'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a__g(X) -> a__h(X)
, a__c() -> d()
, a__h(d()) -> a__g(c())
, mark(g(X)) -> a__g(X)
, mark(h(X)) -> a__h(X)
, mark(c()) -> a__c()
, mark(d()) -> d()
, a__g(X) -> g(X)
, a__h(X) -> h(X)
, a__c() -> c()}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ a__g^#(X) -> c_0(a__h^#(X))
, a__c^#() -> c_1()
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))
, mark^#(h(X)) -> c_4(a__h^#(X))
, mark^#(c()) -> c_5(a__c^#())
, mark^#(d()) -> c_6()
, a__g^#(X) -> c_7()
, a__h^#(X) -> c_8()
, a__c^#() -> c_9()}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{a__g^#(X) -> c_0(a__h^#(X))}
==> {a__h^#(d()) -> c_2(a__g^#(c()))}
{a__g^#(X) -> c_0(a__h^#(X))}
==> {a__h^#(X) -> c_8()}
{a__h^#(d()) -> c_2(a__g^#(c()))}
==> {a__g^#(X) -> c_7()}
{a__h^#(d()) -> c_2(a__g^#(c()))}
==> {a__g^#(X) -> c_0(a__h^#(X))}
{mark^#(g(X)) -> c_3(a__g^#(X))}
==> {a__g^#(X) -> c_7()}
{mark^#(g(X)) -> c_3(a__g^#(X))}
==> {a__g^#(X) -> c_0(a__h^#(X))}
{mark^#(h(X)) -> c_4(a__h^#(X))}
==> {a__h^#(X) -> c_8()}
{mark^#(h(X)) -> c_4(a__h^#(X))}
==> {a__h^#(d()) -> c_2(a__g^#(c()))}
{mark^#(c()) -> c_5(a__c^#())}
==> {a__c^#() -> c_9()}
{mark^#(c()) -> c_5(a__c^#())}
==> {a__c^#() -> c_1()}
We consider the following path(s):
1) { mark^#(g(X)) -> c_3(a__g^#(X))
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__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__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))}
Weak Rules: {mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__g^#(X) -> c_0(a__h^#(X))}
and weakly orienting the rules
{mark^#(g(X)) -> c_3(a__g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__g^#(X) -> c_0(a__h^#(X))}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [8]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
a__h^#(x1) = [1] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h^#(d()) -> c_2(a__g^#(c()))}
and weakly orienting the rules
{ a__g^#(X) -> c_0(a__h^#(X))
, mark^#(g(X)) -> c_3(a__g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(d()) -> c_2(a__g^#(c()))}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [8]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [5]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a__h^#(x1) = [1] x1 + [1]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [12]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(d()) -> c_2(a__g^#(c()))
, a__g^#(X) -> c_0(a__h^#(X))
, mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
The given problem does not contain any strict rules
2) { mark^#(h(X)) -> c_4(a__h^#(X))
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__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__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))}
Weak Rules: {mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__g^#(X) -> c_0(a__h^#(X))}
and weakly orienting the rules
{mark^#(h(X)) -> c_4(a__h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__g^#(X) -> c_0(a__h^#(X))}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
a__g^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
a__h^#(x1) = [1] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [1]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a__h^#(d()) -> c_2(a__g^#(c()))}
and weakly orienting the rules
{ a__g^#(X) -> c_0(a__h^#(X))
, mark^#(h(X)) -> c_4(a__h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(d()) -> c_2(a__g^#(c()))}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [8]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [4]
a__g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
a__h^#(x1) = [1] x1 + [1]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [3]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(d()) -> c_2(a__g^#(c()))
, a__g^#(X) -> c_0(a__h^#(X))
, mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
The given problem does not contain any strict rules
3) { mark^#(g(X)) -> c_3(a__g^#(X))
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, a__g^#(X) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_7()}
Weak Rules:
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__g^#(X) -> c_7()}
and weakly orienting the rules
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__g^#(X) -> c_7()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [8]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [4]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
a__h^#(x1) = [1] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_7()
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
The given problem does not contain any strict rules
4) { mark^#(h(X)) -> c_4(a__h^#(X))
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, a__h^#(X) -> c_8()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(X) -> c_8()}
Weak Rules:
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_8()}
and weakly orienting the rules
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(h(X)) -> c_4(a__h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_8()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [8]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
a__g^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
a__h^#(x1) = [1] x1 + [8]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [4]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(X) -> c_8()
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
The given problem does not contain any strict rules
5) { mark^#(g(X)) -> c_3(a__g^#(X))
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, a__h^#(X) -> c_8()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(X) -> c_8()}
Weak Rules:
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_8()}
and weakly orienting the rules
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_8()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [8]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
a__h^#(x1) = [1] x1 + [8]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [4]
mark^#(x1) = [1] x1 + [13]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(X) -> c_8()
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
The given problem does not contain any strict rules
6) { mark^#(h(X)) -> c_4(a__h^#(X))
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, a__g^#(X) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_7()}
Weak Rules:
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__g^#(X) -> c_7()}
and weakly orienting the rules
{ a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(h(X)) -> c_4(a__h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__g^#(X) -> c_7()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [8]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [8]
a__g^#(x1) = [1] x1 + [5]
c_0(x1) = [1] x1 + [1]
a__h^#(x1) = [1] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
mark^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_7()
, a__g^#(X) -> c_0(a__h^#(X))
, a__h^#(d()) -> c_2(a__g^#(c()))
, mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
The given problem does not contain any strict rules
7) { mark^#(g(X)) -> c_3(a__g^#(X))
, a__g^#(X) -> c_7()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_7()}
Weak Rules: {mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__g^#(X) -> c_7()}
and weakly orienting the rules
{mark^#(g(X)) -> c_3(a__g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__g^#(X) -> c_7()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__g^#(X) -> c_7()
, mark^#(g(X)) -> c_3(a__g^#(X))}
Details:
The given problem does not contain any strict rules
8) { mark^#(c()) -> c_5(a__c^#())
, a__c^#() -> c_9()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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_9()}
Weak Rules: {mark^#(c()) -> c_5(a__c^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__c^#() -> c_9()}
and weakly orienting the rules
{mark^#(c()) -> c_5(a__c^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#() -> c_9()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [1]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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_9()
, mark^#(c()) -> c_5(a__c^#())}
Details:
The given problem does not contain any strict rules
9) {mark^#(g(X)) -> c_3(a__g^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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(a__g^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(g(X)) -> c_3(a__g^#(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(a__g^#(X))}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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(a__g^#(X))}
Details:
The given problem does not contain any strict rules
10)
{mark^#(h(X)) -> c_4(a__h^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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^#(h(X)) -> c_4(a__h^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(h(X)) -> c_4(a__h^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [1] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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^#(h(X)) -> c_4(a__h^#(X))}
Details:
The given problem does not contain any strict rules
11)
{ mark^#(h(X)) -> c_4(a__h^#(X))
, a__h^#(X) -> c_8()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(X) -> c_8()}
Weak Rules: {mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__h^#(X) -> c_8()}
and weakly orienting the rules
{mark^#(h(X)) -> c_4(a__h^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__h^#(X) -> c_8()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [1] x1 + [1]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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__h^#(X) -> c_8()
, mark^#(h(X)) -> c_4(a__h^#(X))}
Details:
The given problem does not contain any strict rules
12)
{ mark^#(c()) -> c_5(a__c^#())
, a__c^#() -> 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__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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_1()}
Weak Rules: {mark^#(c()) -> c_5(a__c^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a__c^#() -> c_1()}
and weakly orienting the rules
{mark^#(c()) -> c_5(a__c^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a__c^#() -> c_1()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [1]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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_1()
, mark^#(c()) -> c_5(a__c^#())}
Details:
The given problem does not contain any strict rules
13)
{mark^#(c()) -> c_5(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__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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_5(a__c^#())}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(c()) -> c_5(a__c^#())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(c()) -> c_5(a__c^#())}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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_5(a__c^#())}
Details:
The given problem does not contain any strict rules
14)
{mark^#(d()) -> 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__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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^#(d()) -> c_6()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{mark^#(d()) -> c_6()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{mark^#(d()) -> c_6()}
Details:
Interpretation Functions:
a__g(x1) = [0] x1 + [0]
a__h(x1) = [0] x1 + [0]
a__c() = [0]
d() = [0]
c() = [0]
mark(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
a__g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
a__h^#(x1) = [0] x1 + [0]
a__c^#() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
mark^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6() = [0]
c_7() = [0]
c_8() = [0]
c_9() = [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^#(d()) -> c_6()}
Details:
The given problem does not contain any strict rules