'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(a, empty()) -> g(a, empty())
, f(a, cons(x, k)) -> f(cons(x, a), k)
, g(empty(), d) -> d
, g(cons(x, k), d) -> g(k, cons(x, d))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))
, g^#(empty(), d) -> c_2()
, g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{f^#(a, empty()) -> c_0(g^#(a, empty()))}
==> {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
{f^#(a, empty()) -> c_0(g^#(a, empty()))}
==> {g^#(empty(), d) -> c_2()}
{f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
==> {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
{f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
==> {f^#(a, empty()) -> c_0(g^#(a, empty()))}
{g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
==> {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
{g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
==> {g^#(empty(), d) -> c_2()}
We consider the following path(s):
1) { f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(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: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
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: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
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: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
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: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
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: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
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: {g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [4]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [3]
f^#(x1, x2) = [7] x1 + [7] x2 + [0]
c_0(x1) = [1] x1 + [6]
g^#(x1, x2) = [4] x1 + [3] x2 + [7]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
2) {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(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: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
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: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
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: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
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: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
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: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
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: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Weak Rules: {}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [5]
f^#(x1, x2) = [1] x1 + [2] x2 + [3]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
3) { f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))
, g^#(empty(), d) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(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: {g^#(empty(), d) -> c_2()}
Weak Rules:
{ g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(empty(), d) -> c_2()}
and weakly orienting the rules
{ g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(empty(), d) -> c_2()}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
g^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(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:
{ g^#(empty(), d) -> c_2()
, g^#(cons(x, k), d) -> c_3(g^#(k, cons(x, d)))
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
The given problem does not contain any strict rules
4) { f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, g^#(empty(), d) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(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: {g^#(empty(), d) -> c_2()}
Weak Rules:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(empty(), d) -> c_2()}
and weakly orienting the rules
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(empty(), d) -> c_2()}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
g^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(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:
{ g^#(empty(), d) -> c_2()
, f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
The given problem does not contain any strict rules
5) { f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))
, f^#(a, empty()) -> c_0(g^#(a, empty()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(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: {f^#(a, empty()) -> c_0(g^#(a, empty()))}
Weak Rules: {f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(a, empty()) -> c_0(g^#(a, empty()))}
and weakly orienting the rules
{f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(a, empty()) -> c_0(g^#(a, empty()))}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
empty() = [0]
g(x1, x2) = [0] x1 + [0] x2 + [0]
cons(x1, x2) = [1] x1 + [1] x2 + [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
g^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(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:
{ f^#(a, empty()) -> c_0(g^#(a, empty()))
, f^#(a, cons(x, k)) -> c_1(f^#(cons(x, a), k))}
Details:
The given problem does not contain any strict rules