1)))) -> a(b(a(a(x1))))
11.09/3.71	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
11.09/3.71	
11.09/3.71	Q is empty.
11.09/3.71	We have to consider all minimal (P,Q,R)-chains.
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(9) PisEmptyProof (EQUIVALENT)
11.09/3.71	The TRS P is empty. Hence, there is no (P,Q,R) chain.
11.09/3.71	----------------------------------------
11.09/3.71	
11.09/3.71	(10)
11.09/3.71	YES
11.09/3.79	EOF