30.82/9.03	YES
32.72/9.36	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
32.72/9.36	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
32.72/9.36	
32.72/9.36	
32.72/9.36	Termination w.r.t. Q of the given QTRS could be proven:
32.72/9.36	
32.72/9.36	(0) QTRS
32.72/9.36	(1) QTRS Reverse [EQUIVALENT, 0 ms]
32.72/9.36	(2) QTRS
32.72/9.36	(3) DependencyPairsProof [EQUIVALENT, 16 ms]
32.72/9.36	(4) QDP
32.72/9.36	(5) MRRProof [EQUIVALENT, 56 ms]
32.72/9.36	(6) QDP
32.72/9.36	(7) MRRProof [EQUIVALENT, 1 ms]
32.72/9.36	(8) QDP
32.72/9.36	(9) PisEmptyProof [EQUIVALENT, 0 ms]
32.72/9.36	(10) YES
32.72/9.36	
32.72/9.36	
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(0)
32.72/9.36	Obligation:
32.72/9.36	Q restricted rewrite system:
32.72/9.36	The TRS R consists of the following rules:
32.72/9.36	
32.72/9.36	   a(a(a(a(x1)))) -> a(b(a(b(x1))))
32.72/9.36	   a(a(b(a(x1)))) -> a(b(a(a(x1))))
32.72/9.36	   b(a(b(b(x1)))) -> b(b(a(a(x1))))
32.72/9.36	
32.72/9.36	Q is empty.
32.72/9.36	
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(1) QTRS Reverse (EQUIVALENT)
32.72/9.36	We applied the QTRS Reverse Processor [REVERSE].
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(2)
32.72/9.36	Obligation:
32.72/9.36	Q restricted rewrite system:
32.72/9.36	The TRS R consists of the following rules:
32.72/9.36	
32.72/9.36	   a(a(a(a(x1)))) -> b(a(b(a(x1))))
32.72/9.36	   a(b(a(a(x1)))) -> a(a(b(a(x1))))
32.72/9.36	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
32.72/9.36	
32.72/9.36	Q is empty.
32.72/9.36	
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(3) DependencyPairsProof (EQUIVALENT)
32.72/9.36	Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(4)
32.72/9.36	Obligation:
32.72/9.36	Q DP problem:
32.72/9.36	The TRS P consists of the following rules:
32.72/9.36	
32.72/9.36	   A(a(a(a(x1)))) -> B(a(b(a(x1))))
32.72/9.36	   A(a(a(a(x1)))) -> A(b(a(x1)))
32.72/9.36	   A(a(a(a(x1)))) -> B(a(x1))
32.72/9.36	   A(b(a(a(x1)))) -> A(a(b(a(x1))))
32.72/9.36	   A(b(a(a(x1)))) -> A(b(a(x1)))
32.72/9.36	   A(b(a(a(x1)))) -> B(a(x1))
32.72/9.36	   B(b(a(b(x1)))) -> A(a(b(b(x1))))
32.72/9.36	   B(b(a(b(x1)))) -> A(b(b(x1)))
32.72/9.36	   B(b(a(b(x1)))) -> B(b(x1))
32.72/9.36	
32.72/9.36	The TRS R consists of the following rules:
32.72/9.36	
32.72/9.36	   a(a(a(a(x1)))) -> b(a(b(a(x1))))
32.72/9.36	   a(b(a(a(x1)))) -> a(a(b(a(x1))))
32.72/9.36	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
32.72/9.36	
32.72/9.36	Q is empty.
32.72/9.36	We have to consider all minimal (P,Q,R)-chains.
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(5) MRRProof (EQUIVALENT)
32.72/9.36	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.
32.72/9.36	
32.72/9.36	Strictly oriented dependency pairs:
32.72/9.36	
32.72/9.36	   A(a(a(a(x1)))) -> A(b(a(x1)))
32.72/9.36	   A(a(a(a(x1)))) -> B(a(x1))
32.72/9.36	   A(b(a(a(x1)))) -> A(b(a(x1)))
32.72/9.36	   A(b(a(a(x1)))) -> B(a(x1))
32.72/9.36	   B(b(a(b(x1)))) -> A(b(b(x1)))
32.72/9.36	   B(b(a(b(x1)))) -> B(b(x1))
32.72/9.36	
32.72/9.36	
32.72/9.36	Used ordering: Polynomial interpretation [POLO]:
32.72/9.36	
32.72/9.36	   POL(A(x_1)) = 2 + 2*x_1
32.72/9.36	   POL(B(x_1)) = 2 + 2*x_1
32.72/9.36	   POL(a(x_1)) = 1 + x_1
32.72/9.36	   POL(b(x_1)) = 1 + x_1
32.72/9.36	
32.72/9.36	
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(6)
32.72/9.36	Obligation:
32.72/9.36	Q DP problem:
32.72/9.36	The TRS P consists of the following rules:
32.72/9.36	
32.72/9.36	   A(a(a(a(x1)))) -> B(a(b(a(x1))))
32.72/9.36	   A(b(a(a(x1)))) -> A(a(b(a(x1))))
32.72/9.36	   B(b(a(b(x1)))) -> A(a(b(b(x1))))
32.72/9.36	
32.72/9.36	The TRS R consists of the following rules:
32.72/9.36	
32.72/9.36	   a(a(a(a(x1)))) -> b(a(b(a(x1))))
32.72/9.36	   a(b(a(a(x1)))) -> a(a(b(a(x1))))
32.72/9.36	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
32.72/9.36	
32.72/9.36	Q is empty.
32.72/9.36	We have to consider all minimal (P,Q,R)-chains.
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(7) MRRProof (EQUIVALENT)
32.72/9.36	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.
32.72/9.36	
32.72/9.36	Strictly oriented dependency pairs:
32.72/9.36	
32.72/9.36	   A(a(a(a(x1)))) -> B(a(b(a(x1))))
32.72/9.36	   A(b(a(a(x1)))) -> A(a(b(a(x1))))
32.72/9.36	   B(b(a(b(x1)))) -> A(a(b(b(x1))))
32.72/9.36	
32.72/9.36	Strictly oriented rules of the TRS R:
32.72/9.36	
32.72/9.36	   a(a(a(a(x1)))) -> b(a(b(a(x1))))
32.72/9.36	   a(b(a(a(x1)))) -> a(a(b(a(x1))))
32.72/9.36	   b(b(a(b(x1)))) -> a(a(b(b(x1))))
32.72/9.36	
32.72/9.36	Used ordering: Polynomial interpretation [POLO]:
32.72/9.36	
32.72/9.36	   POL(A(x_1)) = 2*x_1
32.72/9.36	   POL(B(x_1)) = 2*x_1
32.72/9.36	   POL(a(x_1)) = 2 + 2*x_1
32.72/9.36	   POL(b(x_1)) = 2*x_1
32.72/9.36	
32.72/9.36	
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(8)
32.72/9.36	Obligation:
32.72/9.36	Q DP problem:
32.72/9.36	P is empty.
32.72/9.36	R is empty.
32.72/9.36	Q is empty.
32.72/9.36	We have to consider all minimal (P,Q,R)-chains.
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(9) PisEmptyProof (EQUIVALENT)
32.72/9.36	The TRS P is empty. Hence, there is no (P,Q,R) chain.
32.72/9.36	----------------------------------------
32.72/9.36	
32.72/9.36	(10)
32.72/9.36	YES
32.77/10.27	EOF