'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ app(nil(), YS) -> YS
, app(cons(X), YS) -> cons(X)
, from(X) -> cons(X)
, zWadr(nil(), YS) -> nil()
, zWadr(XS, nil()) -> nil()
, zWadr(cons(X), cons(Y)) -> cons(app(Y, cons(X)))
, prefix(L) -> cons(nil())}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ app^#(nil(), YS) -> c_0()
, app^#(cons(X), YS) -> c_1()
, from^#(X) -> c_2()
, zWadr^#(nil(), YS) -> c_3()
, zWadr^#(XS, nil()) -> c_4()
, zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))
, prefix^#(L) -> c_6()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
==> {app^#(cons(X), YS) -> c_1()}
{zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
==> {app^#(nil(), YS) -> c_0()}
We consider the following path(s):
1) { zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))
, app^#(nil(), YS) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {app^#(nil(), YS) -> c_0()}
Weak Rules: {zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{app^#(nil(), YS) -> c_0()}
and weakly orienting the rules
{zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{app^#(nil(), YS) -> c_0()}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [1] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
prefix^#(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:
{ app^#(nil(), YS) -> c_0()
, zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Details:
The given problem does not contain any strict rules
2) { zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))
, app^#(cons(X), YS) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {app^#(cons(X), YS) -> c_1()}
Weak Rules: {zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{app^#(cons(X), YS) -> c_1()}
and weakly orienting the rules
{zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{app^#(cons(X), YS) -> c_1()}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [1] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
prefix^#(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:
{ app^#(cons(X), YS) -> c_1()
, zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Details:
The given problem does not contain any strict rules
3) {zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [1] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
prefix^#(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: {zWadr^#(cons(X), cons(Y)) -> c_5(app^#(Y, cons(X)))}
Details:
The given problem does not contain any strict rules
4) {zWadr^#(XS, nil()) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {zWadr^#(XS, nil()) -> c_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{zWadr^#(XS, nil()) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{zWadr^#(XS, nil()) -> c_4()}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {zWadr^#(XS, nil()) -> c_4()}
Details:
The given problem does not contain any strict rules
5) {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:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{from^#(X) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{from^#(X) -> c_2()}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [1] x1 + [4]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {from^#(X) -> c_2()}
Details:
The given problem does not contain any strict rules
6) {prefix^#(L) -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {prefix^#(L) -> c_6()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{prefix^#(L) -> c_6()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{prefix^#(L) -> c_6()}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(x1) = [1] x1 + [4]
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: {prefix^#(L) -> c_6()}
Details:
The given problem does not contain any strict rules
7) {zWadr^#(nil(), YS) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {zWadr^#(nil(), YS) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{zWadr^#(nil(), YS) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{zWadr^#(nil(), YS) -> c_3()}
Details:
Interpretation Functions:
app(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1) = [0] x1 + [0]
from(x1) = [0] x1 + [0]
zWadr(x1, x2) = [0] x1 + [0] x2 + [0]
prefix(x1) = [0] x1 + [0]
app^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1() = [0]
from^#(x1) = [0] x1 + [0]
c_2() = [0]
zWadr^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_3() = [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
prefix^#(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: {zWadr^#(nil(), YS) -> c_3()}
Details:
The given problem does not contain any strict rules