'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()
, f(s(x)) -> f(id(x))
, f(0()) -> 0()}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ id^#(s(x)) -> c_0(id^#(x))
, id^#(0()) -> c_1()
, f^#(s(x)) -> c_2(f^#(id(x)))
, f^#(0()) -> c_3()}
The usable rules are:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
The estimated dependency graph contains the following edges:
{id^#(s(x)) -> c_0(id^#(x))}
==> {id^#(0()) -> c_1()}
{id^#(s(x)) -> c_0(id^#(x))}
==> {id^#(s(x)) -> c_0(id^#(x))}
{f^#(s(x)) -> c_2(f^#(id(x)))}
==> {f^#(0()) -> c_3()}
{f^#(s(x)) -> c_2(f^#(id(x)))}
==> {f^#(s(x)) -> c_2(f^#(id(x)))}
We consider the following path(s):
1) { f^#(s(x)) -> c_2(f^#(id(x)))
, f^#(0()) -> c_3()}
The usable rules for this path are the following:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()
, f^#(s(x)) -> c_2(f^#(id(x)))
, f^#(0()) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ id(0()) -> 0()
, f^#(0()) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ id(0()) -> 0()
, f^#(0()) -> c_3()}
Details:
Interpretation Functions:
0() = [0]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
id(x1) = [1] x1 + [1]
id^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(s(x)) -> c_2(f^#(id(x)))}
and weakly orienting the rules
{ id(0()) -> 0()
, f^#(0()) -> c_3()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(x)) -> c_2(f^#(id(x)))}
Details:
Interpretation Functions:
0() = [9]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
id(x1) = [1] x1 + [1]
id^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [2]
c_3() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()
, f^#(0()) -> c_3()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()
, f^#(0()) -> c_3()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()
, f^#(0()) -> c_3()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()
, f^#(0()) -> c_3()}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()
, f^#(0()) -> c_3()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()
, f^#(0()) -> c_3()}
Details:
Interpretation Functions:
0() = [0]
[1]
[1]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [1]
[0 0 1] [1]
id(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
id^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
f^#(x1) = [1 1 0] x1 + [1]
[0 0 0] [1]
[0 1 0] [1]
c_2(x1) = [1 0 0] x1 + [1]
[0 0 0] [1]
[0 0 0] [1]
c_3() = [0]
[0]
[0]
2) {f^#(s(x)) -> c_2(f^#(id(x)))}
The usable rules for this path are the following:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ id(s(x)) -> s(id(x))
, id(0()) -> 0()
, f^#(s(x)) -> c_2(f^#(id(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{id(0()) -> 0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{id(0()) -> 0()}
Details:
Interpretation Functions:
0() = [0]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
id(x1) = [1] x1 + [1]
id^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(s(x)) -> c_2(f^#(id(x)))}
and weakly orienting the rules
{id(0()) -> 0()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(x)) -> c_2(f^#(id(x)))}
Details:
Interpretation Functions:
0() = [1]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
id(x1) = [1] x1 + [1]
id^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [2]
c_3() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()}
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^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {id(s(x)) -> s(id(x))}
Weak Rules:
{ f^#(s(x)) -> c_2(f^#(id(x)))
, id(0()) -> 0()}
Details:
Interpretation Functions:
0() = [1]
[1]
[0]
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
s(x1) = [1 1 0] x1 + [0]
[0 0 1] [1]
[0 0 1] [1]
id(x1) = [1 0 1] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
id^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1() = [0]
[0]
[0]
f^#(x1) = [1 1 0] x1 + [1]
[0 1 0] [0]
[0 0 0] [1]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 1] [0]
[0 0 0] [1]
c_3() = [0]
[0]
[0]
3) { id^#(s(x)) -> c_0(id^#(x))
, id^#(0()) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0() = [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {id^#(0()) -> c_1()}
Weak Rules: {id^#(s(x)) -> c_0(id^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{id^#(0()) -> c_1()}
and weakly orienting the rules
{id^#(s(x)) -> c_0(id^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{id^#(0()) -> c_1()}
Details:
Interpretation Functions:
0() = [0]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
id(x1) = [0] x1 + [0]
id^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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:
{ id^#(0()) -> c_1()
, id^#(s(x)) -> c_0(id^#(x))}
Details:
The given problem does not contain any strict rules
4) {id^#(s(x)) -> c_0(id^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
0() = [0]
f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
id(x1) = [0] x1 + [0]
id^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {id^#(s(x)) -> c_0(id^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{id^#(s(x)) -> c_0(id^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{id^#(s(x)) -> c_0(id^#(x))}
Details:
Interpretation Functions:
0() = [0]
f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
id(x1) = [0] x1 + [0]
id^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [3]
c_1() = [0]
f^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {id^#(s(x)) -> c_0(id^#(x))}
Details:
The given problem does not contain any strict rules