'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ rev(xs) -> revtl(xs, nil())
, revtl(nil(), ys) -> ys
, revtl(cons(x, xs), ys) -> revtl(xs, cons(x, ys))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ rev^#(xs) -> c_0(revtl^#(xs, nil()))
, revtl^#(nil(), ys) -> c_1()
, revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{rev^#(xs) -> c_0(revtl^#(xs, nil()))}
==> {revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
{rev^#(xs) -> c_0(revtl^#(xs, nil()))}
==> {revtl^#(nil(), ys) -> c_1()}
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
==> {revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
==> {revtl^#(nil(), ys) -> c_1()}
We consider the following path(s):
1) { rev^#(xs) -> c_0(revtl^#(xs, nil()))
, revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
rev^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
revtl^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(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:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
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:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
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:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
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:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
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:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
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:
{revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
Interpretation Functions:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [1] x2 + [7]
rev^#(x1) = [7] x1 + [7]
c_0(x1) = [1] x1 + [0]
revtl^#(x1, x2) = [7] x1 + [5] x2 + [7]
c_1() = [0]
c_2(x1) = [1] x1 + [7]
2) { rev^#(xs) -> c_0(revtl^#(xs, nil()))
, revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))
, revtl^#(nil(), 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:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
rev^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
revtl^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(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: {revtl^#(nil(), ys) -> c_1()}
Weak Rules:
{ revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))
, rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
We apply the weight gap principle, strictly orienting the rules
{revtl^#(nil(), ys) -> c_1()}
and weakly orienting the rules
{ revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))
, rev^#(xs) -> c_0(revtl^#(xs, nil()))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{revtl^#(nil(), ys) -> c_1()}
Details:
Interpretation Functions:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
rev^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [3]
revtl^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2(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:
{ revtl^#(nil(), ys) -> c_1()
, revtl^#(cons(x, xs), ys) -> c_2(revtl^#(xs, cons(x, ys)))
, rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
The given problem does not contain any strict rules
3) { rev^#(xs) -> c_0(revtl^#(xs, nil()))
, revtl^#(nil(), 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:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
rev^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
revtl^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(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: {revtl^#(nil(), ys) -> c_1()}
Weak Rules: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
We apply the weight gap principle, strictly orienting the rules
{revtl^#(nil(), ys) -> c_1()}
and weakly orienting the rules
{rev^#(xs) -> c_0(revtl^#(xs, nil()))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{revtl^#(nil(), ys) -> c_1()}
Details:
Interpretation Functions:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
rev^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
revtl^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1() = [0]
c_2(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:
{ revtl^#(nil(), ys) -> c_1()
, rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
The given problem does not contain any strict rules
4) {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
rev^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
revtl^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1() = [0]
c_2(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: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{rev^#(xs) -> c_0(revtl^#(xs, nil()))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
Interpretation Functions:
rev(x1) = [0] x1 + [0]
revtl(x1, x2) = [0] x1 + [0] x2 + [0]
nil() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
rev^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
revtl^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1() = [0]
c_2(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: {rev^#(xs) -> c_0(revtl^#(xs, nil()))}
Details:
The given problem does not contain any strict rules