'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ filter(cons(X), 0(), M) -> cons(0())
, filter(cons(X), s(N), M) -> cons(X)
, sieve(cons(0())) -> cons(0())
, sieve(cons(s(N))) -> cons(s(N))
, nats(N) -> cons(N)
, zprimes() -> sieve(nats(s(s(0()))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ filter^#(cons(X), 0(), M) -> c_0()
, filter^#(cons(X), s(N), M) -> c_1()
, sieve^#(cons(0())) -> c_2()
, sieve^#(cons(s(N))) -> c_3()
, nats^#(N) -> c_4()
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
The usable rules are:
{nats(N) -> cons(N)}
The estimated dependency graph contains the following edges:
{zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
==> {sieve^#(cons(s(N))) -> c_3()}
{zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
==> {sieve^#(cons(0())) -> c_2()}
We consider the following path(s):
1) { zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))
, sieve^#(cons(0())) -> c_2()}
The usable rules for this path are the following:
{nats(N) -> cons(N)}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [1] x1 + [4]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {sieve^#(cons(0())) -> c_2()}
Weak Rules:
{ nats(N) -> cons(N)
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{sieve^#(cons(0())) -> c_2()}
and weakly orienting the rules
{ nats(N) -> cons(N)
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{sieve^#(cons(0())) -> c_2()}
Details:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [1] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [8]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ sieve^#(cons(0())) -> c_2()
, nats(N) -> cons(N)
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Details:
The given problem does not contain any strict rules
2) { zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))
, sieve^#(cons(s(N))) -> c_3()}
The usable rules for this path are the following:
{nats(N) -> cons(N)}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [1] x1 + [4]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {sieve^#(cons(s(N))) -> c_3()}
Weak Rules:
{ nats(N) -> cons(N)
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{sieve^#(cons(s(N))) -> c_3()}
and weakly orienting the rules
{ nats(N) -> cons(N)
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{sieve^#(cons(s(N))) -> c_3()}
Details:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [1] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [8]
c_5(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ sieve^#(cons(s(N))) -> c_3()
, nats(N) -> cons(N)
, zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Details:
The given problem does not contain any strict rules
3) {zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
The usable rules for this path are the following:
{nats(N) -> cons(N)}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [1] x1 + [4]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Weak Rules: {nats(N) -> cons(N)}
Details:
We apply the weight gap principle, strictly orienting the rules
{zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
and weakly orienting the rules
{nats(N) -> cons(N)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))}
Details:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [1] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [8]
c_5(x1) = [1] x1 + [1]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ zprimes^#() -> c_5(sieve^#(nats(s(s(0())))))
, nats(N) -> cons(N)}
Details:
The given problem does not contain any strict rules
4) {filter^#(cons(X), s(N), M) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {filter^#(cons(X), s(N), M) -> c_1()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{filter^#(cons(X), s(N), M) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{filter^#(cons(X), s(N), M) -> c_1()}
Details:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {filter^#(cons(X), s(N), M) -> c_1()}
Details:
The given problem does not contain any strict rules
5) {filter^#(cons(X), 0(), M) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {filter^#(cons(X), 0(), M) -> c_0()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{filter^#(cons(X), 0(), M) -> c_0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{filter^#(cons(X), 0(), M) -> c_0()}
Details:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {filter^#(cons(X), 0(), M) -> c_0()}
Details:
The given problem does not contain any strict rules
6) {nats^#(N) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [0] x1 + [0]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {nats^#(N) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{nats^#(N) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{nats^#(N) -> c_4()}
Details:
Interpretation Functions:
filter(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
cons(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
sieve(x1) = [0] x1 + [0]
nats(x1) = [0] x1 + [0]
zprimes() = [0]
filter^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0() = [0]
c_1() = [0]
sieve^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
nats^#(x1) = [1] x1 + [4]
c_4() = [0]
zprimes^#() = [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {nats^#(N) -> c_4()}
Details:
The given problem does not contain any strict rules