'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ average(s(x), y) -> average(x, s(y))
, average(x, s(s(s(y)))) -> s(average(s(x), y))
, average(0(), 0()) -> 0()
, average(0(), s(0())) -> 0()
, average(0(), s(s(0()))) -> s(0())}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
, average^#(0(), 0()) -> c_2()
, average^#(0(), s(0())) -> c_3()
, average^#(0(), s(s(0()))) -> c_4()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{average^#(s(x), y) -> c_0(average^#(x, s(y)))}
==> {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
{average^#(s(x), y) -> c_0(average^#(x, s(y)))}
==> {average^#(0(), s(s(0()))) -> c_4()}
{average^#(s(x), y) -> c_0(average^#(x, s(y)))}
==> {average^#(0(), s(0())) -> c_3()}
{average^#(s(x), y) -> c_0(average^#(x, s(y)))}
==> {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
{average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
==> {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
{average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
==> {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
We consider the following path(s):
1) { average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
average^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [8]
0() = [0]
average^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_0(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [0]
Finally we apply the subprocessor
'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: {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
Weak Rules: {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), 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: {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
Weak Rules: {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
Weak Rules: {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), 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: {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
Weak Rules: {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), 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: {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
Weak Rules: {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), 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: {average^#(s(x), y) -> c_0(average^#(x, s(y)))}
Weak Rules: {average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [1]
0() = [0]
average^#(x1, x2) = [4] x1 + [2] x2 + [2]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
c_2() = [0]
c_3() = [0]
c_4() = [0]
2) { average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
, average^#(0(), s(s(0()))) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
average^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {average^#(0(), s(s(0()))) -> c_4()}
Weak Rules:
{ average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{average^#(0(), s(s(0()))) -> c_4()}
and weakly orienting the rules
{ average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{average^#(0(), s(s(0()))) -> c_4()}
Details:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
average^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ average^#(0(), s(s(0()))) -> c_4()
, average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
The given problem does not contain any strict rules
3) { average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))
, average^#(0(), s(0())) -> c_3()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
average^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {average^#(0(), s(0())) -> c_3()}
Weak Rules:
{ average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{average^#(0(), s(0())) -> c_3()}
and weakly orienting the rules
{ average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{average^#(0(), s(0())) -> c_3()}
Details:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
average^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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:
{ average^#(0(), s(0())) -> c_3()
, average^#(s(x), y) -> c_0(average^#(x, s(y)))
, average^#(x, s(s(s(y)))) -> c_1(average^#(s(x), y))}
Details:
The given problem does not contain any strict rules
4) {average^#(0(), 0()) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
average^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {average^#(0(), 0()) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{average^#(0(), 0()) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{average^#(0(), 0()) -> c_2()}
Details:
Interpretation Functions:
average(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
average^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4() = [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: {average^#(0(), 0()) -> c_2()}
Details:
The given problem does not contain any strict rules