5.02/3.11	YES
5.25/3.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
5.25/3.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
5.25/3.13	
5.25/3.13	
5.25/3.13	Termination w.r.t. Q of the given QTRS could be proven:
5.25/3.13	
5.25/3.13	(0) QTRS
5.25/3.13	(1) QTRSRRRProof [EQUIVALENT, 94 ms]
5.25/3.13	(2) QTRS
5.25/3.13	(3) QTRSRRRProof [EQUIVALENT, 22 ms]
5.25/3.13	(4) QTRS
5.25/3.13	(5) QTRSRRRProof [EQUIVALENT, 15 ms]
5.25/3.13	(6) QTRS
5.25/3.13	(7) QTRSRRRProof [EQUIVALENT, 8 ms]
5.25/3.13	(8) QTRS
5.25/3.13	(9) QTRSRRRProof [EQUIVALENT, 13 ms]
5.25/3.13	(10) QTRS
5.25/3.13	(11) QTRSRRRProof [EQUIVALENT, 0 ms]
5.25/3.13	(12) QTRS
5.25/3.13	(13) RisEmptyProof [EQUIVALENT, 0 ms]
5.25/3.13	(14) YES
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(0)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	The TRS R consists of the following rules:
5.25/3.13	
5.25/3.13	   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
5.25/3.13	   active(__(X, nil)) -> mark(X)
5.25/3.13	   active(__(nil, X)) -> mark(X)
5.25/3.13	   active(U11(tt)) -> mark(U12(tt))
5.25/3.13	   active(U12(tt)) -> mark(tt)
5.25/3.13	   active(isNePal(__(I, __(P, I)))) -> mark(U11(tt))
5.25/3.13	   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
5.25/3.13	   mark(nil) -> active(nil)
5.25/3.13	   mark(U11(X)) -> active(U11(mark(X)))
5.25/3.13	   mark(tt) -> active(tt)
5.25/3.13	   mark(U12(X)) -> active(U12(mark(X)))
5.25/3.13	   mark(isNePal(X)) -> active(isNePal(mark(X)))
5.25/3.13	   __(mark(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, mark(X2)) -> __(X1, X2)
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(mark(X)) -> U11(X)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(mark(X)) -> U12(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(mark(X)) -> isNePal(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(1) QTRSRRRProof (EQUIVALENT)
5.25/3.13	Used ordering:
5.25/3.13	Polynomial interpretation [POLO]:
5.25/3.13	
5.25/3.13	   POL(U11(x_1)) = 2 + x_1
5.25/3.13	   POL(U12(x_1)) = 2*x_1
5.25/3.13	   POL(__(x_1, x_2)) = 2 + x_1 + x_2
5.25/3.13	   POL(active(x_1)) = x_1
5.25/3.13	   POL(isNePal(x_1)) = x_1
5.25/3.13	   POL(mark(x_1)) = x_1
5.25/3.13	   POL(nil) = 0
5.25/3.13	   POL(tt) = 1
5.25/3.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.25/3.13	
5.25/3.13	   active(__(X, nil)) -> mark(X)
5.25/3.13	   active(__(nil, X)) -> mark(X)
5.25/3.13	   active(U11(tt)) -> mark(U12(tt))
5.25/3.13	   active(U12(tt)) -> mark(tt)
5.25/3.13	   active(isNePal(__(I, __(P, I)))) -> mark(U11(tt))
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(2)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	The TRS R consists of the following rules:
5.25/3.13	
5.25/3.13	   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
5.25/3.13	   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
5.25/3.13	   mark(nil) -> active(nil)
5.25/3.13	   mark(U11(X)) -> active(U11(mark(X)))
5.25/3.13	   mark(tt) -> active(tt)
5.25/3.13	   mark(U12(X)) -> active(U12(mark(X)))
5.25/3.13	   mark(isNePal(X)) -> active(isNePal(mark(X)))
5.25/3.13	   __(mark(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, mark(X2)) -> __(X1, X2)
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(mark(X)) -> U11(X)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(mark(X)) -> U12(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(mark(X)) -> isNePal(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(3) QTRSRRRProof (EQUIVALENT)
5.25/3.13	Used ordering:
5.25/3.13	Polynomial interpretation [POLO]:
5.25/3.13	
5.25/3.13	   POL(U11(x_1)) = 2*x_1
5.25/3.13	   POL(U12(x_1)) = x_1
5.25/3.13	   POL(__(x_1, x_2)) = 2 + 2*x_1 + x_2
5.25/3.13	   POL(active(x_1)) = x_1
5.25/3.13	   POL(isNePal(x_1)) = x_1
5.25/3.13	   POL(mark(x_1)) = x_1
5.25/3.13	   POL(nil) = 0
5.25/3.13	   POL(tt) = 0
5.25/3.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.25/3.13	
5.25/3.13	   active(__(__(X, Y), Z)) -> mark(__(X, __(Y, Z)))
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(4)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	The TRS R consists of the following rules:
5.25/3.13	
5.25/3.13	   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
5.25/3.13	   mark(nil) -> active(nil)
5.25/3.13	   mark(U11(X)) -> active(U11(mark(X)))
5.25/3.13	   mark(tt) -> active(tt)
5.25/3.13	   mark(U12(X)) -> active(U12(mark(X)))
5.25/3.13	   mark(isNePal(X)) -> active(isNePal(mark(X)))
5.25/3.13	   __(mark(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, mark(X2)) -> __(X1, X2)
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(mark(X)) -> U11(X)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(mark(X)) -> U12(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(mark(X)) -> isNePal(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(5) QTRSRRRProof (EQUIVALENT)
5.25/3.13	Used ordering:
5.25/3.13	Polynomial interpretation [POLO]:
5.25/3.13	
5.25/3.13	   POL(U11(x_1)) = 1 + 2*x_1
5.25/3.13	   POL(U12(x_1)) = 1 + 2*x_1
5.25/3.13	   POL(__(x_1, x_2)) = 1 + 2*x_1 + x_2
5.25/3.13	   POL(active(x_1)) = x_1
5.25/3.13	   POL(isNePal(x_1)) = 2*x_1
5.25/3.13	   POL(mark(x_1)) = 2*x_1
5.25/3.13	   POL(nil) = 1
5.25/3.13	   POL(tt) = 1
5.25/3.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.25/3.13	
5.25/3.13	   mark(__(X1, X2)) -> active(__(mark(X1), mark(X2)))
5.25/3.13	   mark(nil) -> active(nil)
5.25/3.13	   mark(U11(X)) -> active(U11(mark(X)))
5.25/3.13	   mark(tt) -> active(tt)
5.25/3.13	   mark(U12(X)) -> active(U12(mark(X)))
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(6)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	The TRS R consists of the following rules:
5.25/3.13	
5.25/3.13	   mark(isNePal(X)) -> active(isNePal(mark(X)))
5.25/3.13	   __(mark(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, mark(X2)) -> __(X1, X2)
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(mark(X)) -> U11(X)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(mark(X)) -> U12(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(mark(X)) -> isNePal(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(7) QTRSRRRProof (EQUIVALENT)
5.25/3.13	Used ordering:
5.25/3.13	Polynomial interpretation [POLO]:
5.25/3.13	
5.25/3.13	   POL(U11(x_1)) = 2*x_1
5.25/3.13	   POL(U12(x_1)) = 2*x_1
5.25/3.13	   POL(__(x_1, x_2)) = x_1 + x_2
5.25/3.13	   POL(active(x_1)) = x_1
5.25/3.13	   POL(isNePal(x_1)) = 2 + 2*x_1
5.25/3.13	   POL(mark(x_1)) = 2 + 2*x_1
5.25/3.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.25/3.13	
5.25/3.13	   __(mark(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, mark(X2)) -> __(X1, X2)
5.25/3.13	   U11(mark(X)) -> U11(X)
5.25/3.13	   U12(mark(X)) -> U12(X)
5.25/3.13	   isNePal(mark(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(8)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	The TRS R consists of the following rules:
5.25/3.13	
5.25/3.13	   mark(isNePal(X)) -> active(isNePal(mark(X)))
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(9) QTRSRRRProof (EQUIVALENT)
5.25/3.13	Used ordering:
5.25/3.13	mark/1(YES)
5.25/3.13	isNePal/1(YES)
5.25/3.13	active/1)YES(
5.25/3.13	__/2(YES,YES)
5.25/3.13	U11/1)YES(
5.25/3.13	U12/1(YES)
5.25/3.13	
5.25/3.13	Quasi precedence:
5.25/3.13	mark_1 > isNePal_1
5.25/3.13	
5.25/3.13	
5.25/3.13	Status:
5.25/3.13	mark_1: multiset status
5.25/3.13	isNePal_1: multiset status
5.25/3.13	___2: [1,2]
5.25/3.13	U12_1: multiset status
5.25/3.13	
5.25/3.13	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.25/3.13	
5.25/3.13	   mark(isNePal(X)) -> active(isNePal(mark(X)))
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(10)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	The TRS R consists of the following rules:
5.25/3.13	
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(11) QTRSRRRProof (EQUIVALENT)
5.25/3.13	Used ordering:
5.25/3.13	Knuth-Bendix order [KBO] with precedence:active_1 > isNePal_1 > U12_1 > U11_1 > ___2
5.25/3.13	
5.25/3.13	and weight map:
5.25/3.13	
5.25/3.13	   active_1=0
5.25/3.13	   U11_1=1
5.25/3.13	   U12_1=1
5.25/3.13	   isNePal_1=1
5.25/3.13	   ___2=0
5.25/3.13	
5.25/3.13	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
5.25/3.13	
5.25/3.13	   __(active(X1), X2) -> __(X1, X2)
5.25/3.13	   __(X1, active(X2)) -> __(X1, X2)
5.25/3.13	   U11(active(X)) -> U11(X)
5.25/3.13	   U12(active(X)) -> U12(X)
5.25/3.13	   isNePal(active(X)) -> isNePal(X)
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(12)
5.25/3.13	Obligation:
5.25/3.13	Q restricted rewrite system:
5.25/3.13	R is empty.
5.25/3.13	The set Q consists of the following terms:
5.25/3.13	
5.25/3.13	   active(__(__(x0, x1), x2))
5.25/3.13	   active(__(x0, nil))
5.25/3.13	   active(__(nil, x0))
5.25/3.13	   active(U11(tt))
5.25/3.13	   active(U12(tt))
5.25/3.13	   active(isNePal(__(x0, __(x1, x0))))
5.25/3.13	   mark(__(x0, x1))
5.25/3.13	   mark(nil)
5.25/3.13	   mark(U11(x0))
5.25/3.13	   mark(tt)
5.25/3.13	   mark(U12(x0))
5.25/3.13	   mark(isNePal(x0))
5.25/3.13	   __(mark(x0), x1)
5.25/3.13	   __(x0, mark(x1))
5.25/3.13	   __(active(x0), x1)
5.25/3.13	   __(x0, active(x1))
5.25/3.13	   U11(mark(x0))
5.25/3.13	   U11(active(x0))
5.25/3.13	   U12(mark(x0))
5.25/3.13	   U12(active(x0))
5.25/3.13	   isNePal(mark(x0))
5.25/3.13	   isNePal(active(x0))
5.25/3.13	
5.25/3.13	
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(13) RisEmptyProof (EQUIVALENT)
5.25/3.13	The TRS R is empty. Hence, termination is trivially proven.
5.25/3.13	----------------------------------------
5.25/3.13	
5.25/3.13	(14)
5.25/3.13	YES
5.25/3.17	EOF