18.03/5.65	YES
18.20/5.75	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
18.20/5.75	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
18.20/5.75	
18.20/5.75	
18.20/5.75	Termination w.r.t. Q of the given QTRS could be proven:
18.20/5.75	
18.20/5.75	(0) QTRS
18.20/5.75	(1) DependencyPairsProof [EQUIVALENT, 3 ms]
18.20/5.75	(2) QDP
18.20/5.75	(3) DependencyGraphProof [EQUIVALENT, 2 ms]
18.20/5.75	(4) AND
18.20/5.75	    (5) QDP
18.20/5.75	        (6) QDPSizeChangeProof [EQUIVALENT, 0 ms]
18.20/5.75	        (7) YES
18.20/5.75	    (8) QDP
18.20/5.75	        (9) QDPOrderProof [EQUIVALENT, 54 ms]
18.20/5.75	        (10) QDP
18.20/5.75	        (11) PisEmptyProof [EQUIVALENT, 0 ms]
18.20/5.75	        (12) YES
18.20/5.75	
18.20/5.75	
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(0)
18.20/5.75	Obligation:
18.20/5.75	Q restricted rewrite system:
18.20/5.75	The TRS R consists of the following rules:
18.20/5.75	
18.20/5.75	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(b(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
18.20/5.75	
18.20/5.75	Q is empty.
18.20/5.75	
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(1) DependencyPairsProof (EQUIVALENT)
18.20/5.75	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(2)
18.20/5.75	Obligation:
18.20/5.75	Q DP problem:
18.20/5.75	The TRS P consists of the following rules:
18.20/5.75	
18.20/5.75	   B(a(a(a(x1)))) -> B(b(a(a(x1))))
18.20/5.75	   B(a(a(a(x1)))) -> B(a(a(x1)))
18.20/5.75	   A(b(b(b(x1)))) -> B(b(b(a(x1))))
18.20/5.75	   A(b(b(b(x1)))) -> B(b(a(x1)))
18.20/5.75	   A(b(b(b(x1)))) -> B(a(x1))
18.20/5.75	   A(b(b(b(x1)))) -> A(x1)
18.20/5.75	   A(b(b(b(x1)))) -> B(b(a(a(x1))))
18.20/5.75	   A(b(b(b(x1)))) -> B(a(a(x1)))
18.20/5.75	   A(b(b(b(x1)))) -> A(a(x1))
18.20/5.75	
18.20/5.75	The TRS R consists of the following rules:
18.20/5.75	
18.20/5.75	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(b(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
18.20/5.75	
18.20/5.75	Q is empty.
18.20/5.75	We have to consider all minimal (P,Q,R)-chains.
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(3) DependencyGraphProof (EQUIVALENT)
18.20/5.75	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 6 less nodes.
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(4)
18.20/5.75	Complex Obligation (AND)
18.20/5.75	
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(5)
18.20/5.75	Obligation:
18.20/5.75	Q DP problem:
18.20/5.75	The TRS P consists of the following rules:
18.20/5.75	
18.20/5.75	   B(a(a(a(x1)))) -> B(a(a(x1)))
18.20/5.75	
18.20/5.75	The TRS R consists of the following rules:
18.20/5.75	
18.20/5.75	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(b(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
18.20/5.75	
18.20/5.75	Q is empty.
18.20/5.75	We have to consider all minimal (P,Q,R)-chains.
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(6) QDPSizeChangeProof (EQUIVALENT)
18.20/5.75	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
18.20/5.75	
18.20/5.75	From the DPs we obtained the following set of size-change graphs:
18.20/5.75	*B(a(a(a(x1)))) -> B(a(a(x1)))
18.20/5.75	The graph contains the following edges 1 > 1
18.20/5.75	
18.20/5.75	
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(7)
18.20/5.75	YES
18.20/5.75	
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(8)
18.20/5.75	Obligation:
18.20/5.75	Q DP problem:
18.20/5.75	The TRS P consists of the following rules:
18.20/5.75	
18.20/5.75	   A(b(b(b(x1)))) -> A(a(x1))
18.20/5.75	   A(b(b(b(x1)))) -> A(x1)
18.20/5.75	
18.20/5.75	The TRS R consists of the following rules:
18.20/5.75	
18.20/5.75	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(b(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
18.20/5.75	
18.20/5.75	Q is empty.
18.20/5.75	We have to consider all minimal (P,Q,R)-chains.
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(9) QDPOrderProof (EQUIVALENT)
18.20/5.75	We use the reduction pair processor [LPAR04,JAR06].
18.20/5.75	
18.20/5.75	
18.20/5.75	The following pairs can be oriented strictly and are deleted.
18.20/5.75	
18.20/5.75	   A(b(b(b(x1)))) -> A(a(x1))
18.20/5.75	   A(b(b(b(x1)))) -> A(x1)
18.20/5.75	The remaining pairs can at least be oriented weakly.
18.20/5.75	Used ordering:  Polynomial Order [NEGPOLO,POLO] with Interpretation:
18.20/5.75	
18.20/5.75	POL( A_1(x_1) ) = max{0, 2x_1 - 2}
18.20/5.75	POL( a_1(x_1) ) = 2x_1 + 2
18.20/5.75	POL( b_1(x_1) ) = 2x_1 + 2
18.20/5.75	
18.20/5.75	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
18.20/5.75	
18.20/5.75	   a(b(b(b(x1)))) -> b(b(b(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
18.20/5.75	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
18.20/5.75	
18.20/5.75	
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(10)
18.20/5.75	Obligation:
18.20/5.75	Q DP problem:
18.20/5.75	P is empty.
18.20/5.75	The TRS R consists of the following rules:
18.20/5.75	
18.20/5.75	   b(a(a(a(x1)))) -> b(b(a(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(b(a(x1))))
18.20/5.75	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
18.20/5.75	
18.20/5.75	Q is empty.
18.20/5.75	We have to consider all minimal (P,Q,R)-chains.
18.20/5.75	----------------------------------------
18.20/5.75	
18.20/5.75	(11) PisEmptyProof (EQUIVALENT)
18.20/5.76	The TRS P is empty. Hence, there is no (P,Q,R) chain.
18.20/5.76	----------------------------------------
18.20/5.76	
18.20/5.76	(12)
18.20/5.76	YES
18.55/5.81	EOF