'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ zeros() -> cons(0(), n__zeros())
, tail(cons(X, XS)) -> activate(XS)
, zeros() -> n__zeros()
, activate(n__zeros()) -> zeros()
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ zeros^#() -> c_0()
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))
, zeros^#() -> c_2()
, activate^#(n__zeros()) -> c_3(zeros^#())
, activate^#(X) -> c_4()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
==> {activate^#(n__zeros()) -> c_3(zeros^#())}
{tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
==> {activate^#(X) -> c_4()}
{activate^#(n__zeros()) -> c_3(zeros^#())}
==> {zeros^#() -> c_2()}
{activate^#(n__zeros()) -> c_3(zeros^#())}
==> {zeros^#() -> c_0()}
We consider the following path(s):
1) { tail^#(cons(X, XS)) -> c_1(activate^#(XS))
, activate^#(n__zeros()) -> c_3(zeros^#())
, zeros^#() -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {zeros^#() -> c_2()}
Weak Rules:
{ activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
We apply the weight gap principle, strictly orienting the rules
{zeros^#() -> c_2()}
and weakly orienting the rules
{ activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{zeros^#() -> c_2()}
Details:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [1]
c_0() = [0]
tail^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4() = [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:
{ zeros^#() -> c_2()
, activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
The given problem does not contain any strict rules
2) { tail^#(cons(X, XS)) -> c_1(activate^#(XS))
, activate^#(n__zeros()) -> c_3(zeros^#())
, zeros^#() -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {zeros^#() -> c_0()}
Weak Rules:
{ activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
We apply the weight gap principle, strictly orienting the rules
{zeros^#() -> c_0()}
and weakly orienting the rules
{ activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{zeros^#() -> c_0()}
Details:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [1]
c_0() = [0]
tail^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4() = [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:
{ zeros^#() -> c_0()
, activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
The given problem does not contain any strict rules
3) { tail^#(cons(X, XS)) -> c_1(activate^#(XS))
, activate^#(n__zeros()) -> c_3(zeros^#())}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {activate^#(n__zeros()) -> c_3(zeros^#())}
Weak Rules: {tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__zeros()) -> c_3(zeros^#())}
and weakly orienting the rules
{tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__zeros()) -> c_3(zeros^#())}
Details:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4() = [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:
{ activate^#(n__zeros()) -> c_3(zeros^#())
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
The given problem does not contain any strict rules
4) {tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
The given problem does not contain any strict rules
5) { tail^#(cons(X, XS)) -> c_1(activate^#(XS))
, activate^#(X) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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: {activate^#(X) -> c_4()}
Weak Rules: {tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_4()}
and weakly orienting the rules
{tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_4()}
Details:
Interpretation Functions:
zeros() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
0() = [0]
n__zeros() = [0]
tail(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
zeros^#() = [0]
c_0() = [0]
tail^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4() = [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:
{ activate^#(X) -> c_4()
, tail^#(cons(X, XS)) -> c_1(activate^#(XS))}
Details:
The given problem does not contain any strict rules