3.73/1.88	YES
3.73/1.89	proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
3.73/1.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.73/1.89	
3.73/1.89	
3.73/1.89	Termination w.r.t. Q of the given QTRS could be proven:
3.73/1.89	
3.73/1.89	(0) QTRS
3.73/1.89	(1) QTRSToCSRProof [SOUND, 0 ms]
3.73/1.89	(2) CSR
3.73/1.89	(3) CSRRRRProof [EQUIVALENT, 54 ms]
3.73/1.89	(4) CSR
3.73/1.89	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.73/1.89	(6) CSR
3.73/1.89	(7) CSRRRRProof [EQUIVALENT, 4 ms]
3.73/1.89	(8) CSR
3.73/1.89	(9) RisEmptyProof [EQUIVALENT, 0 ms]
3.73/1.89	(10) YES
3.73/1.89	
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(0)
3.73/1.89	Obligation:
3.73/1.89	Q restricted rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   active(eq(0, 0)) -> mark(true)
3.73/1.89	   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
3.73/1.89	   active(eq(X, Y)) -> mark(false)
3.73/1.89	   active(inf(X)) -> mark(cons(X, inf(s(X))))
3.73/1.89	   active(take(0, X)) -> mark(nil)
3.73/1.89	   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
3.73/1.89	   active(length(nil)) -> mark(0)
3.73/1.89	   active(length(cons(X, L))) -> mark(s(length(L)))
3.73/1.89	   active(inf(X)) -> inf(active(X))
3.73/1.89	   active(take(X1, X2)) -> take(active(X1), X2)
3.73/1.89	   active(take(X1, X2)) -> take(X1, active(X2))
3.73/1.89	   active(length(X)) -> length(active(X))
3.73/1.89	   inf(mark(X)) -> mark(inf(X))
3.73/1.89	   take(mark(X1), X2) -> mark(take(X1, X2))
3.73/1.89	   take(X1, mark(X2)) -> mark(take(X1, X2))
3.73/1.89	   length(mark(X)) -> mark(length(X))
3.73/1.89	   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
3.73/1.89	   proper(0) -> ok(0)
3.73/1.89	   proper(true) -> ok(true)
3.73/1.89	   proper(s(X)) -> s(proper(X))
3.73/1.89	   proper(false) -> ok(false)
3.73/1.89	   proper(inf(X)) -> inf(proper(X))
3.73/1.89	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.73/1.89	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
3.73/1.89	   proper(nil) -> ok(nil)
3.73/1.89	   proper(length(X)) -> length(proper(X))
3.73/1.89	   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
3.73/1.89	   s(ok(X)) -> ok(s(X))
3.73/1.89	   inf(ok(X)) -> ok(inf(X))
3.73/1.89	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.73/1.89	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
3.73/1.89	   length(ok(X)) -> ok(length(X))
3.73/1.89	   top(mark(X)) -> top(proper(X))
3.73/1.89	   top(ok(X)) -> top(active(X))
3.73/1.89	
3.73/1.89	The set Q consists of the following terms:
3.73/1.89	
3.73/1.89	   active(eq(x0, x1))
3.73/1.89	   active(inf(x0))
3.73/1.89	   active(take(x0, x1))
3.73/1.89	   active(length(x0))
3.73/1.89	   inf(mark(x0))
3.73/1.89	   take(mark(x0), x1)
3.73/1.89	   take(x0, mark(x1))
3.73/1.89	   length(mark(x0))
3.73/1.89	   proper(eq(x0, x1))
3.73/1.89	   proper(0)
3.73/1.89	   proper(true)
3.73/1.89	   proper(s(x0))
3.73/1.89	   proper(false)
3.73/1.89	   proper(inf(x0))
3.73/1.89	   proper(cons(x0, x1))
3.73/1.89	   proper(take(x0, x1))
3.73/1.89	   proper(nil)
3.73/1.89	   proper(length(x0))
3.73/1.89	   eq(ok(x0), ok(x1))
3.73/1.89	   s(ok(x0))
3.73/1.89	   inf(ok(x0))
3.73/1.89	   cons(ok(x0), ok(x1))
3.73/1.89	   take(ok(x0), ok(x1))
3.73/1.89	   length(ok(x0))
3.73/1.89	   top(mark(x0))
3.73/1.89	   top(ok(x0))
3.73/1.89	
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(1) QTRSToCSRProof (SOUND)
3.73/1.89	The following Q TRS is given: Q restricted rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   active(eq(0, 0)) -> mark(true)
3.73/1.89	   active(eq(s(X), s(Y))) -> mark(eq(X, Y))
3.73/1.89	   active(eq(X, Y)) -> mark(false)
3.73/1.89	   active(inf(X)) -> mark(cons(X, inf(s(X))))
3.73/1.89	   active(take(0, X)) -> mark(nil)
3.73/1.89	   active(take(s(X), cons(Y, L))) -> mark(cons(Y, take(X, L)))
3.73/1.89	   active(length(nil)) -> mark(0)
3.73/1.89	   active(length(cons(X, L))) -> mark(s(length(L)))
3.73/1.89	   active(inf(X)) -> inf(active(X))
3.73/1.89	   active(take(X1, X2)) -> take(active(X1), X2)
3.73/1.89	   active(take(X1, X2)) -> take(X1, active(X2))
3.73/1.89	   active(length(X)) -> length(active(X))
3.73/1.89	   inf(mark(X)) -> mark(inf(X))
3.73/1.89	   take(mark(X1), X2) -> mark(take(X1, X2))
3.73/1.89	   take(X1, mark(X2)) -> mark(take(X1, X2))
3.73/1.89	   length(mark(X)) -> mark(length(X))
3.73/1.89	   proper(eq(X1, X2)) -> eq(proper(X1), proper(X2))
3.73/1.89	   proper(0) -> ok(0)
3.73/1.89	   proper(true) -> ok(true)
3.73/1.89	   proper(s(X)) -> s(proper(X))
3.73/1.89	   proper(false) -> ok(false)
3.73/1.89	   proper(inf(X)) -> inf(proper(X))
3.73/1.89	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.73/1.89	   proper(take(X1, X2)) -> take(proper(X1), proper(X2))
3.73/1.89	   proper(nil) -> ok(nil)
3.73/1.89	   proper(length(X)) -> length(proper(X))
3.73/1.89	   eq(ok(X1), ok(X2)) -> ok(eq(X1, X2))
3.73/1.89	   s(ok(X)) -> ok(s(X))
3.73/1.89	   inf(ok(X)) -> ok(inf(X))
3.73/1.89	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.73/1.89	   take(ok(X1), ok(X2)) -> ok(take(X1, X2))
3.73/1.89	   length(ok(X)) -> ok(length(X))
3.73/1.89	   top(mark(X)) -> top(proper(X))
3.73/1.89	   top(ok(X)) -> top(active(X))
3.73/1.89	
3.73/1.89	The set Q consists of the following terms:
3.73/1.89	
3.73/1.89	   active(eq(x0, x1))
3.73/1.89	   active(inf(x0))
3.73/1.89	   active(take(x0, x1))
3.73/1.89	   active(length(x0))
3.73/1.89	   inf(mark(x0))
3.73/1.89	   take(mark(x0), x1)
3.73/1.89	   take(x0, mark(x1))
3.73/1.89	   length(mark(x0))
3.73/1.89	   proper(eq(x0, x1))
3.73/1.89	   proper(0)
3.73/1.89	   proper(true)
3.73/1.89	   proper(s(x0))
3.73/1.89	   proper(false)
3.73/1.89	   proper(inf(x0))
3.73/1.89	   proper(cons(x0, x1))
3.73/1.89	   proper(take(x0, x1))
3.73/1.89	   proper(nil)
3.73/1.89	   proper(length(x0))
3.73/1.89	   eq(ok(x0), ok(x1))
3.73/1.89	   s(ok(x0))
3.73/1.89	   inf(ok(x0))
3.73/1.89	   cons(ok(x0), ok(x1))
3.73/1.89	   take(ok(x0), ok(x1))
3.73/1.89	   length(ok(x0))
3.73/1.89	   top(mark(x0))
3.73/1.89	   top(ok(x0))
3.73/1.89	
3.73/1.89	Special symbols used for the transformation (see [GM04]):
3.73/1.89	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	0: empty set
3.73/1.89	true: empty set
3.73/1.89	s: empty set
3.73/1.89	false: empty set
3.73/1.89	inf: {1}
3.73/1.89	cons: empty set
3.73/1.89	take: {1, 2}
3.73/1.89	nil: empty set
3.73/1.89	length: {1}
3.73/1.89	The QTRS contained just a subset of rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is sound, but not necessarily complete.
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(2)
3.73/1.89	Obligation:
3.73/1.89	Context-sensitive rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   eq(0, 0) -> true
3.73/1.89	   eq(s(X), s(Y)) -> eq(X, Y)
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	   inf(X) -> cons(X, inf(s(X)))
3.73/1.89	   take(0, X) -> nil
3.73/1.89	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.73/1.89	   length(nil) -> 0
3.73/1.89	   length(cons(X, L)) -> s(length(L))
3.73/1.89	
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	0: empty set
3.73/1.89	true: empty set
3.73/1.89	s: empty set
3.73/1.89	false: empty set
3.73/1.89	inf: {1}
3.73/1.89	cons: empty set
3.73/1.89	take: {1, 2}
3.73/1.89	nil: empty set
3.73/1.89	length: {1}
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(3) CSRRRRProof (EQUIVALENT)
3.73/1.89	The following CSR is given: Context-sensitive rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   eq(0, 0) -> true
3.73/1.89	   eq(s(X), s(Y)) -> eq(X, Y)
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	   inf(X) -> cons(X, inf(s(X)))
3.73/1.89	   take(0, X) -> nil
3.73/1.89	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.73/1.89	   length(nil) -> 0
3.73/1.89	   length(cons(X, L)) -> s(length(L))
3.73/1.89	
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	0: empty set
3.73/1.89	true: empty set
3.73/1.89	s: empty set
3.73/1.89	false: empty set
3.73/1.89	inf: {1}
3.73/1.89	cons: empty set
3.73/1.89	take: {1, 2}
3.73/1.89	nil: empty set
3.73/1.89	length: {1}
3.73/1.89	Used ordering:
3.73/1.89	Polynomial interpretation [POLO]:
3.73/1.89	
3.73/1.89	   POL(0) = 1
3.73/1.89	   POL(cons(x_1, x_2)) = 1 + x_1
3.73/1.89	   POL(eq(x_1, x_2)) = 0
3.73/1.89	   POL(false) = 0
3.73/1.89	   POL(inf(x_1)) = 2 + x_1
3.73/1.89	   POL(length(x_1)) = 2 + x_1
3.73/1.89	   POL(nil) = 1
3.73/1.89	   POL(s(x_1)) = 0
3.73/1.89	   POL(take(x_1, x_2)) = 2*x_1 + 2*x_2
3.73/1.89	   POL(true) = 0
3.73/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.73/1.89	
3.73/1.89	   inf(X) -> cons(X, inf(s(X)))
3.73/1.89	   take(0, X) -> nil
3.73/1.89	   take(s(X), cons(Y, L)) -> cons(Y, take(X, L))
3.73/1.89	   length(nil) -> 0
3.73/1.89	   length(cons(X, L)) -> s(length(L))
3.73/1.89	
3.73/1.89	
3.73/1.89	
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(4)
3.73/1.89	Obligation:
3.73/1.89	Context-sensitive rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   eq(0, 0) -> true
3.73/1.89	   eq(s(X), s(Y)) -> eq(X, Y)
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	0: empty set
3.73/1.89	true: empty set
3.73/1.89	s: empty set
3.73/1.89	false: empty set
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(5) CSRRRRProof (EQUIVALENT)
3.73/1.89	The following CSR is given: Context-sensitive rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   eq(0, 0) -> true
3.73/1.89	   eq(s(X), s(Y)) -> eq(X, Y)
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	0: empty set
3.73/1.89	true: empty set
3.73/1.89	s: empty set
3.73/1.89	false: empty set
3.73/1.89	Used ordering:
3.73/1.89	Polynomial interpretation [POLO]:
3.73/1.89	
3.73/1.89	   POL(0) = 1
3.73/1.89	   POL(eq(x_1, x_2)) = x_1 + x_2
3.73/1.89	   POL(false) = 0
3.73/1.89	   POL(s(x_1)) = 1 + x_1
3.73/1.89	   POL(true) = 0
3.73/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.73/1.89	
3.73/1.89	   eq(0, 0) -> true
3.73/1.89	   eq(s(X), s(Y)) -> eq(X, Y)
3.73/1.89	
3.73/1.89	
3.73/1.89	
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(6)
3.73/1.89	Obligation:
3.73/1.89	Context-sensitive rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	false: empty set
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(7) CSRRRRProof (EQUIVALENT)
3.73/1.89	The following CSR is given: Context-sensitive rewrite system:
3.73/1.89	The TRS R consists of the following rules:
3.73/1.89	
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	
3.73/1.89	The replacement map contains the following entries:
3.73/1.89	
3.73/1.89	eq: empty set
3.73/1.89	false: empty set
3.73/1.89	Used ordering:
3.73/1.89	Polynomial interpretation [POLO]:
3.73/1.89	
3.73/1.89	   POL(eq(x_1, x_2)) = 1
3.73/1.89	   POL(false) = 0
3.73/1.89	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.73/1.89	
3.73/1.89	   eq(X, Y) -> false
3.73/1.89	
3.73/1.89	
3.73/1.89	
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(8)
3.73/1.89	Obligation:
3.73/1.89	Context-sensitive rewrite system:
3.73/1.89	R is empty.
3.73/1.89	
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(9) RisEmptyProof (EQUIVALENT)
3.73/1.89	The CSR R is empty. Hence, termination is trivially proven.
3.73/1.89	----------------------------------------
3.73/1.89	
3.73/1.89	(10)
3.73/1.89	YES
3.73/1.91	EOF