'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ first(0(), X) -> nil()
, first(s(X), cons(Y, Z)) -> cons(Y, n__first(X, activate(Z)))
, from(X) -> cons(X, n__from(s(X)))
, first(X1, X2) -> n__first(X1, X2)
, from(X) -> n__from(X)
, activate(n__first(X1, X2)) -> first(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ first^#(0(), X) -> c_0()
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, from^#(X) -> c_2()
, first^#(X1, X2) -> c_3()
, from^#(X) -> c_4()
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, activate^#(n__from(X)) -> c_6(from^#(X))
, activate^#(X) -> c_7()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))}
==> {activate^#(n__from(X)) -> c_6(from^#(X))}
{first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))}
==> {activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
{first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))}
==> {activate^#(X) -> c_7()}
{activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
==> {first^#(X1, X2) -> c_3()}
{activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
==> {first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))}
{activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
==> {first^#(0(), X) -> c_0()}
{activate^#(n__from(X)) -> c_6(from^#(X))}
==> {from^#(X) -> c_4()}
{activate^#(n__from(X)) -> c_6(from^#(X))}
==> {from^#(X) -> c_2()}
We consider the following path(s):
1) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))}
and weakly orienting the rules
{activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules
2) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, activate^#(n__from(X)) -> c_6(from^#(X))
, from^#(X) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {from^#(X) -> c_2()}
Weak Rules:
{ activate^#(n__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{from^#(X) -> c_2()}
and weakly orienting the rules
{ activate^#(n__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{from^#(X) -> c_2()}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [1] x1 + [0]
c_7() = [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:
{ from^#(X) -> c_2()
, activate^#(n__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules
3) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, activate^#(n__from(X)) -> c_6(from^#(X))
, from^#(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:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {from^#(X) -> c_4()}
Weak Rules:
{ activate^#(n__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{from^#(X) -> c_4()}
and weakly orienting the rules
{ activate^#(n__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{from^#(X) -> c_4()}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [1] x1 + [0]
c_7() = [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:
{ from^#(X) -> c_4()
, activate^#(n__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules
4) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, activate^#(n__from(X)) -> c_6(from^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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__from(X)) -> c_6(from^#(X))}
Weak Rules:
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__from(X)) -> c_6(from^#(X))}
and weakly orienting the rules
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__from(X)) -> c_6(from^#(X))}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [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__from(X)) -> c_6(from^#(X))
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules
5) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, first^#(0(), X) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {first^#(0(), X) -> c_0()}
Weak Rules:
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{first^#(0(), X) -> c_0()}
and weakly orienting the rules
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{first^#(0(), X) -> c_0()}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [4]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ first^#(0(), X) -> c_0()
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules
6) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, first^#(X1, X2) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {first^#(X1, X2) -> c_3()}
Weak Rules:
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{first^#(X1, X2) -> c_3()}
and weakly orienting the rules
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{first^#(X1, X2) -> c_3()}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [8]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [3]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ first^#(X1, X2) -> c_3()
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules
7) { first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))
, activate^#(X) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__first(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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_7()}
Weak Rules:
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_7()}
and weakly orienting the rules
{ first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_7()}
Details:
Interpretation Functions:
first(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
nil() = [0]
s(x1) = [1] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__first(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
first^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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_7()
, first^#(s(X), cons(Y, Z)) -> c_1(activate^#(Z))
, activate^#(n__first(X1, X2)) -> c_5(first^#(X1, X2))}
Details:
The given problem does not contain any strict rules