11.14/3.71	YES
11.51/3.78	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
11.51/3.78	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.51/3.78	
11.51/3.78	
11.51/3.78	Termination w.r.t. Q of the given QTRS could be proven:
11.51/3.78	
11.51/3.78	(0) QTRS
11.51/3.78	(1) QTRS Reverse [EQUIVALENT, 0 ms]
11.51/3.78	(2) QTRS
11.51/3.78	(3) QTRSRRRProof [EQUIVALENT, 13 ms]
11.51/3.78	(4) QTRS
11.51/3.78	(5) DependencyPairsProof [EQUIVALENT, 2 ms]
11.51/3.78	(6) QDP
11.51/3.78	(7) DependencyGraphProof [EQUIVALENT, 1 ms]
11.51/3.78	(8) QDP
11.51/3.78	(9) QDPOrderProof [EQUIVALENT, 21 ms]
11.51/3.78	(10) QDP
11.51/3.78	(11) DependencyGraphProof [EQUIVALENT, 0 ms]
11.51/3.78	(12) TRUE
11.51/3.78	
11.51/3.78	
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(0)
11.51/3.78	Obligation:
11.51/3.78	Q restricted rewrite system:
11.51/3.78	The TRS R consists of the following rules:
11.51/3.78	
11.51/3.78	   a(b(b(a(x1)))) -> b(a(b(a(x1))))
11.51/3.78	   b(b(b(a(x1)))) -> a(a(b(b(x1))))
11.51/3.78	   b(b(a(b(x1)))) -> b(b(b(a(x1))))
11.51/3.78	
11.51/3.78	Q is empty.
11.51/3.78	
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(1) QTRS Reverse (EQUIVALENT)
11.51/3.78	We applied the QTRS Reverse Processor [REVERSE].
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(2)
11.51/3.78	Obligation:
11.51/3.78	Q restricted rewrite system:
11.51/3.78	The TRS R consists of the following rules:
11.51/3.78	
11.51/3.78	   a(b(b(a(x1)))) -> a(b(a(b(x1))))
11.51/3.78	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
11.51/3.78	   b(a(b(b(x1)))) -> a(b(b(b(x1))))
11.51/3.78	
11.51/3.78	Q is empty.
11.51/3.78	
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(3) QTRSRRRProof (EQUIVALENT)
11.51/3.78	Used ordering:
11.51/3.78	Polynomial interpretation [POLO]:
11.51/3.78	
11.51/3.78	   POL(a(x_1)) = x_1
11.51/3.78	   POL(b(x_1)) = 1 + x_1
11.51/3.78	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
11.51/3.78	
11.51/3.78	   a(b(b(b(x1)))) -> b(b(a(a(x1))))
11.51/3.78	
11.51/3.78	
11.51/3.78	
11.51/3.78	
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(4)
11.51/3.78	Obligation:
11.51/3.78	Q restricted rewrite system:
11.51/3.78	The TRS R consists of the following rules:
11.51/3.78	
11.51/3.78	   a(b(b(a(x1)))) -> a(b(a(b(x1))))
11.51/3.78	   b(a(b(b(x1)))) -> a(b(b(b(x1))))
11.51/3.78	
11.51/3.78	Q is empty.
11.51/3.78	
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(5) DependencyPairsProof (EQUIVALENT)
11.51/3.78	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(6)
11.51/3.78	Obligation:
11.51/3.78	Q DP problem:
11.51/3.78	The TRS P consists of the following rules:
11.51/3.78	
11.51/3.78	   A(b(b(a(x1)))) -> A(b(a(b(x1))))
11.51/3.78	   A(b(b(a(x1)))) -> B(a(b(x1)))
11.51/3.78	   A(b(b(a(x1)))) -> A(b(x1))
11.51/3.78	   A(b(b(a(x1)))) -> B(x1)
11.51/3.78	   B(a(b(b(x1)))) -> A(b(b(b(x1))))
11.51/3.78	   B(a(b(b(x1)))) -> B(b(b(x1)))
11.51/3.78	
11.51/3.78	The TRS R consists of the following rules:
11.51/3.78	
11.51/3.78	   a(b(b(a(x1)))) -> a(b(a(b(x1))))
11.51/3.78	   b(a(b(b(x1)))) -> a(b(b(b(x1))))
11.51/3.78	
11.51/3.78	Q is empty.
11.51/3.78	We have to consider all minimal (P,Q,R)-chains.
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(7) DependencyGraphProof (EQUIVALENT)
11.51/3.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(8)
11.51/3.78	Obligation:
11.51/3.78	Q DP problem:
11.51/3.78	The TRS P consists of the following rules:
11.51/3.78	
11.51/3.78	   A(b(b(a(x1)))) -> B(a(b(x1)))
11.51/3.78	   B(a(b(b(x1)))) -> A(b(b(b(x1))))
11.51/3.78	   A(b(b(a(x1)))) -> A(b(x1))
11.51/3.78	   A(b(b(a(x1)))) -> B(x1)
11.51/3.78	   B(a(b(b(x1)))) -> B(b(b(x1)))
11.51/3.78	
11.51/3.78	The TRS R consists of the following rules:
11.51/3.78	
11.51/3.78	   a(b(b(a(x1)))) -> a(b(a(b(x1))))
11.51/3.78	   b(a(b(b(x1)))) -> a(b(b(b(x1))))
11.51/3.78	
11.51/3.78	Q is empty.
11.51/3.78	We have to consider all minimal (P,Q,R)-chains.
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(9) QDPOrderProof (EQUIVALENT)
11.51/3.78	We use the reduction pair processor [LPAR04,JAR06].
11.51/3.78	
11.51/3.78	
11.51/3.78	The following pairs can be oriented strictly and are deleted.
11.51/3.78	
11.51/3.78	   A(b(b(a(x1)))) -> B(a(b(x1)))
11.51/3.78	   A(b(b(a(x1)))) -> A(b(x1))
11.51/3.78	   A(b(b(a(x1)))) -> B(x1)
11.51/3.78	   B(a(b(b(x1)))) -> B(b(b(x1)))
11.51/3.78	The remaining pairs can at least be oriented weakly.
11.51/3.78	Used ordering:  Polynomial interpretation [POLO]:
11.51/3.78	
11.51/3.78	   POL(A(x_1)) = 1 + x_1
11.51/3.78	   POL(B(x_1)) = x_1
11.51/3.78	   POL(a(x_1)) = 1 + x_1
11.51/3.78	   POL(b(x_1)) = x_1
11.51/3.78	
11.51/3.78	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
11.51/3.78	
11.51/3.78	   b(a(b(b(x1)))) -> a(b(b(b(x1))))
11.51/3.78	   a(b(b(a(x1)))) -> a(b(a(b(x1))))
11.51/3.78	
11.51/3.78	
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(10)
11.51/3.78	Obligation:
11.51/3.78	Q DP problem:
11.51/3.78	The TRS P consists of the following rules:
11.51/3.78	
11.51/3.78	   B(a(b(b(x1)))) -> A(b(b(b(x1))))
11.51/3.78	
11.51/3.78	The TRS R consists of the following rules:
11.51/3.78	
11.51/3.78	   a(b(b(a(x1)))) -> a(b(a(b(x1))))
11.51/3.78	   b(a(b(b(x1)))) -> a(b(b(b(x1))))
11.51/3.78	
11.51/3.78	Q is empty.
11.51/3.78	We have to consider all minimal (P,Q,R)-chains.
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(11) DependencyGraphProof (EQUIVALENT)
11.51/3.78	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.
11.51/3.78	----------------------------------------
11.51/3.78	
11.51/3.78	(12)
11.51/3.78	TRUE
11.87/3.91	EOF