'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ +(0(), y) -> y
, +(s(x), y) -> s(+(x, y))
, +(s(x), y) -> +(x, s(y))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ +^#(0(), y) -> c_0()
, +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{+^#(s(x), y) -> c_1(+^#(x, y))}
==> {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
{+^#(s(x), y) -> c_1(+^#(x, y))}
==> {+^#(s(x), y) -> c_1(+^#(x, y))}
{+^#(s(x), y) -> c_1(+^#(x, y))}
==> {+^#(0(), y) -> c_0()}
{+^#(s(x), y) -> c_2(+^#(x, s(y)))}
==> {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
{+^#(s(x), y) -> c_2(+^#(x, s(y)))}
==> {+^#(s(x), y) -> c_1(+^#(x, y))}
{+^#(s(x), y) -> c_2(+^#(x, s(y)))}
==> {+^#(0(), y) -> c_0()}
We consider the following path(s):
1) { +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [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:
{ +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{+^#(s(x), y) -> c_1(+^#(x, y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{+^#(s(x), y) -> c_1(+^#(x, y))}
Details:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [8]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [8]
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: {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
Weak Rules: {+^#(s(x), y) -> c_1(+^#(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: {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
Weak Rules: {+^#(s(x), y) -> c_1(+^#(x, y))}
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_2(+^#(x, s(y)))}
Weak Rules: {+^#(s(x), y) -> c_1(+^#(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: {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
Weak Rules: {+^#(s(x), y) -> c_1(+^#(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: {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
Weak Rules: {+^#(s(x), y) -> c_1(+^#(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: {+^#(s(x), y) -> c_2(+^#(x, s(y)))}
Weak Rules: {+^#(s(x), y) -> c_1(+^#(x, y))}
Details:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [5]
+^#(x1, x2) = [6] x1 + [4] x2 + [2]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [5]
2) { +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))
, +^#(0(), y) -> c_0()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [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: {+^#(0(), y) -> c_0()}
Weak Rules:
{ +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{+^#(0(), y) -> c_0()}
and weakly orienting the rules
{ +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{+^#(0(), y) -> c_0()}
Details:
Interpretation Functions:
+(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [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:
{ +^#(0(), y) -> c_0()
, +^#(s(x), y) -> c_1(+^#(x, y))
, +^#(s(x), y) -> c_2(+^#(x, s(y)))}
Details:
The given problem does not contain any strict rules