'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__f(f(a())) -> a__f(g(f(a())))
, mark(f(X)) -> a__f(X)
, mark(a()) -> a()
, mark(g(X)) -> g(mark(X))
, a__f(X) -> f(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(a()) -> c_2()
, mark^#(g(X)) -> c_3(mark^#(X))
, a__f^#(X) -> c_4()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
==> {a__f^#(X) -> c_4()}
{mark^#(f(X)) -> c_1(a__f^#(X))}
==> {a__f^#(X) -> c_4()}
{mark^#(f(X)) -> c_1(a__f^#(X))}
==> {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
{mark^#(g(X)) -> c_3(mark^#(X))}
==> {mark^#(g(X)) -> c_3(mark^#(X))}
{mark^#(g(X)) -> c_3(mark^#(X))}
==> {mark^#(a()) -> c_2()}
{mark^#(g(X)) -> c_3(mark^#(X))}
==> {mark^#(f(X)) -> c_1(a__f^#(X))}
We consider the following path(s):
1) { mark^#(g(X)) -> c_3(mark^#(X))
, mark^#(f(X)) -> c_1(a__f^#(X))
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, a__f^#(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:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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: {a__f^#(X) -> c_4()}
Weak Rules:
{ a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X) -> c_4()}
and weakly orienting the rules
{ a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X) -> c_4()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [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:
{ a__f^#(X) -> c_4()
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))
, mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The given problem does not contain any strict rules
2) { mark^#(g(X)) -> c_3(mark^#(X))
, mark^#(f(X)) -> c_1(a__f^#(X))
, a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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: {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(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: {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {a__f^#(f(a())) -> c_0(a__f^#(g(f(a()))))}
Weak Rules:
{ mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 2
, f_0(3) -> 2
, f_0(4) -> 2
, f_1(12) -> 11
, a_0() -> 3
, a_1() -> 12
, g_0(2) -> 4
, g_0(3) -> 4
, g_0(4) -> 4
, g_1(11) -> 10
, a__f^#_0(2) -> 6
, a__f^#_0(3) -> 6
, a__f^#_0(4) -> 6
, a__f^#_1(10) -> 9
, c_0_1(9) -> 6
, mark^#_0(2) -> 8
, mark^#_0(3) -> 8
, mark^#_0(4) -> 8
, c_1_0(6) -> 8
, c_3_0(8) -> 8}
3) { mark^#(g(X)) -> c_3(mark^#(X))
, mark^#(f(X)) -> c_1(a__f^#(X))
, a__f^#(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:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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: {a__f^#(X) -> c_4()}
Weak Rules:
{ mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X) -> c_4()}
and weakly orienting the rules
{ mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X) -> c_4()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [15]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [1]
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:
{ a__f^#(X) -> c_4()
, mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The given problem does not contain any strict rules
4) { mark^#(g(X)) -> c_3(mark^#(X))
, mark^#(a()) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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: {mark^#(a()) -> c_2()}
Weak Rules: {mark^#(g(X)) -> c_3(mark^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(a()) -> c_2()}
and weakly orienting the rules
{mark^#(g(X)) -> c_3(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(a()) -> c_2()}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [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:
{ mark^#(a()) -> c_2()
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The given problem does not contain any strict rules
5) { mark^#(g(X)) -> c_3(mark^#(X))
, mark^#(f(X)) -> c_1(a__f^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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: {mark^#(f(X)) -> c_1(a__f^#(X))}
Weak Rules: {mark^#(g(X)) -> c_3(mark^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_1(a__f^#(X))}
and weakly orienting the rules
{mark^#(g(X)) -> c_3(mark^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_1(a__f^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [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:
{ mark^#(f(X)) -> c_1(a__f^#(X))
, mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The given problem does not contain any strict rules
6) {mark^#(g(X)) -> c_3(mark^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [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: {mark^#(g(X)) -> c_3(mark^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(g(X)) -> c_3(mark^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(g(X)) -> c_3(mark^#(X))}
Details:
Interpretation Functions:
a__f(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [8]
mark(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [1] x1 + [3]
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: {mark^#(g(X)) -> c_3(mark^#(X))}
Details:
The given problem does not contain any strict rules