'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__c() -> a__f(g(c()))
, a__f(g(X)) -> g(X)
, mark(c()) -> a__c()
, mark(f(X)) -> a__f(X)
, mark(g(X)) -> g(X)
, a__c() -> c()
, a__f(X) -> f(X)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__c^#() -> c_0(a__f^#(g(c())))
, a__f^#(g(X)) -> c_1()
, mark^#(c()) -> c_2(a__c^#())
, mark^#(f(X)) -> c_3(a__f^#(X))
, mark^#(g(X)) -> c_4()
, a__c^#() -> c_5()
, a__f^#(X) -> c_6()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{a__c^#() -> c_0(a__f^#(g(c())))}
==> {a__f^#(X) -> c_6()}
{a__c^#() -> c_0(a__f^#(g(c())))}
==> {a__f^#(g(X)) -> c_1()}
{mark^#(c()) -> c_2(a__c^#())}
==> {a__c^#() -> c_5()}
{mark^#(c()) -> c_2(a__c^#())}
==> {a__c^#() -> c_0(a__f^#(g(c())))}
{mark^#(f(X)) -> c_3(a__f^#(X))}
==> {a__f^#(X) -> c_6()}
{mark^#(f(X)) -> c_3(a__f^#(X))}
==> {a__f^#(g(X)) -> c_1()}
We consider the following path(s):
1) { mark^#(c()) -> c_2(a__c^#())
, a__c^#() -> c_0(a__f^#(g(c())))
, a__f^#(g(X)) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(g(X)) -> c_1()}
Weak Rules:
{ a__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__f^#(g(X)) -> c_1()}
and weakly orienting the rules
{ a__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(g(X)) -> c_1()}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [1]
c_0(x1) = [1] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(g(X)) -> c_1()
, a__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
Details:
The given problem does not contain any strict rules
2) { mark^#(c()) -> c_2(a__c^#())
, a__c^#() -> c_0(a__f^#(g(c())))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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__c^#() -> c_0(a__f^#(g(c())))}
Weak Rules: {mark^#(c()) -> c_2(a__c^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__c^#() -> c_0(a__f^#(g(c())))}
and weakly orienting the rules
{mark^#(c()) -> c_2(a__c^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#() -> c_0(a__f^#(g(c())))}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [8]
c_0(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
Details:
The given problem does not contain any strict rules
3) { mark^#(c()) -> c_2(a__c^#())
, a__c^#() -> c_0(a__f^#(g(c())))
, a__f^#(X) -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_6()}
Weak Rules:
{ a__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X) -> c_6()}
and weakly orienting the rules
{ a__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X) -> c_6()}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [9]
c_0(x1) = [1] x1 + [1]
a__f^#(x1) = [1] x1 + [8]
c_1() = [0]
mark^#(x1) = [1] x1 + [9]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_6()
, a__c^#() -> c_0(a__f^#(g(c())))
, mark^#(c()) -> c_2(a__c^#())}
Details:
The given problem does not contain any strict rules
4) { mark^#(f(X)) -> c_3(a__f^#(X))
, a__f^#(g(X)) -> c_1()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(g(X)) -> c_1()}
Weak Rules: {mark^#(f(X)) -> c_3(a__f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__f^#(g(X)) -> c_1()}
and weakly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(g(X)) -> c_1()}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(g(X)) -> c_1()
, mark^#(f(X)) -> c_3(a__f^#(X))}
Details:
The given problem does not contain any strict rules
5) { mark^#(c()) -> c_2(a__c^#())
, a__c^#() -> c_5()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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__c^#() -> c_5()}
Weak Rules: {mark^#(c()) -> c_2(a__c^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__c^#() -> c_5()}
and weakly orienting the rules
{mark^#(c()) -> c_2(a__c^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#() -> c_5()}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [1]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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__c^#() -> c_5()
, mark^#(c()) -> c_2(a__c^#())}
Details:
The given problem does not contain any strict rules
6) {mark^#(c()) -> c_2(a__c^#())}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(c()) -> c_2(a__c^#())}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(c()) -> c_2(a__c^#())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(c()) -> c_2(a__c^#())}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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^#(c()) -> c_2(a__c^#())}
Details:
The given problem does not contain any strict rules
7) { mark^#(f(X)) -> c_3(a__f^#(X))
, a__f^#(X) -> c_6()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_6()}
Weak Rules: {mark^#(f(X)) -> c_3(a__f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__f^#(X) -> c_6()}
and weakly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__f^#(X) -> c_6()}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [1] x1 + [1]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_6()
, mark^#(f(X)) -> c_3(a__f^#(X))}
Details:
The given problem does not contain any strict rules
8) {mark^#(f(X)) -> c_3(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__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_3(a__f^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(f(X)) -> c_3(a__f^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(f(X)) -> c_3(a__f^#(X))}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [1] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [1] x1 + [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_3(a__f^#(X))}
Details:
The given problem does not contain any strict rules
9) {mark^#(g(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__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_4()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(g(X)) -> c_4()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(g(X)) -> c_4()}
Details:
Interpretation Functions:
a__c() = [0]
a__f(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
c() = [0]
mark(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
a__c^#() = [0]
c_0(x1) = [0] x1 + [0]
a__f^#(x1) = [0] x1 + [0]
c_1() = [0]
mark^#(x1) = [1] x1 + [1]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
c_6() = [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_4()}
Details:
The given problem does not contain any strict rules