'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))
, f^#(X) -> c_1()
, a^#() -> c_2()
, g^#(X) -> c_3()
, activate^#(n__f(X)) -> c_4(f^#(X))
, activate^#(n__a()) -> c_5(a^#())
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, activate^#(X) -> c_7()}
The usable rules are:
{ activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)}
The estimated dependency graph contains the following edges:
{f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))}
==> {f^#(X) -> c_1()}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(X) -> c_1()}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))}
{activate^#(n__a()) -> c_5(a^#())}
==> {a^#() -> c_2()}
{activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
==> {g^#(X) -> c_3()}
We consider the following path(s):
1) { activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, g^#(X) -> c_3()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)}
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:
{ activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, g^#(X) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a() -> n__a()}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [1]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate(n__f(X)) -> f(X)}
and weakly orienting the rules
{ a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(n__f(X)) -> f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__a() = [1]
n__g(x1) = [1] x1 + [0]
a() = [1]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ activate(n__f(X)) -> f(X)
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [9]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g^#(X) -> c_3()}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(X) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [4]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [4]
g(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [9]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [3]
c_3() = [0]
activate^#(x1) = [1] x1 + [12]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [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:
{ activate(n__g(X)) -> g(activate(X))
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))}
Weak Rules:
{ g^#(X) -> c_3()
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
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:
{ activate(n__g(X)) -> g(activate(X))
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))}
Weak Rules:
{ g^#(X) -> c_3()
, g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 4
, f_1(2) -> 5
, f_1(7) -> 4
, f_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(2) -> 5
, n__f_1(7) -> 4
, n__f_1(7) -> 5
, n__f_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(8) -> 7
, a_0() -> 4
, a_1() -> 5
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, g^#_0(2) -> 1
, g^#_0(4) -> 3
, g^#_1(5) -> 6
, c_3_0() -> 1
, c_3_0() -> 3
, c_3_1() -> 6
, activate^#_0(2) -> 1
, c_6_0(3) -> 1
, c_6_1(6) -> 1}
2) {activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)}
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:
{ activate(n__f(X)) -> f(X)
, activate(n__a()) -> a()
, activate(n__g(X)) -> g(activate(X))
, activate(X) -> X
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))
, f(X) -> n__f(X)
, a() -> n__a()
, g(X) -> n__g(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{a() -> n__a()}
and weakly orienting the rules
{ activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a() -> n__a()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [1]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
and weakly orienting the rules
{ a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__g(X)) -> c_6(g^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate(n__f(X)) -> f(X)}
and weakly orienting the rules
{ activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(n__f(X)) -> f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [5]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [1]
c_7() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{g(X) -> n__g(X)}
and weakly orienting the rules
{ activate(n__f(X)) -> f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g(X) -> n__g(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [8]
g(x1) = [1] x1 + [9]
activate(x1) = [1] x1 + [8]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [1] x1 + [3]
c_3() = [0]
activate^#(x1) = [1] x1 + [13]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [0]
c_7() = [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:
{ activate(n__g(X)) -> g(activate(X))
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))}
Weak Rules:
{ g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
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:
{ activate(n__g(X)) -> g(activate(X))
, f(n__f(n__a())) -> f(n__g(n__f(n__a())))}
Weak Rules:
{ g(X) -> n__g(X)
, activate(n__f(X)) -> f(X)
, activate^#(n__g(X)) -> c_6(g^#(activate(X)))
, a() -> n__a()
, activate(n__a()) -> a()
, activate(X) -> X
, f(X) -> n__f(X)}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 4
, f_1(2) -> 5
, f_1(7) -> 4
, f_1(7) -> 5
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(2) -> 5
, n__f_1(7) -> 4
, n__f_1(7) -> 5
, n__f_1(9) -> 8
, n__a_0() -> 2
, n__a_0() -> 4
, n__a_0() -> 5
, n__a_1() -> 5
, n__a_1() -> 9
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(5) -> 4
, n__g_1(5) -> 5
, n__g_1(8) -> 7
, a_0() -> 4
, a_1() -> 5
, g_1(5) -> 4
, g_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, g^#_0(2) -> 1
, g^#_0(4) -> 3
, g^#_1(5) -> 6
, activate^#_0(2) -> 1
, c_6_0(3) -> 1
, c_6_1(6) -> 1}
3) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(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: {f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(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: {f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ n__f_0(2) -> 2
, n__f_1(6) -> 5
, n__a_0() -> 2
, n__a_1() -> 6
, n__g_0(2) -> 2
, n__g_1(5) -> 4
, f^#_0(2) -> 1
, f^#_1(4) -> 3
, c_0_1(3) -> 1
, activate^#_0(2) -> 1
, c_4_0(1) -> 1}
4) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))
, f^#(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:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {f^#(X) -> c_1()}
Weak Rules:
{ f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_1()}
and weakly orienting the rules
{ f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [1] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ f^#(X) -> c_1()
, f^#(n__f(n__a())) -> c_0(f^#(n__g(n__f(n__a()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
5) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(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:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {f^#(X) -> c_1()}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_1()}
and weakly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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:
{ f^#(X) -> c_1()
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
6) {activate^#(n__f(X)) -> c_4(f^#(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) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {activate^#(n__f(X)) -> c_4(f^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
7) {activate^#(n__a()) -> c_5(a^#())}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {activate^#(n__a()) -> c_5(a^#())}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__a()) -> c_5(a^#())}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__a()) -> c_5(a^#())}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {activate^#(n__a()) -> c_5(a^#())}
Details:
The given problem does not contain any strict rules
8) { activate^#(n__a()) -> c_5(a^#())
, 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:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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_2()}
Weak Rules: {activate^#(n__a()) -> c_5(a^#())}
Details:
We apply the weight gap principle, strictly orienting the rules
{a^#() -> c_2()}
and weakly orienting the rules
{activate^#(n__a()) -> c_5(a^#())}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{a^#() -> c_2()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [1]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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_2()
, activate^#(n__a()) -> c_5(a^#())}
Details:
The given problem does not contain any strict rules
9) {activate^#(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:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {activate^#(X) -> c_7()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_7()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_7()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__a() = [0]
n__g(x1) = [0] x1 + [0]
a() = [0]
g(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1() = [0]
a^#() = [0]
c_2() = [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7() = [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: {activate^#(X) -> c_7()}
Details:
The given problem does not contain any strict rules