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