'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ h(x, c(y, z)) -> h(c(s(y), x), z)
, h(c(s(x), c(s(0()), y)), z) -> h(y, c(s(0()), c(x, z)))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))
, h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
==> {h^#(c(s(x), c(s(0()), y)), z) ->
c_1(h^#(y, c(s(0()), c(x, z))))}
{h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
==> {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
==> {h^#(c(s(x), c(s(0()), y)), z) ->
c_1(h^#(y, c(s(0()), c(x, z))))}
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
==> {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
We consider the following path(s):
1) { h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))
, h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
h^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))
, h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Details:
Interpretation Functions:
h(x1, x2) = [0] x1 + [0] x2 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [8]
0() = [0]
h^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] 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 DP runtime-complexity with respect to
Strict Rules: {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
Weak Rules:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
Weak Rules:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
Weak Rules:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
Weak Rules:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
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 DP runtime-complexity with respect to
Strict Rules: {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
Weak Rules:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {h^#(x, c(y, z)) -> c_0(h^#(c(s(y), x), z))}
Weak Rules:
{h^#(c(s(x), c(s(0()), y)), z) -> c_1(h^#(y, c(s(0()), c(x, z))))}
Details:
Interpretation Functions:
h(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]
c(x1, x2) = [1 0 0] x1 + [1 0 0] x2 + [0]
[0 1 1] [0 1 0] [1]
[0 0 0] [0 0 0] [0]
s(x1) = [0 1 1] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
0() = [0]
[1]
[1]
h^#(x1, x2) = [1 0 0] x1 + [0 1 0] x2 + [0]
[0 0 0] [0 0 0] [0]
[0 0 0] [0 0 0] [0]
c_0(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]