'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 2nd(cons(X, n__cons(Y, Z))) -> activate(Y)
, from(X) -> cons(X, n__from(s(X)))
, cons(X1, X2) -> n__cons(X1, X2)
, from(X) -> n__from(X)
, activate(n__cons(X1, X2)) -> cons(X1, X2)
, activate(n__from(X)) -> from(X)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, cons^#(X1, X2) -> c_2()
, from^#(X) -> c_3()
, activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))
, activate^#(n__from(X)) -> c_5(from^#(X))
, activate^#(X) -> c_6()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
==> {activate^#(n__from(X)) -> c_5(from^#(X))}
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
==> {activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))}
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
==> {activate^#(X) -> c_6()}
{from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
==> {cons^#(X1, X2) -> c_2()}
{activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))}
==> {cons^#(X1, X2) -> c_2()}
{activate^#(n__from(X)) -> c_5(from^#(X))}
==> {from^#(X) -> c_3()}
{activate^#(n__from(X)) -> c_5(from^#(X))}
==> {from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
We consider the following path(s):
1) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
The usable rules for this path are empty.
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
a) We first check the conditional [Fail]:
We are not considering a strict trs contains single rule TRS.
b) We continue with the else-branch:
The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
--------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [4] x1 + [4] x2 + [1]
n__cons(x1, x2) = [1] x1 + [1] x2 + [6]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
2nd^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [0]
activate^#(x1) = [4] x1 + [4]
from^#(x1) = [4] x1 + [2]
c_1(x1) = [1] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Weak Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Weak Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
a) We first check the conditional [Success]:
We are considering a strict trs contains single rule TRS.
b) We continue with the then-branch:
The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
--------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Weak Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Weak Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [4] x1 + [4] x2 + [0]
n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [2]
s(x1) = [1] x1 + [1]
2nd^#(x1) = [4] x1 + [1]
c_0(x1) = [4] x1 + [0]
activate^#(x1) = [4] x1 + [0]
from^#(x1) = [4] x1 + [6]
c_1(x1) = [2] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6() = [0]
2) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, cons^#(X1, X2) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
a) We first check the conditional [Success]:
We are considering a strict trs contains single rule TRS.
b) We continue with the then-branch:
The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
--------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ from^#(X) -> c_1(cons^#(X, n__from(s(X))))
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [6] x2 + [0]
n__cons(x1, x2) = [1] x1 + [0] x2 + [2]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [3]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [6] x1 + [1]
c_0(x1) = [7] x1 + [1]
activate^#(x1) = [5] x1 + [0]
from^#(x1) = [2] x1 + [5]
c_1(x1) = [1] x1 + [1]
cons^#(x1, x2) = [1] x1 + [0] x2 + [4]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [4]
c_6() = [0]
3) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(n__from(X)) -> c_5(from^#(X))
, from^#(X) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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_3()}
Weak Rules:
{ activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{from^#(X) -> c_3()}
and weakly orienting the rules
{ activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{from^#(X) -> c_3()}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
n__cons(x1, x2) = [1] x1 + [1] x2 + [1]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [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_3()
, activate^#(n__from(X)) -> c_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The given problem does not contain any strict rules
4) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))
, cons^#(X1, X2) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {cons^#(X1, X2) -> c_2()}
Weak Rules:
{ activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{cons^#(X1, X2) -> c_2()}
and weakly orienting the rules
{ activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{cons^#(X1, X2) -> c_2()}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [8]
n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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:
{ cons^#(X1, X2) -> c_2()
, activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The given problem does not contain any strict rules
5) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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__cons(X1, X2)) -> c_4(cons^#(X1, X2))}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))}
and weakly orienting the rules
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__cons(X1, X2)) -> c_4(cons^#(X1, X2))}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [7]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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__cons(X1, X2)) -> c_4(cons^#(X1, X2))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The given problem does not contain any strict rules
6) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(n__from(X)) -> c_5(from^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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_5(from^#(X))}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__from(X)) -> c_5(from^#(X))}
and weakly orienting the rules
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__from(X)) -> c_5(from^#(X))}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [1] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [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_5(from^#(X))
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The given problem does not contain any strict rules
7) { 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))
, activate^#(X) -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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_6()}
Weak Rules: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_6()}
and weakly orienting the rules
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_6()}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [1]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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_6()
, 2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The given problem does not contain any strict rules
8) {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__cons(x1, x2) = [0] x1 + [0] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
activate^#(x1) = [0] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
Interpretation Functions:
2nd(x1) = [0] x1 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
n__cons(x1, x2) = [1] x1 + [1] x2 + [0]
activate(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
n__from(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
2nd^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
activate^#(x1) = [1] x1 + [0]
from^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
cons^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [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: {2nd^#(cons(X, n__cons(Y, Z))) -> c_0(activate^#(Y))}
Details:
The given problem does not contain any strict rules