'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ double(0()) -> 0()
, double(s(x)) -> s(s(double(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, +(s(x), y) -> s(+(x, y))
, double(x) -> +(x, x)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ double^#(0()) -> c_0()
, double^#(s(x)) -> c_1(double^#(x))
, +^#(x, 0()) -> c_2()
, +^#(x, s(y)) -> c_3(+^#(x, y))
, +^#(s(x), y) -> c_4(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{double^#(s(x)) -> c_1(double^#(x))}
==> {double^#(x) -> c_5(+^#(x, x))}
{double^#(s(x)) -> c_1(double^#(x))}
==> {double^#(s(x)) -> c_1(double^#(x))}
{double^#(s(x)) -> c_1(double^#(x))}
==> {double^#(0()) -> c_0()}
{+^#(x, s(y)) -> c_3(+^#(x, y))}
==> {+^#(s(x), y) -> c_4(+^#(x, y))}
{+^#(x, s(y)) -> c_3(+^#(x, y))}
==> {+^#(x, s(y)) -> c_3(+^#(x, y))}
{+^#(x, s(y)) -> c_3(+^#(x, y))}
==> {+^#(x, 0()) -> c_2()}
{+^#(s(x), y) -> c_4(+^#(x, y))}
==> {+^#(s(x), y) -> c_4(+^#(x, y))}
{+^#(s(x), y) -> c_4(+^#(x, y))}
==> {+^#(x, s(y)) -> c_3(+^#(x, y))}
{+^#(s(x), y) -> c_4(+^#(x, y))}
==> {+^#(x, 0()) -> c_2()}
{double^#(x) -> c_5(+^#(x, x))}
==> {+^#(s(x), y) -> c_4(+^#(x, y))}
{double^#(x) -> c_5(+^#(x, x))}
==> {+^#(x, s(y)) -> c_3(+^#(x, y))}
{double^#(x) -> c_5(+^#(x, x))}
==> {+^#(x, 0()) -> c_2()}
We consider the following path(s):
1) { double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))
, +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(s(x), y) -> c_4(+^#(x, y))}
Weak Rules:
{ +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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: {+^#(s(x), y) -> c_4(+^#(x, y))}
Weak Rules:
{ +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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: {+^#(s(x), y) -> c_4(+^#(x, y))}
Weak Rules:
{ +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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: {+^#(s(x), y) -> c_4(+^#(x, y))}
Weak Rules:
{ +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [5]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [5] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
+^#(x1, x2) = [4] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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: {+^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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: {+^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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: {+^#(x, s(y)) -> c_3(+^#(x, y))}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [3]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [4] x1 + [3]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [1] x2 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [2]
2) { double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))
, +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, +^#(x, 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:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ +^#(s(x), y) -> c_4(+^#(x, y))
, +^#(x, s(y)) -> c_3(+^#(x, y))
, double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [1] x1 + [0]
3) { double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))}
The usable rules for this path are empty.
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {double^#(s(x)) -> c_1(double^#(x))}
Weak Rules: {double^#(x) -> c_5(+^#(x, 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 relative runtime-complexity with respect to
Strict Rules: {double^#(s(x)) -> c_1(double^#(x))}
Weak Rules: {double^#(x) -> c_5(+^#(x, 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 relative runtime-complexity with respect to
Strict Rules: {double^#(s(x)) -> c_1(double^#(x))}
Weak Rules: {double^#(x) -> c_5(+^#(x, x))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {double^#(s(x)) -> c_1(double^#(x))}
Weak Rules: {double^#(x) -> c_5(+^#(x, x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [7]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [4] x1 + [5]
c_0() = [0]
c_1(x1) = [1] x1 + [4]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {double^#(x) -> c_5(+^#(x, x))}
Weak Rules: {double^#(s(x)) -> c_1(double^#(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 relative runtime-complexity with respect to
Strict Rules: {double^#(x) -> c_5(+^#(x, x))}
Weak Rules: {double^#(s(x)) -> c_1(double^#(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 relative runtime-complexity with respect to
Strict Rules: {double^#(x) -> c_5(+^#(x, x))}
Weak Rules: {double^#(s(x)) -> c_1(double^#(x))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules: {double^#(x) -> c_5(+^#(x, x))}
Weak Rules: {double^#(s(x)) -> c_1(double^#(x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [3]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [7] x1 + [4]
c_0() = [0]
c_1(x1) = [1] x1 + [2]
+^#(x1, x2) = [4] x1 + [2] x2 + [2]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
4) { double^#(s(x)) -> c_1(double^#(x))
, double^#(x) -> c_5(+^#(x, x))
, +^#(x, 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:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(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(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {+^#(x, 0()) -> c_2()}
Weak Rules:
{ double^#(x) -> c_5(+^#(x, x))
, double^#(s(x)) -> c_1(double^#(x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [7]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [4] x1 + [1]
5) { double^#(s(x)) -> c_1(double^#(x))
, double^#(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:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {double^#(0()) -> c_0()}
Weak Rules: {double^#(s(x)) -> c_1(double^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{double^#(0()) -> c_0()}
and weakly orienting the rules
{double^#(s(x)) -> c_1(double^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{double^#(0()) -> c_0()}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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:
{ double^#(0()) -> c_0()
, double^#(s(x)) -> c_1(double^#(x))}
Details:
The given problem does not contain any strict rules
6) {double^#(s(x)) -> c_1(double^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {double^#(s(x)) -> c_1(double^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{double^#(s(x)) -> c_1(double^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{double^#(s(x)) -> c_1(double^#(x))}
Details:
Interpretation Functions:
double(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [8]
+(x1, x2) = [0] x1 + [0] x2 + [0]
double^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [3]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {double^#(s(x)) -> c_1(double^#(x))}
Details:
The given problem does not contain any strict rules