'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(g(x), s(0()), y) -> f(g(s(0())), y, g(x))
, g(s(x)) -> s(g(x))
, g(0()) -> 0()}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))
, g^#(s(x)) -> c_1(g^#(x))
, g^#(0()) -> c_2()}
The usable rules are:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()}
The estimated dependency graph contains the following edges:
{f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))}
==> {f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))}
{g^#(s(x)) -> c_1(g^#(x))}
==> {g^#(0()) -> c_2()}
{g^#(s(x)) -> c_1(g^#(x))}
==> {g^#(s(x)) -> c_1(g^#(x))}
We consider the following path(s):
1) {f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))}
The usable rules for this path are the following:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()}
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:
{ g(s(x)) -> s(g(x))
, g(0()) -> 0()
, f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g(0()) -> 0()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(0()) -> 0()}
Details:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [0]
c_0(x1) = [1] x1 + [1]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ g(s(x)) -> s(g(x))
, f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))}
Weak Rules: {g(0()) -> 0()}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost relative runtime-complexity with respect to
Strict Rules:
{ g(s(x)) -> s(g(x))
, f^#(g(x), s(0()), y) -> c_0(f^#(g(s(0())), y, g(x)))}
Weak Rules: {g(0()) -> 0()}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ s_0(2) -> 2
, 0_0() -> 2
, f^#_0(2, 2, 2) -> 1}
2) { g^#(s(x)) -> c_1(g^#(x))
, g^#(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:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [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: {g^#(0()) -> c_2()}
Weak Rules: {g^#(s(x)) -> c_1(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(0()) -> c_2()}
and weakly orienting the rules
{g^#(s(x)) -> c_1(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(0()) -> c_2()}
Details:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [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:
{ g^#(0()) -> c_2()
, g^#(s(x)) -> c_1(g^#(x))}
Details:
The given problem does not contain any strict rules
3) {g^#(s(x)) -> c_1(g^#(x))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [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: {g^#(s(x)) -> c_1(g^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(s(x)) -> c_1(g^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(s(x)) -> c_1(g^#(x))}
Details:
Interpretation Functions:
f(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
g(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [8]
0() = [0]
f^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_0(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [3]
c_2() = [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: {g^#(s(x)) -> c_1(g^#(x))}
Details:
The given problem does not contain any strict rules