3.74/1.75	YES
3.74/1.76	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.74/1.76	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.74/1.76	
3.74/1.76	
3.74/1.76	Termination w.r.t. Q of the given QTRS could be proven:
3.74/1.76	
3.74/1.76	(0) QTRS
3.74/1.76	(1) QTRSRRRProof [EQUIVALENT, 119 ms]
3.74/1.76	(2) QTRS
3.74/1.76	(3) QTRSRRRProof [EQUIVALENT, 0 ms]
3.74/1.76	(4) QTRS
3.74/1.76	(5) RisEmptyProof [EQUIVALENT, 0 ms]
3.74/1.76	(6) YES
3.74/1.76	
3.74/1.76	
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(0)
3.74/1.76	Obligation:
3.74/1.76	Q restricted rewrite system:
3.74/1.76	The TRS R consists of the following rules:
3.74/1.76	
3.74/1.76	   a__eq(0, 0) -> true
3.74/1.76	   a__eq(s(X), s(Y)) -> a__eq(X, Y)
3.74/1.76	   a__eq(X, Y) -> false
3.74/1.76	   a__inf(X) -> cons(X, inf(s(X)))
3.74/1.76	   a__take(0, X) -> nil
3.74/1.76	   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.74/1.76	   a__length(nil) -> 0
3.74/1.76	   a__length(cons(X, L)) -> s(length(L))
3.74/1.76	   mark(eq(X1, X2)) -> a__eq(X1, X2)
3.74/1.76	   mark(inf(X)) -> a__inf(mark(X))
3.74/1.76	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.74/1.76	   mark(length(X)) -> a__length(mark(X))
3.74/1.76	   mark(0) -> 0
3.74/1.76	   mark(true) -> true
3.74/1.76	   mark(s(X)) -> s(X)
3.74/1.76	   mark(false) -> false
3.74/1.76	   mark(cons(X1, X2)) -> cons(X1, X2)
3.74/1.76	   mark(nil) -> nil
3.74/1.76	   a__eq(X1, X2) -> eq(X1, X2)
3.74/1.76	   a__inf(X) -> inf(X)
3.74/1.76	   a__take(X1, X2) -> take(X1, X2)
3.74/1.76	   a__length(X) -> length(X)
3.74/1.76	
3.74/1.76	The set Q consists of the following terms:
3.74/1.76	
3.74/1.76	   a__eq(x0, x1)
3.74/1.76	   a__inf(x0)
3.74/1.76	   mark(eq(x0, x1))
3.74/1.76	   mark(inf(x0))
3.74/1.76	   mark(take(x0, x1))
3.74/1.76	   mark(length(x0))
3.74/1.76	   mark(0)
3.74/1.76	   mark(true)
3.74/1.76	   mark(s(x0))
3.74/1.76	   mark(false)
3.74/1.76	   mark(cons(x0, x1))
3.74/1.76	   mark(nil)
3.74/1.76	   a__take(x0, x1)
3.74/1.76	   a__length(x0)
3.74/1.76	
3.74/1.76	
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(1) QTRSRRRProof (EQUIVALENT)
3.74/1.76	Used ordering:
3.74/1.76	Quasi precedence:
3.74/1.76	[a__eq_2, a__length_1, length_1, mark_1] > true > [s_1, cons_2, inf_1]
3.74/1.76	[a__eq_2, a__length_1, length_1, mark_1] > false > [s_1, cons_2, inf_1]
3.74/1.76	[a__eq_2, a__length_1, length_1, mark_1] > a__inf_1 > [s_1, cons_2, inf_1]
3.74/1.76	[a__eq_2, a__length_1, length_1, mark_1] > a__take_2 > nil > 0 > [s_1, cons_2, inf_1]
3.74/1.76	[a__eq_2, a__length_1, length_1, mark_1] > a__take_2 > take_2 > [s_1, cons_2, inf_1]
3.74/1.76	[a__eq_2, a__length_1, length_1, mark_1] > eq_2 > [s_1, cons_2, inf_1]
3.74/1.76	
3.74/1.76	
3.74/1.76	Status:
3.74/1.76	a__eq_2: multiset status
3.74/1.76	0: multiset status
3.74/1.76	true: multiset status
3.74/1.76	s_1: multiset status
3.74/1.76	false: multiset status
3.74/1.76	a__inf_1: multiset status
3.74/1.76	cons_2: multiset status
3.74/1.76	inf_1: multiset status
3.74/1.76	a__take_2: multiset status
3.74/1.76	nil: multiset status
3.74/1.76	take_2: multiset status
3.74/1.76	a__length_1: multiset status
3.74/1.76	length_1: multiset status
3.74/1.76	mark_1: multiset status
3.74/1.76	eq_2: multiset status
3.74/1.76	
3.74/1.76	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.74/1.76	
3.74/1.76	   a__eq(0, 0) -> true
3.74/1.76	   a__eq(s(X), s(Y)) -> a__eq(X, Y)
3.74/1.76	   a__eq(X, Y) -> false
3.74/1.76	   a__inf(X) -> cons(X, inf(s(X)))
3.74/1.76	   a__take(0, X) -> nil
3.74/1.76	   a__take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.74/1.76	   a__length(nil) -> 0
3.74/1.76	   a__length(cons(X, L)) -> s(length(L))
3.74/1.76	   mark(eq(X1, X2)) -> a__eq(X1, X2)
3.74/1.76	   mark(inf(X)) -> a__inf(mark(X))
3.74/1.76	   mark(take(X1, X2)) -> a__take(mark(X1), mark(X2))
3.74/1.76	   mark(0) -> 0
3.74/1.76	   mark(true) -> true
3.74/1.76	   mark(s(X)) -> s(X)
3.74/1.76	   mark(false) -> false
3.74/1.76	   mark(cons(X1, X2)) -> cons(X1, X2)
3.74/1.76	   mark(nil) -> nil
3.74/1.76	   a__eq(X1, X2) -> eq(X1, X2)
3.74/1.76	   a__inf(X) -> inf(X)
3.74/1.76	   a__take(X1, X2) -> take(X1, X2)
3.74/1.76	
3.74/1.76	
3.74/1.76	
3.74/1.76	
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(2)
3.74/1.76	Obligation:
3.74/1.76	Q restricted rewrite system:
3.74/1.76	The TRS R consists of the following rules:
3.74/1.76	
3.74/1.76	   mark(length(X)) -> a__length(mark(X))
3.74/1.76	   a__length(X) -> length(X)
3.74/1.76	
3.74/1.76	The set Q consists of the following terms:
3.74/1.76	
3.74/1.76	   a__eq(x0, x1)
3.74/1.76	   a__inf(x0)
3.74/1.76	   mark(eq(x0, x1))
3.74/1.76	   mark(inf(x0))
3.74/1.76	   mark(take(x0, x1))
3.74/1.76	   mark(length(x0))
3.74/1.76	   mark(0)
3.74/1.76	   mark(true)
3.74/1.76	   mark(s(x0))
3.74/1.76	   mark(false)
3.74/1.76	   mark(cons(x0, x1))
3.74/1.76	   mark(nil)
3.74/1.76	   a__take(x0, x1)
3.74/1.76	   a__length(x0)
3.74/1.76	
3.74/1.76	
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(3) QTRSRRRProof (EQUIVALENT)
3.74/1.76	Used ordering:
3.74/1.76	Knuth-Bendix order [KBO] with precedence:mark_1 > a__length_1 > length_1
3.74/1.76	
3.74/1.76	and weight map:
3.74/1.76	
3.74/1.76	   mark_1=2
3.74/1.76	   length_1=1
3.74/1.76	   a__length_1=1
3.74/1.76	
3.74/1.76	The variable weight is 1With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.74/1.76	
3.74/1.76	   mark(length(X)) -> a__length(mark(X))
3.74/1.76	   a__length(X) -> length(X)
3.74/1.76	
3.74/1.76	
3.74/1.76	
3.74/1.76	
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(4)
3.74/1.76	Obligation:
3.74/1.76	Q restricted rewrite system:
3.74/1.76	R is empty.
3.74/1.76	The set Q consists of the following terms:
3.74/1.76	
3.74/1.76	   a__eq(x0, x1)
3.74/1.76	   a__inf(x0)
3.74/1.76	   mark(eq(x0, x1))
3.74/1.76	   mark(inf(x0))
3.74/1.76	   mark(take(x0, x1))
3.74/1.76	   mark(length(x0))
3.74/1.76	   mark(0)
3.74/1.76	   mark(true)
3.74/1.76	   mark(s(x0))
3.74/1.76	   mark(false)
3.74/1.76	   mark(cons(x0, x1))
3.74/1.76	   mark(nil)
3.74/1.76	   a__take(x0, x1)
3.74/1.76	   a__length(x0)
3.74/1.76	
3.74/1.76	
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(5) RisEmptyProof (EQUIVALENT)
3.74/1.76	The TRS R is empty. Hence, termination is trivially proven.
3.74/1.76	----------------------------------------
3.74/1.76	
3.74/1.76	(6)
3.74/1.76	YES
3.74/1.79	EOF