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