WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 417 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 126 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 77 ms]
        (18) typed CpxTrs
        (19) RewriteLemmaProof [LOWER BOUND(ID), 73 ms]
        (20) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(filter(cons(X, Y), 0, M)) -> mark(cons(0, filter(Y, M, M)))
   active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
   active(sieve(cons(0, Y))) -> mark(cons(0, sieve(Y)))
   active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
   active(nats(N)) -> mark(cons(N, nats(s(N))))
   active(zprimes) -> mark(sieve(nats(s(s(0)))))
   active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
   active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
   active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(sieve(X)) -> sieve(active(X))
   active(nats(X)) -> nats(active(X))
   filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
   filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
   filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   sieve(mark(X)) -> mark(sieve(X))
   nats(mark(X)) -> mark(nats(X))
   proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0) -> ok(0)
   proper(s(X)) -> s(proper(X))
   proper(sieve(X)) -> sieve(proper(X))
   proper(nats(X)) -> nats(proper(X))
   proper(zprimes) -> ok(zprimes)
   filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   sieve(ok(X)) -> ok(sieve(X))
   nats(ok(X)) -> ok(nats(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
   active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
   active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
   active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
   active(nats(N)) -> mark(cons(N, nats(s(N))))
   active(zprimes) -> mark(sieve(nats(s(s(0')))))
   active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
   active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
   active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
   active(cons(X1, X2)) -> cons(active(X1), X2)
   active(s(X)) -> s(active(X))
   active(sieve(X)) -> sieve(active(X))
   active(nats(X)) -> nats(active(X))
   filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
   filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
   filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
   cons(mark(X1), X2) -> mark(cons(X1, X2))
   s(mark(X)) -> mark(s(X))
   sieve(mark(X)) -> mark(sieve(X))
   nats(mark(X)) -> mark(nats(X))
   proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
   proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
   proper(0') -> ok(0')
   proper(s(X)) -> s(proper(X))
   proper(sieve(X)) -> sieve(proper(X))
   proper(nats(X)) -> nats(proper(X))
   proper(zprimes) -> ok(zprimes)
   filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
   cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
   s(ok(X)) -> ok(s(X))
   sieve(ok(X)) -> ok(sieve(X))
   nats(ok(X)) -> ok(nats(X))
   top(mark(X)) -> top(proper(X))
   top(ok(X)) -> top(active(X))

S is empty.
Rewrite Strategy: FULL
----------------------------------------

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok

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(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
active, cons, filter, sieve, s, nats, proper, top

They will be analysed ascendingly in the following order:
cons < active
filter < active
sieve < active
s < active
nats < active
active < top
cons < proper
filter < proper
sieve < proper
s < proper
nats < proper
proper < top

----------------------------------------

(6)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, filter, sieve, s, nats, proper, top

They will be analysed ascendingly in the following order:
cons < active
filter < active
sieve < active
s < active
nats < active
active < top
cons < proper
filter < proper
sieve < proper
s < proper
nats < proper
proper < top

----------------------------------------

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) -> *4_0, rt in Omega(n5_0)

Induction Base:
cons(gen_0':mark:zprimes:ok3_0(+(1, 0)), gen_0':mark:zprimes:ok3_0(b))

Induction Step:
cons(gen_0':mark:zprimes:ok3_0(+(1, +(n5_0, 1))), gen_0':mark:zprimes:ok3_0(b)) ->_R^Omega(1)
mark(cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(8)
Complex Obligation (BEST)

----------------------------------------

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
cons, active, filter, sieve, s, nats, proper, top

They will be analysed ascendingly in the following order:
cons < active
filter < active
sieve < active
s < active
nats < active
active < top
cons < proper
filter < proper
sieve < proper
s < proper
nats < proper
proper < top

----------------------------------------

(10) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(11)
BOUNDS(n^1, INF)

----------------------------------------

(12)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) -> *4_0, rt in Omega(n5_0)


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
filter, active, sieve, s, nats, proper, top

They will be analysed ascendingly in the following order:
filter < active
sieve < active
s < active
nats < active
active < top
filter < proper
sieve < proper
s < proper
nats < proper
proper < top

----------------------------------------

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
filter(gen_0':mark:zprimes:ok3_0(+(1, n916_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) -> *4_0, rt in Omega(n916_0)

Induction Base:
filter(gen_0':mark:zprimes:ok3_0(+(1, 0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c))

Induction Step:
filter(gen_0':mark:zprimes:ok3_0(+(1, +(n916_0, 1))), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) ->_R^Omega(1)
mark(filter(gen_0':mark:zprimes:ok3_0(+(1, n916_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
filter(gen_0':mark:zprimes:ok3_0(+(1, n916_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) -> *4_0, rt in Omega(n916_0)


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
sieve, active, s, nats, proper, top

They will be analysed ascendingly in the following order:
sieve < active
s < active
nats < active
active < top
sieve < proper
s < proper
nats < proper
proper < top

----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3572_0))) -> *4_0, rt in Omega(n3572_0)

Induction Base:
sieve(gen_0':mark:zprimes:ok3_0(+(1, 0)))

Induction Step:
sieve(gen_0':mark:zprimes:ok3_0(+(1, +(n3572_0, 1)))) ->_R^Omega(1)
mark(sieve(gen_0':mark:zprimes:ok3_0(+(1, n3572_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(16)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
filter(gen_0':mark:zprimes:ok3_0(+(1, n916_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) -> *4_0, rt in Omega(n916_0)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3572_0))) -> *4_0, rt in Omega(n3572_0)


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
s, active, nats, proper, top

They will be analysed ascendingly in the following order:
s < active
nats < active
active < top
s < proper
nats < proper
proper < top

----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
s(gen_0':mark:zprimes:ok3_0(+(1, n4290_0))) -> *4_0, rt in Omega(n4290_0)

Induction Base:
s(gen_0':mark:zprimes:ok3_0(+(1, 0)))

Induction Step:
s(gen_0':mark:zprimes:ok3_0(+(1, +(n4290_0, 1)))) ->_R^Omega(1)
mark(s(gen_0':mark:zprimes:ok3_0(+(1, n4290_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
filter(gen_0':mark:zprimes:ok3_0(+(1, n916_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) -> *4_0, rt in Omega(n916_0)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3572_0))) -> *4_0, rt in Omega(n3572_0)
s(gen_0':mark:zprimes:ok3_0(+(1, n4290_0))) -> *4_0, rt in Omega(n4290_0)


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
nats, active, proper, top

They will be analysed ascendingly in the following order:
nats < active
active < top
nats < proper
proper < top

----------------------------------------

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
nats(gen_0':mark:zprimes:ok3_0(+(1, n5109_0))) -> *4_0, rt in Omega(n5109_0)

Induction Base:
nats(gen_0':mark:zprimes:ok3_0(+(1, 0)))

Induction Step:
nats(gen_0':mark:zprimes:ok3_0(+(1, +(n5109_0, 1)))) ->_R^Omega(1)
mark(nats(gen_0':mark:zprimes:ok3_0(+(1, n5109_0)))) ->_IH
mark(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
TRS:
Rules:
active(filter(cons(X, Y), 0', M)) -> mark(cons(0', filter(Y, M, M)))
active(filter(cons(X, Y), s(N), M)) -> mark(cons(X, filter(Y, N, M)))
active(sieve(cons(0', Y))) -> mark(cons(0', sieve(Y)))
active(sieve(cons(s(N), Y))) -> mark(cons(s(N), sieve(filter(Y, N, N))))
active(nats(N)) -> mark(cons(N, nats(s(N))))
active(zprimes) -> mark(sieve(nats(s(s(0')))))
active(filter(X1, X2, X3)) -> filter(active(X1), X2, X3)
active(filter(X1, X2, X3)) -> filter(X1, active(X2), X3)
active(filter(X1, X2, X3)) -> filter(X1, X2, active(X3))
active(cons(X1, X2)) -> cons(active(X1), X2)
active(s(X)) -> s(active(X))
active(sieve(X)) -> sieve(active(X))
active(nats(X)) -> nats(active(X))
filter(mark(X1), X2, X3) -> mark(filter(X1, X2, X3))
filter(X1, mark(X2), X3) -> mark(filter(X1, X2, X3))
filter(X1, X2, mark(X3)) -> mark(filter(X1, X2, X3))
cons(mark(X1), X2) -> mark(cons(X1, X2))
s(mark(X)) -> mark(s(X))
sieve(mark(X)) -> mark(sieve(X))
nats(mark(X)) -> mark(nats(X))
proper(filter(X1, X2, X3)) -> filter(proper(X1), proper(X2), proper(X3))
proper(cons(X1, X2)) -> cons(proper(X1), proper(X2))
proper(0') -> ok(0')
proper(s(X)) -> s(proper(X))
proper(sieve(X)) -> sieve(proper(X))
proper(nats(X)) -> nats(proper(X))
proper(zprimes) -> ok(zprimes)
filter(ok(X1), ok(X2), ok(X3)) -> ok(filter(X1, X2, X3))
cons(ok(X1), ok(X2)) -> ok(cons(X1, X2))
s(ok(X)) -> ok(s(X))
sieve(ok(X)) -> ok(sieve(X))
nats(ok(X)) -> ok(nats(X))
top(mark(X)) -> top(proper(X))
top(ok(X)) -> top(active(X))

Types:
active :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
filter :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
cons :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok -> 0':mark:zprimes:ok
0' :: 0':mark:zprimes:ok
mark :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
s :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
sieve :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
nats :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
zprimes :: 0':mark:zprimes:ok
proper :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
ok :: 0':mark:zprimes:ok -> 0':mark:zprimes:ok
top :: 0':mark:zprimes:ok -> top
hole_0':mark:zprimes:ok1_0 :: 0':mark:zprimes:ok
hole_top2_0 :: top
gen_0':mark:zprimes:ok3_0 :: Nat -> 0':mark:zprimes:ok


Lemmas:
cons(gen_0':mark:zprimes:ok3_0(+(1, n5_0)), gen_0':mark:zprimes:ok3_0(b)) -> *4_0, rt in Omega(n5_0)
filter(gen_0':mark:zprimes:ok3_0(+(1, n916_0)), gen_0':mark:zprimes:ok3_0(b), gen_0':mark:zprimes:ok3_0(c)) -> *4_0, rt in Omega(n916_0)
sieve(gen_0':mark:zprimes:ok3_0(+(1, n3572_0))) -> *4_0, rt in Omega(n3572_0)
s(gen_0':mark:zprimes:ok3_0(+(1, n4290_0))) -> *4_0, rt in Omega(n4290_0)
nats(gen_0':mark:zprimes:ok3_0(+(1, n5109_0))) -> *4_0, rt in Omega(n5109_0)


Generator Equations:
gen_0':mark:zprimes:ok3_0(0) <=> 0'
gen_0':mark:zprimes:ok3_0(+(x, 1)) <=> mark(gen_0':mark:zprimes:ok3_0(x))


The following defined symbols remain to be analysed:
active, proper, top

They will be analysed ascendingly in the following order:
active < top
proper < top