'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits(0()) -> 0()
, bits(s(x)) -> s(bits(half(s(x))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ half^#(0()) -> c_0()
, half^#(s(0())) -> c_1()
, half^#(s(s(x))) -> c_2(half^#(x))
, bits^#(0()) -> c_3()
, bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
The usable rules are:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
The estimated dependency graph contains the following edges:
{half^#(s(s(x))) -> c_2(half^#(x))}
==> {half^#(s(s(x))) -> c_2(half^#(x))}
{half^#(s(s(x))) -> c_2(half^#(x))}
==> {half^#(s(0())) -> c_1()}
{half^#(s(s(x))) -> c_2(half^#(x))}
==> {half^#(0()) -> c_0()}
{bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
==> {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
{bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
==> {bits^#(0()) -> c_3()}
We consider the following path(s):
1) {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
The usable rules for this path are the following:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
half(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [8]
bits(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
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: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
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: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
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: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
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: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
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: {bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
Details:
Interpretation Functions:
half(x1) = [0 1 0] x1 + [0]
[0 1 0] [0]
[0 0 1] [0]
0() = [0]
[0]
[1]
s(x1) = [0 1 1] x1 + [1]
[0 1 1] [0]
[0 0 0] [1]
bits(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
half^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_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]
bits^#(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 1] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [0]
[0 0 0] [1]
[0 0 1] [0]
2) { bits^#(s(x)) -> c_4(bits^#(half(s(x))))
, bits^#(0()) -> c_3()}
The usable rules for this path are the following:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
half(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [8]
bits(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {bits^#(0()) -> c_3()}
Weak Rules:
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{bits^#(0()) -> c_3()}
and weakly orienting the rules
{ half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{bits^#(0()) -> c_3()}
Details:
Interpretation Functions:
half(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
bits(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
bits^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(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:
{ bits^#(0()) -> c_3()
, half(0()) -> 0()
, half(s(0())) -> 0()
, half(s(s(x))) -> s(half(x))
, bits^#(s(x)) -> c_4(bits^#(half(s(x))))}
Details:
The given problem does not contain any strict rules
3) { half^#(s(s(x))) -> c_2(half^#(x))
, half^#(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:
half(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
bits(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {half^#(s(0())) -> c_1()}
Weak Rules: {half^#(s(s(x))) -> c_2(half^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{half^#(s(0())) -> c_1()}
and weakly orienting the rules
{half^#(s(s(x))) -> c_2(half^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{half^#(s(0())) -> c_1()}
Details:
Interpretation Functions:
half(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
bits(x1) = [0] x1 + [0]
half^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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:
{ half^#(s(0())) -> c_1()
, half^#(s(s(x))) -> c_2(half^#(x))}
Details:
The given problem does not contain any strict rules
4) { half^#(s(s(x))) -> c_2(half^#(x))
, half^#(0()) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
half(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
bits(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {half^#(0()) -> c_0()}
Weak Rules: {half^#(s(s(x))) -> c_2(half^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{half^#(0()) -> c_0()}
and weakly orienting the rules
{half^#(s(s(x))) -> c_2(half^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{half^#(0()) -> c_0()}
Details:
Interpretation Functions:
half(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
bits(x1) = [0] x1 + [0]
half^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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:
{ half^#(0()) -> c_0()
, half^#(s(s(x))) -> c_2(half^#(x))}
Details:
The given problem does not contain any strict rules
5) {half^#(s(s(x))) -> c_2(half^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
half(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
bits(x1) = [0] x1 + [0]
half^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {half^#(s(s(x))) -> c_2(half^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{half^#(s(s(x))) -> c_2(half^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{half^#(s(s(x))) -> c_2(half^#(x))}
Details:
Interpretation Functions:
half(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [8]
bits(x1) = [0] x1 + [0]
half^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [11]
bits^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {half^#(s(s(x))) -> c_2(half^#(x))}
Details:
The given problem does not contain any strict rules