3.62/1.73	YES
3.77/1.74	proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
3.77/1.74	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
3.77/1.74	
3.77/1.74	
3.77/1.74	Termination w.r.t. Q of the given QTRS could be proven:
3.77/1.74	
3.77/1.74	(0) QTRS
3.77/1.74	(1) QTRSToCSRProof [SOUND, 0 ms]
3.77/1.74	(2) CSR
3.77/1.74	(3) CSRRRRProof [EQUIVALENT, 87 ms]
3.77/1.74	(4) CSR
3.77/1.74	(5) CSRRRRProof [EQUIVALENT, 0 ms]
3.77/1.74	(6) CSR
3.77/1.74	(7) RisEmptyProof [EQUIVALENT, 0 ms]
3.77/1.74	(8) YES
3.77/1.74	
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(0)
3.77/1.74	Obligation:
3.77/1.74	Q restricted rewrite system:
3.77/1.74	The TRS R consists of the following rules:
3.77/1.74	
3.77/1.74	   active(filter(cons(X, Y), 0, M)) -> mark(cons(0, filter(Y, M, M)))
3.77/1.74	   active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
3.77/1.74	   active(sieve(cons(0, Y))) -> mark(cons(0, sieve(Y)))
3.77/1.74	   active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
3.77/1.74	   active(nats(N)) -> mark(cons(N, nats(s(N))))
3.77/1.74	   active(zprimes) -> mark(sieve(nats(s(s(0)))))
3.77/1.74	   active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
3.77/1.74	   active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
3.77/1.74	   active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
3.77/1.74	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.77/1.74	   active(s(X)) -> s(active(X))
3.77/1.74	   active(sieve(X)) -> sieve(active(X))
3.77/1.74	   active(nats(X)) -> nats(active(X))
3.77/1.74	   filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
3.77/1.74	   filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
3.77/1.74	   filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
3.77/1.74	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.77/1.74	   s(mark(X)) -> mark(s(X))
3.77/1.74	   sieve(mark(X)) -> mark(sieve(X))
3.77/1.74	   nats(mark(X)) -> mark(nats(X))
3.77/1.74	   proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
3.77/1.74	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.77/1.74	   proper(0) -> ok(0)
3.77/1.74	   proper(s(X)) -> s(proper(X))
3.77/1.74	   proper(sieve(X)) -> sieve(proper(X))
3.77/1.74	   proper(nats(X)) -> nats(proper(X))
3.77/1.74	   proper(zprimes) -> ok(zprimes)
3.77/1.74	   filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
3.77/1.74	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.77/1.74	   s(ok(X)) -> ok(s(X))
3.77/1.74	   sieve(ok(X)) -> ok(sieve(X))
3.77/1.74	   nats(ok(X)) -> ok(nats(X))
3.77/1.74	   top(mark(X)) -> top(proper(X))
3.77/1.74	   top(ok(X)) -> top(active(X))
3.77/1.74	
3.77/1.74	The set Q consists of the following terms:
3.77/1.74	
3.77/1.74	   active(nats(x0))
3.77/1.74	   active(zprimes)
3.77/1.74	   active(filter(x0, x1, x2))
3.77/1.74	   active(cons(x0, x1))
3.77/1.74	   active(s(x0))
3.77/1.74	   active(sieve(x0))
3.77/1.74	   filter(mark(x0), x1, x2)
3.77/1.74	   filter(x0, mark(x1), x2)
3.77/1.74	   filter(x0, x1, mark(x2))
3.77/1.74	   cons(mark(x0), x1)
3.77/1.74	   s(mark(x0))
3.77/1.74	   sieve(mark(x0))
3.77/1.74	   nats(mark(x0))
3.77/1.74	   proper(filter(x0, x1, x2))
3.77/1.74	   proper(cons(x0, x1))
3.77/1.74	   proper(0)
3.77/1.74	   proper(s(x0))
3.77/1.74	   proper(sieve(x0))
3.77/1.74	   proper(nats(x0))
3.77/1.74	   proper(zprimes)
3.77/1.74	   filter(ok(x0), ok(x1), ok(x2))
3.77/1.74	   cons(ok(x0), ok(x1))
3.77/1.74	   s(ok(x0))
3.77/1.74	   sieve(ok(x0))
3.77/1.74	   nats(ok(x0))
3.77/1.74	   top(mark(x0))
3.77/1.74	   top(ok(x0))
3.77/1.74	
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(1) QTRSToCSRProof (SOUND)
3.77/1.74	The following Q TRS is given: Q restricted rewrite system:
3.77/1.74	The TRS R consists of the following rules:
3.77/1.74	
3.77/1.74	   active(filter(cons(X, Y), 0, M)) -> mark(cons(0, filter(Y, M, M)))
3.77/1.74	   active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
3.77/1.74	   active(sieve(cons(0, Y))) -> mark(cons(0, sieve(Y)))
3.77/1.74	   active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
3.77/1.74	   active(nats(N)) -> mark(cons(N, nats(s(N))))
3.77/1.74	   active(zprimes) -> mark(sieve(nats(s(s(0)))))
3.77/1.74	   active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
3.77/1.74	   active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
3.77/1.74	   active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
3.77/1.74	   active(cons(X1, X2)) -> cons(active(X1), X2)
3.77/1.74	   active(s(X)) -> s(active(X))
3.77/1.74	   active(sieve(X)) -> sieve(active(X))
3.77/1.74	   active(nats(X)) -> nats(active(X))
3.77/1.74	   filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
3.77/1.74	   filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
3.77/1.74	   filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
3.77/1.74	   cons(mark(X1), X2) -> mark(cons(X1, X2))
3.77/1.74	   s(mark(X)) -> mark(s(X))
3.77/1.74	   sieve(mark(X)) -> mark(sieve(X))
3.77/1.74	   nats(mark(X)) -> mark(nats(X))
3.77/1.74	   proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
3.77/1.74	   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
3.77/1.74	   proper(0) -> ok(0)
3.77/1.74	   proper(s(X)) -> s(proper(X))
3.77/1.74	   proper(sieve(X)) -> sieve(proper(X))
3.77/1.74	   proper(nats(X)) -> nats(proper(X))
3.77/1.74	   proper(zprimes) -> ok(zprimes)
3.77/1.74	   filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
3.77/1.74	   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
3.77/1.74	   s(ok(X)) -> ok(s(X))
3.77/1.74	   sieve(ok(X)) -> ok(sieve(X))
3.77/1.74	   nats(ok(X)) -> ok(nats(X))
3.77/1.74	   top(mark(X)) -> top(proper(X))
3.77/1.74	   top(ok(X)) -> top(active(X))
3.77/1.74	
3.77/1.74	The set Q consists of the following terms:
3.77/1.74	
3.77/1.74	   active(nats(x0))
3.77/1.74	   active(zprimes)
3.77/1.74	   active(filter(x0, x1, x2))
3.77/1.74	   active(cons(x0, x1))
3.77/1.74	   active(s(x0))
3.77/1.74	   active(sieve(x0))
3.77/1.74	   filter(mark(x0), x1, x2)
3.77/1.74	   filter(x0, mark(x1), x2)
3.77/1.74	   filter(x0, x1, mark(x2))
3.77/1.74	   cons(mark(x0), x1)
3.77/1.74	   s(mark(x0))
3.77/1.74	   sieve(mark(x0))
3.77/1.74	   nats(mark(x0))
3.77/1.74	   proper(filter(x0, x1, x2))
3.77/1.74	   proper(cons(x0, x1))
3.77/1.74	   proper(0)
3.77/1.74	   proper(s(x0))
3.77/1.74	   proper(sieve(x0))
3.77/1.74	   proper(nats(x0))
3.77/1.74	   proper(zprimes)
3.77/1.74	   filter(ok(x0), ok(x1), ok(x2))
3.77/1.74	   cons(ok(x0), ok(x1))
3.77/1.74	   s(ok(x0))
3.77/1.74	   sieve(ok(x0))
3.77/1.74	   nats(ok(x0))
3.77/1.74	   top(mark(x0))
3.77/1.74	   top(ok(x0))
3.77/1.74	
3.77/1.74	Special symbols used for the transformation (see [GM04]):
3.77/1.74	top: top_1, active: active_1, mark: mark_1, ok: ok_1, proper: proper_1
3.77/1.74	The replacement map contains the following entries:
3.77/1.74	
3.77/1.74	filter: {1, 2, 3}
3.77/1.74	cons: {1}
3.77/1.74	0: empty set
3.77/1.74	s: {1}
3.77/1.74	sieve: {1}
3.77/1.74	nats: {1}
3.77/1.74	zprimes: empty set
3.77/1.74	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.
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(2)
3.77/1.74	Obligation:
3.77/1.74	Context-sensitive rewrite system:
3.77/1.74	The TRS R consists of the following rules:
3.77/1.74	
3.77/1.74	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.77/1.74	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.77/1.74	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.77/1.74	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.77/1.74	   nats(N) -> cons(N, nats(s(N)))
3.77/1.74	   zprimes -> sieve(nats(s(s(0))))
3.77/1.74	
3.77/1.74	The replacement map contains the following entries:
3.77/1.74	
3.77/1.74	filter: {1, 2, 3}
3.77/1.74	cons: {1}
3.77/1.74	0: empty set
3.77/1.74	s: {1}
3.77/1.74	sieve: {1}
3.77/1.74	nats: {1}
3.77/1.74	zprimes: empty set
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(3) CSRRRRProof (EQUIVALENT)
3.77/1.74	The following CSR is given: Context-sensitive rewrite system:
3.77/1.74	The TRS R consists of the following rules:
3.77/1.74	
3.77/1.74	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.77/1.74	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.77/1.74	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.77/1.74	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.77/1.74	   nats(N) -> cons(N, nats(s(N)))
3.77/1.74	   zprimes -> sieve(nats(s(s(0))))
3.77/1.74	
3.77/1.74	The replacement map contains the following entries:
3.77/1.74	
3.77/1.74	filter: {1, 2, 3}
3.77/1.74	cons: {1}
3.77/1.74	0: empty set
3.77/1.74	s: {1}
3.77/1.74	sieve: {1}
3.77/1.74	nats: {1}
3.77/1.74	zprimes: empty set
3.77/1.74	Used ordering:
3.77/1.74	Polynomial interpretation [POLO]:
3.77/1.74	
3.77/1.74	   POL(0) = 0
3.77/1.74	   POL(cons(x_1, x_2)) = x_1
3.77/1.74	   POL(filter(x_1, x_2, x_3)) = x_1 + x_2 + x_3
3.77/1.74	   POL(nats(x_1)) = x_1
3.77/1.74	   POL(s(x_1)) = x_1
3.77/1.74	   POL(sieve(x_1)) = 1 + x_1
3.77/1.74	   POL(zprimes) = 1
3.77/1.74	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.77/1.74	
3.77/1.74	   sieve(cons(0, Y)) -> cons(0, sieve(Y))
3.77/1.74	   sieve(cons(s(N), Y)) -> cons(s(N), sieve(filter(Y, N, N)))
3.77/1.74	
3.77/1.74	
3.77/1.74	
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(4)
3.77/1.74	Obligation:
3.77/1.74	Context-sensitive rewrite system:
3.77/1.74	The TRS R consists of the following rules:
3.77/1.74	
3.77/1.74	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.77/1.74	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.77/1.74	   nats(N) -> cons(N, nats(s(N)))
3.77/1.74	   zprimes -> sieve(nats(s(s(0))))
3.77/1.74	
3.77/1.74	The replacement map contains the following entries:
3.77/1.74	
3.77/1.74	filter: {1, 2, 3}
3.77/1.74	cons: {1}
3.77/1.74	0: empty set
3.77/1.74	s: {1}
3.77/1.74	sieve: {1}
3.77/1.74	nats: {1}
3.77/1.74	zprimes: empty set
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(5) CSRRRRProof (EQUIVALENT)
3.77/1.74	The following CSR is given: Context-sensitive rewrite system:
3.77/1.74	The TRS R consists of the following rules:
3.77/1.74	
3.77/1.74	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.77/1.74	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.77/1.74	   nats(N) -> cons(N, nats(s(N)))
3.77/1.74	   zprimes -> sieve(nats(s(s(0))))
3.77/1.74	
3.77/1.74	The replacement map contains the following entries:
3.77/1.74	
3.77/1.74	filter: {1, 2, 3}
3.77/1.74	cons: {1}
3.77/1.74	0: empty set
3.77/1.74	s: {1}
3.77/1.74	sieve: {1}
3.77/1.74	nats: {1}
3.77/1.74	zprimes: empty set
3.77/1.74	Used ordering:
3.77/1.74	Polynomial interpretation [POLO]:
3.77/1.74	
3.77/1.74	   POL(0) = 0
3.77/1.74	   POL(cons(x_1, x_2)) = x_1
3.77/1.74	   POL(filter(x_1, x_2, x_3)) = 1 + x_1 + 2*x_2 + 2*x_3
3.77/1.74	   POL(nats(x_1)) = 1 + x_1
3.77/1.74	   POL(s(x_1)) = x_1
3.77/1.74	   POL(sieve(x_1)) = x_1
3.77/1.74	   POL(zprimes) = 2
3.77/1.74	With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:
3.77/1.74	
3.77/1.74	   filter(cons(X, Y), 0, M) -> cons(0, filter(Y, M, M))
3.77/1.74	   filter(cons(X, Y), s(N), M) -> cons(X, filter(Y, N, M))
3.77/1.74	   nats(N) -> cons(N, nats(s(N)))
3.77/1.74	   zprimes -> sieve(nats(s(s(0))))
3.77/1.74	
3.77/1.74	
3.77/1.74	
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(6)
3.77/1.74	Obligation:
3.77/1.74	Context-sensitive rewrite system:
3.77/1.74	R is empty.
3.77/1.74	
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(7) RisEmptyProof (EQUIVALENT)
3.77/1.74	The CSR R is empty. Hence, termination is trivially proven.
3.77/1.74	----------------------------------------
3.77/1.74	
3.77/1.74	(8)
3.77/1.74	YES
3.77/1.79	EOF