14.85/4.89	YES
15.15/4.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
15.15/4.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
15.15/4.91	
15.15/4.91	
15.15/4.91	Termination w.r.t. Q of the given QTRS could be proven:
15.15/4.91	
15.15/4.91	(0) QTRS
15.15/4.91	(1) QTRSRRRProof [EQUIVALENT, 72 ms]
15.15/4.91	(2) QTRS
15.15/4.91	(3) DependencyPairsProof [EQUIVALENT, 3 ms]
15.15/4.91	(4) QDP
15.15/4.91	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
15.15/4.91	(6) AND
15.15/4.91	    (7) QDP
15.15/4.91	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
15.15/4.91	        (9) QDP
15.15/4.91	        (10) DependencyGraphProof [EQUIVALENT, 0 ms]
15.15/4.91	        (11) QDP
15.15/4.91	        (12) UsableRulesProof [EQUIVALENT, 0 ms]
15.15/4.91	        (13) QDP
15.15/4.91	        (14) MRRProof [EQUIVALENT, 3 ms]
15.15/4.91	        (15) QDP
15.15/4.91	        (16) PisEmptyProof [EQUIVALENT, 0 ms]
15.15/4.91	        (17) YES
15.15/4.91	    (18) QDP
15.15/4.91	        (19) UsableRulesProof [EQUIVALENT, 0 ms]
15.15/4.91	        (20) QDP
15.15/4.91	        (21) QDPSizeChangeProof [EQUIVALENT, 2 ms]
15.15/4.91	        (22) YES
15.15/4.91	
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(0)
15.15/4.91	Obligation:
15.15/4.91	Q restricted rewrite system:
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
15.15/4.91	   b(b(a(a(x1)))) -> a(b(a(b(x1))))
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(1) QTRSRRRProof (EQUIVALENT)
15.15/4.91	Used ordering:
15.15/4.91	Polynomial interpretation [POLO]:
15.15/4.91	
15.15/4.91	   POL(a(x_1)) = x_1
15.15/4.91	   POL(b(x_1)) = 1 + x_1
15.15/4.91	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
15.15/4.91	
15.15/4.91	   b(a(b(b(x1)))) -> a(b(a(b(x1))))
15.15/4.91	
15.15/4.91	
15.15/4.91	
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(2)
15.15/4.91	Obligation:
15.15/4.91	Q restricted rewrite system:
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(b(a(a(x1)))) -> a(b(a(b(x1))))
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(3) DependencyPairsProof (EQUIVALENT)
15.15/4.91	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(4)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(b(a(a(x1)))) -> B(a(b(x1)))
15.15/4.91	   B(b(a(a(x1)))) -> B(x1)
15.15/4.91	   B(a(a(b(x1)))) -> B(a(b(a(x1))))
15.15/4.91	   B(a(a(b(x1)))) -> B(a(x1))
15.15/4.91	
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(b(a(a(x1)))) -> a(b(a(b(x1))))
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(5) DependencyGraphProof (EQUIVALENT)
15.15/4.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(6)
15.15/4.91	Complex Obligation (AND)
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(7)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(a(a(b(x1)))) -> B(a(x1))
15.15/4.91	   B(a(a(b(x1)))) -> B(a(b(a(x1))))
15.15/4.91	
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(b(a(a(x1)))) -> a(b(a(b(x1))))
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(8) UsableRulesProof (EQUIVALENT)
15.15/4.91	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(9)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(a(a(b(x1)))) -> B(a(x1))
15.15/4.91	   B(a(a(b(x1)))) -> B(a(b(a(x1))))
15.15/4.91	
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(10) DependencyGraphProof (EQUIVALENT)
15.15/4.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(11)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(a(a(b(x1)))) -> B(a(x1))
15.15/4.91	
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(12) UsableRulesProof (EQUIVALENT)
15.15/4.91	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(13)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(a(a(b(x1)))) -> B(a(x1))
15.15/4.91	
15.15/4.91	R is empty.
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(14) MRRProof (EQUIVALENT)
15.15/4.91	By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
15.15/4.91	
15.15/4.91	Strictly oriented dependency pairs:
15.15/4.91	
15.15/4.91	   B(a(a(b(x1)))) -> B(a(x1))
15.15/4.91	
15.15/4.91	
15.15/4.91	Used ordering: Polynomial interpretation [POLO]:
15.15/4.91	
15.15/4.91	   POL(B(x_1)) = x_1
15.15/4.91	   POL(a(x_1)) = 2 + x_1
15.15/4.91	   POL(b(x_1)) = 2*x_1
15.15/4.91	
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(15)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	P is empty.
15.15/4.91	R is empty.
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(16) PisEmptyProof (EQUIVALENT)
15.15/4.91	The TRS P is empty. Hence, there is no (P,Q,R) chain.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(17)
15.15/4.91	YES
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(18)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(b(a(a(x1)))) -> B(x1)
15.15/4.91	
15.15/4.91	The TRS R consists of the following rules:
15.15/4.91	
15.15/4.91	   b(b(a(a(x1)))) -> a(b(a(b(x1))))
15.15/4.91	   b(a(a(b(x1)))) -> b(a(b(a(x1))))
15.15/4.91	
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(19) UsableRulesProof (EQUIVALENT)
15.15/4.91	We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(20)
15.15/4.91	Obligation:
15.15/4.91	Q DP problem:
15.15/4.91	The TRS P consists of the following rules:
15.15/4.91	
15.15/4.91	   B(b(a(a(x1)))) -> B(x1)
15.15/4.91	
15.15/4.91	R is empty.
15.15/4.91	Q is empty.
15.15/4.91	We have to consider all minimal (P,Q,R)-chains.
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(21) QDPSizeChangeProof (EQUIVALENT)
15.15/4.91	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. 
15.15/4.91	
15.15/4.91	From the DPs we obtained the following set of size-change graphs:
15.15/4.91	*B(b(a(a(x1)))) -> B(x1)
15.15/4.91	The graph contains the following edges 1 > 1
15.15/4.91	
15.15/4.91	
15.15/4.91	----------------------------------------
15.15/4.91	
15.15/4.91	(22)
15.15/4.91	YES
15.15/4.95	EOF