'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)
, activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(X) -> c_0()
, h^#(X) -> c_1()
, f^#(X) -> c_2()
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate^#(X) -> c_5()}
The usable rules are:
{ activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)}
The estimated dependency graph contains the following edges:
{activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
==> {h^#(X) -> c_1()}
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
==> {f^#(X) -> c_2()}
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
==> {f^#(X) -> c_0()}
We consider the following path(s):
1) { activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, h^#(X) -> c_1()}
The usable rules for this path are the following:
{ activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, h^#(X) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h^#(X) -> c_1()}
and weakly orienting the rules
{activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(X) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [8]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
and weakly orienting the rules
{ h^#(X) -> c_1()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, h^#(X) -> c_1()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(X) -> n__h(X)}
and weakly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, h^#(X) -> c_1()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(X) -> n__h(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [9]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, h^#(X) -> c_1()
, activate(X) -> 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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, h^#(X) -> c_1()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(15) -> 14
, f_1(15) -> 15
, g_0(2) -> 2
, g_0(2) -> 14
, g_0(2) -> 15
, g_0(3) -> 2
, g_0(3) -> 14
, g_0(3) -> 15
, g_0(4) -> 2
, g_0(4) -> 14
, g_0(4) -> 15
, g_1(16) -> 14
, g_1(16) -> 15
, n__h_0(2) -> 3
, n__h_0(2) -> 14
, n__h_0(2) -> 15
, n__h_0(3) -> 3
, n__h_0(3) -> 14
, n__h_0(3) -> 15
, n__h_0(4) -> 3
, n__h_0(4) -> 14
, n__h_0(4) -> 15
, n__h_1(15) -> 14
, n__h_1(15) -> 15
, n__h_1(17) -> 16
, n__f_0(2) -> 4
, n__f_0(2) -> 14
, n__f_0(2) -> 15
, n__f_0(3) -> 4
, n__f_0(3) -> 14
, n__f_0(3) -> 15
, n__f_0(4) -> 4
, n__f_0(4) -> 14
, n__f_0(4) -> 15
, n__f_1(15) -> 14
, n__f_1(15) -> 15
, n__f_1(15) -> 17
, h_1(15) -> 14
, h_1(15) -> 15
, activate_0(2) -> 14
, activate_0(3) -> 14
, activate_0(4) -> 14
, activate_1(2) -> 15
, activate_1(3) -> 15
, activate_1(4) -> 15
, h^#_0(2) -> 9
, h^#_0(3) -> 9
, h^#_0(4) -> 9
, h^#_0(14) -> 13
, h^#_1(15) -> 18
, c_1_0() -> 9
, c_1_0() -> 13
, c_1_1() -> 18
, activate^#_0(2) -> 12
, activate^#_0(3) -> 12
, activate^#_0(4) -> 12
, c_3_0(13) -> 12
, c_3_1(18) -> 12}
2) { activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_2()}
The usable rules for this path are the following:
{ activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_2()}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_2()}
and weakly orienting the rules
{activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_2()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
and weakly orienting the rules
{ f^#(X) -> c_2()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_2()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(X) -> n__h(X)}
and weakly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_2()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(X) -> n__h(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [1]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_2()
, activate(X) -> 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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_2()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_0(2) -> 5
, g_1(6) -> 4
, g_1(6) -> 5
, n__h_0(2) -> 2
, n__h_0(2) -> 4
, n__h_0(2) -> 5
, n__h_1(5) -> 4
, n__h_1(5) -> 5
, n__h_1(7) -> 6
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(5) -> 7
, h_1(5) -> 4
, h_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 8
, c_2_0() -> 1
, c_2_0() -> 3
, c_2_1() -> 8
, activate^#_0(2) -> 1
, c_4_0(3) -> 1
, c_4_1(8) -> 1}
3) { activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_0()}
The usable rules for this path are the following:
{ activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_0()}
and weakly orienting the rules
{activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [8]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
and weakly orienting the rules
{ f^#(X) -> c_0()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_0()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(X) -> n__h(X)}
and weakly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_0()
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(X) -> n__h(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [1]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_0()
, activate(X) -> 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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, f^#(X) -> c_0()
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, g_0(2) -> 2
, g_0(2) -> 4
, g_0(2) -> 5
, g_1(6) -> 4
, g_1(6) -> 5
, n__h_0(2) -> 2
, n__h_0(2) -> 4
, n__h_0(2) -> 5
, n__h_1(5) -> 4
, n__h_1(5) -> 5
, n__h_1(7) -> 6
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 5
, n__f_1(5) -> 4
, n__f_1(5) -> 5
, n__f_1(5) -> 7
, h_1(5) -> 4
, h_1(5) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 8
, c_0_0() -> 1
, c_0_0() -> 3
, c_0_1() -> 8
, activate^#_0(2) -> 1
, c_4_0(3) -> 1
, c_4_1(8) -> 1}
4) {activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
and weakly orienting the rules
{activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__h(X)) -> c_3(h^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(X) -> n__h(X)}
and weakly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(X) -> n__h(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [4]
activate(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [1] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [5]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, activate(X) -> 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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__h(X)) -> c_3(h^#(activate(X)))
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(15) -> 14
, f_1(15) -> 15
, g_0(2) -> 2
, g_0(2) -> 14
, g_0(2) -> 15
, g_0(3) -> 2
, g_0(3) -> 14
, g_0(3) -> 15
, g_0(4) -> 2
, g_0(4) -> 14
, g_0(4) -> 15
, g_1(16) -> 14
, g_1(16) -> 15
, n__h_0(2) -> 3
, n__h_0(2) -> 14
, n__h_0(2) -> 15
, n__h_0(3) -> 3
, n__h_0(3) -> 14
, n__h_0(3) -> 15
, n__h_0(4) -> 3
, n__h_0(4) -> 14
, n__h_0(4) -> 15
, n__h_1(15) -> 14
, n__h_1(15) -> 15
, n__h_1(17) -> 16
, n__f_0(2) -> 4
, n__f_0(2) -> 14
, n__f_0(2) -> 15
, n__f_0(3) -> 4
, n__f_0(3) -> 14
, n__f_0(3) -> 15
, n__f_0(4) -> 4
, n__f_0(4) -> 14
, n__f_0(4) -> 15
, n__f_1(15) -> 14
, n__f_1(15) -> 15
, n__f_1(15) -> 17
, h_1(15) -> 14
, h_1(15) -> 15
, activate_0(2) -> 14
, activate_0(3) -> 14
, activate_0(4) -> 14
, activate_1(2) -> 15
, activate_1(3) -> 15
, activate_1(4) -> 15
, h^#_0(2) -> 9
, h^#_0(3) -> 9
, h^#_0(4) -> 9
, h^#_0(14) -> 13
, h^#_1(15) -> 18
, activate^#_0(2) -> 12
, activate^#_0(3) -> 12
, activate^#_0(4) -> 12
, c_3_0(13) -> 12
, c_3_1(18) -> 12}
5) {activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))
, activate(X) -> X
, f(X) -> g(n__h(n__f(X)))
, h(X) -> n__h(X)
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
and weakly orienting the rules
{activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [9]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
g(x1) = [1] x1 + [1]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h(X) -> n__h(X)}
and weakly orienting the rules
{ f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h(X) -> n__h(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
n__h(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
h(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [5]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
c_5() = [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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(X) -> 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__h(X)) -> h(activate(X))
, activate(n__f(X)) -> f(activate(X))}
Weak Rules:
{ h(X) -> n__h(X)
, f(X) -> g(n__h(n__f(X)))
, f(X) -> n__f(X)
, activate^#(n__f(X)) -> c_4(f^#(activate(X)))
, activate(X) -> X}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_1(15) -> 14
, f_1(15) -> 15
, g_0(2) -> 2
, g_0(2) -> 14
, g_0(2) -> 15
, g_0(3) -> 2
, g_0(3) -> 14
, g_0(3) -> 15
, g_0(4) -> 2
, g_0(4) -> 14
, g_0(4) -> 15
, g_1(16) -> 14
, g_1(16) -> 15
, n__h_0(2) -> 3
, n__h_0(2) -> 14
, n__h_0(2) -> 15
, n__h_0(3) -> 3
, n__h_0(3) -> 14
, n__h_0(3) -> 15
, n__h_0(4) -> 3
, n__h_0(4) -> 14
, n__h_0(4) -> 15
, n__h_1(15) -> 14
, n__h_1(15) -> 15
, n__h_1(17) -> 16
, n__f_0(2) -> 4
, n__f_0(2) -> 14
, n__f_0(2) -> 15
, n__f_0(3) -> 4
, n__f_0(3) -> 14
, n__f_0(3) -> 15
, n__f_0(4) -> 4
, n__f_0(4) -> 14
, n__f_0(4) -> 15
, n__f_1(15) -> 14
, n__f_1(15) -> 15
, n__f_1(15) -> 17
, h_1(15) -> 14
, h_1(15) -> 15
, activate_0(2) -> 14
, activate_0(3) -> 14
, activate_0(4) -> 14
, activate_1(2) -> 15
, activate_1(3) -> 15
, activate_1(4) -> 15
, f^#_0(2) -> 7
, f^#_0(3) -> 7
, f^#_0(4) -> 7
, f^#_0(14) -> 13
, f^#_1(15) -> 18
, activate^#_0(2) -> 12
, activate^#_0(3) -> 12
, activate^#_0(4) -> 12
, c_4_0(13) -> 12
, c_4_1(18) -> 12}
6) {activate^#(X) -> c_5()}
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]
g(x1) = [0] x1 + [0]
n__h(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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_5()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_5()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_5()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
n__h(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
h^#(x1) = [0] x1 + [0]
c_1() = [0]
c_2() = [0]
activate^#(x1) = [1] x1 + [4]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [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_5()}
Details:
The given problem does not contain any strict rules