'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, c(X) -> d(activate(X))
, h(X) -> c(n__d(X))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, f^#(X) -> c_3()
, g^#(X) -> c_4()
, d^#(X) -> c_5()
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate^#(n__g(X)) -> c_7(g^#(X))
, activate^#(n__d(X)) -> c_8(d^#(X))
, activate^#(X) -> c_9()}
The usable rules are:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))}
The estimated dependency graph contains the following edges:
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
==> {c^#(X) -> c_1(d^#(activate(X)))}
{c^#(X) -> c_1(d^#(activate(X)))}
==> {d^#(X) -> c_5()}
{h^#(X) -> c_2(c^#(n__d(X)))}
==> {c^#(X) -> c_1(d^#(activate(X)))}
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
==> {f^#(X) -> c_3()}
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
==> {f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
{activate^#(n__g(X)) -> c_7(g^#(X))}
==> {g^#(X) -> c_4()}
{activate^#(n__d(X)) -> c_8(d^#(X))}
==> {d^#(X) -> c_5()}
We consider the following path(s):
1) { activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, c^#(X) -> c_1(d^#(activate(X)))
, d^#(X) -> c_5()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, d^#(X) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
d^#(x1) = [1] x1 + [7]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [6]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(X) -> c_1(d^#(activate(X)))}
and weakly orienting the rules
{ f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(X) -> c_1(d^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [3]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [13]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [7]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [7]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ c(X) -> d(activate(X))
, d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [8]
f^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [13]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, d^#(X) -> c_5()}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(6) -> 4
, f_1(6) -> 6
, f_2(14) -> 12
, f_2(14) -> 18
, f_3(21) -> 18
, f_3(21) -> 22
, c_1(9) -> 4
, c_1(9) -> 6
, c_2(15) -> 4
, c_2(15) -> 6
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 6
, n__f_1(6) -> 4
, n__f_1(6) -> 6
, n__f_1(10) -> 9
, n__f_1(10) -> 12
, n__f_1(10) -> 18
, n__f_2(6) -> 17
, n__f_2(14) -> 12
, n__f_2(14) -> 18
, n__f_2(16) -> 15
, n__f_2(16) -> 18
, n__f_2(16) -> 22
, n__f_3(21) -> 18
, n__f_3(21) -> 22
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 6
, n__g_1(2) -> 6
, n__g_1(4) -> 10
, n__g_1(4) -> 14
, n__g_2(4) -> 14
, n__g_2(17) -> 16
, n__g_2(17) -> 21
, n__g_3(17) -> 21
, d_0(2) -> 4
, d_1(2) -> 6
, d_1(12) -> 4
, d_1(12) -> 6
, d_2(18) -> 4
, d_2(18) -> 6
, d_3(22) -> 4
, d_3(22) -> 6
, activate_0(2) -> 4
, activate_1(2) -> 6
, activate_1(9) -> 12
, activate_2(9) -> 18
, activate_2(10) -> 14
, activate_2(15) -> 18
, activate_3(15) -> 22
, activate_3(16) -> 21
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 6
, n__d_1(2) -> 6
, n__d_1(12) -> 4
, n__d_1(12) -> 6
, n__d_2(18) -> 4
, n__d_2(18) -> 6
, n__d_3(22) -> 4
, n__d_3(22) -> 6
, g_0(2) -> 4
, g_1(2) -> 6
, g_2(4) -> 14
, g_3(17) -> 21
, f^#_0(2) -> 1
, f^#_0(4) -> 5
, f^#_1(6) -> 11
, c_0_1(8) -> 5
, c_0_1(8) -> 11
, c_0_2(20) -> 11
, c^#_0(2) -> 1
, c^#_1(9) -> 8
, c^#_2(15) -> 20
, c_1_0(3) -> 1
, c_1_1(7) -> 1
, c_1_1(13) -> 8
, c_1_2(19) -> 8
, c_1_3(23) -> 20
, d^#_0(2) -> 1
, d^#_0(4) -> 3
, d^#_1(6) -> 7
, d^#_1(12) -> 13
, d^#_2(18) -> 19
, d^#_3(22) -> 23
, c_5_0() -> 1
, c_5_0() -> 3
, c_5_1() -> 7
, c_5_1() -> 13
, c_5_2() -> 19
, c_5_3() -> 23
, activate^#_0(2) -> 1
, c_6_0(5) -> 1
, c_6_1(11) -> 1}
2) { activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, c^#(X) -> c_1(d^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, c^#(X) -> c_1(d^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> 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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [4]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(X) -> c_1(d^#(activate(X)))}
and weakly orienting the rules
{ f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(X) -> c_1(d^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [9]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [13]
c_6(x1) = [1] x1 + [3]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [7]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ c(X) -> d(activate(X))
, d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [7]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, c^#(X) -> c_1(d^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(6) -> 4
, f_1(6) -> 6
, f_2(14) -> 12
, f_2(14) -> 18
, f_3(21) -> 18
, f_3(21) -> 22
, c_1(9) -> 4
, c_1(9) -> 6
, c_2(15) -> 4
, c_2(15) -> 6
, n__f_0(2) -> 2
, n__f_0(2) -> 4
, n__f_0(2) -> 6
, n__f_1(6) -> 4
, n__f_1(6) -> 6
, n__f_1(10) -> 9
, n__f_1(10) -> 12
, n__f_1(10) -> 18
, n__f_2(6) -> 17
, n__f_2(14) -> 12
, n__f_2(14) -> 18
, n__f_2(16) -> 15
, n__f_2(16) -> 18
, n__f_2(16) -> 22
, n__f_3(21) -> 18
, n__f_3(21) -> 22
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 6
, n__g_1(2) -> 6
, n__g_1(4) -> 10
, n__g_1(4) -> 14
, n__g_2(4) -> 14
, n__g_2(17) -> 16
, n__g_2(17) -> 21
, n__g_3(17) -> 21
, d_0(2) -> 4
, d_1(2) -> 6
, d_1(12) -> 4
, d_1(12) -> 6
, d_2(18) -> 4
, d_2(18) -> 6
, d_3(22) -> 4
, d_3(22) -> 6
, activate_0(2) -> 4
, activate_1(2) -> 6
, activate_1(9) -> 12
, activate_2(9) -> 18
, activate_2(10) -> 14
, activate_2(15) -> 18
, activate_3(15) -> 22
, activate_3(16) -> 21
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 6
, n__d_1(2) -> 6
, n__d_1(12) -> 4
, n__d_1(12) -> 6
, n__d_2(18) -> 4
, n__d_2(18) -> 6
, n__d_3(22) -> 4
, n__d_3(22) -> 6
, g_0(2) -> 4
, g_1(2) -> 6
, g_2(4) -> 14
, g_3(17) -> 21
, f^#_0(2) -> 1
, f^#_0(4) -> 5
, f^#_1(6) -> 11
, c_0_1(8) -> 5
, c_0_1(8) -> 11
, c_0_2(20) -> 11
, c^#_0(2) -> 1
, c^#_1(9) -> 8
, c^#_2(15) -> 20
, c_1_0(3) -> 1
, c_1_1(7) -> 1
, c_1_1(13) -> 8
, c_1_2(19) -> 8
, c_1_3(23) -> 20
, d^#_0(2) -> 1
, d^#_0(4) -> 3
, d^#_1(6) -> 7
, d^#_1(12) -> 13
, d^#_2(18) -> 19
, d^#_3(22) -> 23
, activate^#_0(2) -> 1
, c_6_0(5) -> 1
, c_6_1(11) -> 1}
3) { activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> 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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [4]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
and weakly orienting the rules
{ f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [7]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [7]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [7]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ c(X) -> d(activate(X))
, d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_0(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(f(X)) -> c_0(c^#(n__f(n__g(n__f(X)))))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, f_2(11) -> 10
, f_2(11) -> 15
, f_3(17) -> 15
, f_3(17) -> 18
, c_1(8) -> 4
, c_1(8) -> 5
, c_2(12) -> 4
, c_2(12) -> 5
, 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(9) -> 8
, n__f_1(9) -> 10
, n__f_1(9) -> 15
, n__f_2(5) -> 14
, n__f_2(11) -> 10
, n__f_2(11) -> 15
, n__f_2(13) -> 12
, n__f_2(13) -> 15
, n__f_2(13) -> 18
, n__f_3(17) -> 15
, n__f_3(17) -> 18
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(2) -> 5
, n__g_1(4) -> 9
, n__g_1(4) -> 11
, n__g_2(4) -> 11
, n__g_2(14) -> 13
, n__g_2(14) -> 17
, n__g_3(14) -> 17
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(10) -> 4
, d_1(10) -> 5
, d_2(15) -> 4
, d_2(15) -> 5
, d_3(18) -> 4
, d_3(18) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, activate_1(8) -> 10
, activate_2(8) -> 15
, activate_2(9) -> 11
, activate_2(12) -> 15
, activate_3(12) -> 18
, activate_3(13) -> 17
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 5
, n__d_1(2) -> 5
, n__d_1(10) -> 4
, n__d_1(10) -> 5
, n__d_2(15) -> 4
, n__d_2(15) -> 5
, n__d_3(18) -> 4
, n__d_3(18) -> 5
, g_0(2) -> 4
, g_1(2) -> 5
, g_2(4) -> 11
, g_3(14) -> 17
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 6
, c_0_1(7) -> 3
, c_0_1(7) -> 6
, c_0_2(16) -> 6
, c^#_0(2) -> 1
, c^#_1(8) -> 7
, c^#_2(12) -> 16
, activate^#_0(2) -> 1
, c_6_0(3) -> 1
, c_6_1(6) -> 1}
4) { h^#(X) -> c_2(c^#(n__d(X)))
, c^#(X) -> c_1(d^#(activate(X)))
, d^#(X) -> c_5()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, d^#(X) -> c_5()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [2]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> 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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [2]
g(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h^#(X) -> c_2(c^#(n__d(X)))}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(X) -> c_2(c^#(n__d(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [3]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(X) -> c_1(d^#(activate(X)))}
and weakly orienting the rules
{ h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(X) -> c_1(d^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d^#(X) -> c_5()}
and weakly orienting the rules
{ c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(X) -> c_5()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
h^#(x1) = [1] x1 + [14]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ d^#(X) -> c_5()
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ d(X) -> n__d(X)
, d^#(X) -> c_5()
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [12]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [8]
h^#(x1) = [1] x1 + [14]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ c(X) -> d(activate(X))
, d(X) -> n__d(X)
, d^#(X) -> c_5()
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, d^#(X) -> c_5()
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, d(X) -> n__d(X)
, d^#(X) -> c_5()
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, f_2(14) -> 11
, f_2(14) -> 18
, f_3(21) -> 18
, f_3(21) -> 22
, c_1(9) -> 4
, c_1(9) -> 5
, c_2(15) -> 4
, c_2(15) -> 5
, 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(10) -> 9
, n__f_1(10) -> 11
, n__f_1(10) -> 18
, n__f_2(5) -> 17
, n__f_2(14) -> 11
, n__f_2(14) -> 18
, n__f_2(16) -> 15
, n__f_2(16) -> 18
, n__f_2(16) -> 22
, n__f_3(21) -> 18
, n__f_3(21) -> 22
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(2) -> 5
, n__g_1(4) -> 10
, n__g_1(4) -> 14
, n__g_2(4) -> 14
, n__g_2(17) -> 16
, n__g_2(17) -> 21
, n__g_3(17) -> 21
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(2) -> 13
, d_1(11) -> 4
, d_1(11) -> 5
, d_2(2) -> 20
, d_2(18) -> 4
, d_2(18) -> 5
, d_3(22) -> 4
, d_3(22) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, activate_1(8) -> 13
, activate_1(9) -> 11
, activate_2(8) -> 20
, activate_2(9) -> 18
, activate_2(10) -> 14
, activate_2(15) -> 18
, activate_3(15) -> 22
, activate_3(16) -> 21
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 5
, n__d_1(2) -> 5
, n__d_1(2) -> 8
, n__d_1(2) -> 13
, n__d_1(2) -> 20
, n__d_1(11) -> 4
, n__d_1(11) -> 5
, n__d_2(2) -> 20
, n__d_2(18) -> 4
, n__d_2(18) -> 5
, n__d_3(22) -> 4
, n__d_3(22) -> 5
, g_0(2) -> 4
, g_1(2) -> 5
, g_2(4) -> 14
, g_3(17) -> 21
, c^#_0(2) -> 1
, c^#_1(8) -> 7
, c_1_0(3) -> 1
, c_1_1(6) -> 1
, c_1_1(12) -> 7
, c_1_2(19) -> 7
, d^#_0(2) -> 1
, d^#_0(4) -> 3
, d^#_1(5) -> 6
, d^#_1(13) -> 12
, d^#_2(20) -> 19
, h^#_0(2) -> 1
, c_2_0(1) -> 1
, c_2_1(7) -> 1
, c_5_0() -> 1
, c_5_0() -> 3
, c_5_1() -> 6
, c_5_1() -> 12
, c_5_2() -> 19}
5) { h^#(X) -> c_2(c^#(n__d(X)))
, c^#(X) -> c_1(d^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, h^#(X) -> c_2(c^#(n__d(X)))
, c^#(X) -> c_1(d^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> 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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{h^#(X) -> c_2(c^#(n__d(X)))}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(X) -> c_2(c^#(n__d(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(X) -> c_1(d^#(activate(X)))}
and weakly orienting the rules
{ h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(X) -> c_1(d^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [9]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [8]
n__f(x1) = [1] x1 + [1]
n__g(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [1]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ c(X) -> d(activate(X))
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [1]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ d(X) -> n__d(X)
, c(X) -> d(activate(X))
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [1] x1 + [12]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, c^#(X) -> c_1(d^#(activate(X)))
, h^#(X) -> c_2(c^#(n__d(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, f_2(14) -> 11
, f_2(14) -> 18
, f_3(21) -> 18
, f_3(21) -> 22
, c_1(9) -> 4
, c_1(9) -> 5
, c_2(15) -> 4
, c_2(15) -> 5
, 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(10) -> 9
, n__f_1(10) -> 11
, n__f_1(10) -> 18
, n__f_2(5) -> 17
, n__f_2(14) -> 11
, n__f_2(14) -> 18
, n__f_2(16) -> 15
, n__f_2(16) -> 18
, n__f_2(16) -> 22
, n__f_3(21) -> 18
, n__f_3(21) -> 22
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(2) -> 5
, n__g_1(4) -> 10
, n__g_1(4) -> 14
, n__g_2(4) -> 14
, n__g_2(17) -> 16
, n__g_2(17) -> 21
, n__g_3(17) -> 21
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(2) -> 13
, d_1(11) -> 4
, d_1(11) -> 5
, d_2(2) -> 20
, d_2(18) -> 4
, d_2(18) -> 5
, d_3(22) -> 4
, d_3(22) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, activate_1(8) -> 13
, activate_1(9) -> 11
, activate_2(8) -> 20
, activate_2(9) -> 18
, activate_2(10) -> 14
, activate_2(15) -> 18
, activate_3(15) -> 22
, activate_3(16) -> 21
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 5
, n__d_1(2) -> 5
, n__d_1(2) -> 8
, n__d_1(2) -> 13
, n__d_1(2) -> 20
, n__d_1(11) -> 4
, n__d_1(11) -> 5
, n__d_2(2) -> 20
, n__d_2(18) -> 4
, n__d_2(18) -> 5
, n__d_3(22) -> 4
, n__d_3(22) -> 5
, g_0(2) -> 4
, g_1(2) -> 5
, g_2(4) -> 14
, g_3(17) -> 21
, c^#_0(2) -> 1
, c^#_1(8) -> 7
, c_1_0(3) -> 1
, c_1_1(6) -> 1
, c_1_1(12) -> 7
, c_1_2(19) -> 7
, d^#_0(2) -> 1
, d^#_0(4) -> 3
, d^#_1(5) -> 6
, d^#_1(13) -> 12
, d^#_2(20) -> 19
, h^#_0(2) -> 1
, c_2_0(1) -> 1
, c_2_1(7) -> 1}
6) { activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(X) -> c_3()}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, f^#(X) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
and weakly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [3]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> 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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [7]
d(x1) = [1] x1 + [4]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [3]
g(x1) = [1] x1 + [8]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_3()}
and weakly orienting the rules
{ d(X) -> n__d(X)
, g(X) -> n__g(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [2]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [1]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [12]
c_6(x1) = [1] x1 + [9]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ f^#(X) -> c_3()
, d(X) -> n__d(X)
, g(X) -> n__g(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [9]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [7]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [6]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ c(X) -> d(activate(X))
, f^#(X) -> c_3()
, d(X) -> n__d(X)
, g(X) -> n__g(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, f^#(X) -> c_3()
, d(X) -> n__d(X)
, g(X) -> n__g(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(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__f(X)) -> f(activate(X))}
Weak Rules:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, c(X) -> d(activate(X))
, f^#(X) -> c_3()
, d(X) -> n__d(X)
, g(X) -> n__g(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, f_2(10) -> 9
, f_2(10) -> 14
, f_3(15) -> 14
, f_3(15) -> 16
, c_1(7) -> 4
, c_1(7) -> 5
, c_2(11) -> 4
, c_2(11) -> 5
, 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(8) -> 7
, n__f_1(8) -> 9
, n__f_1(8) -> 14
, n__f_2(5) -> 13
, n__f_2(10) -> 9
, n__f_2(10) -> 14
, n__f_2(12) -> 11
, n__f_2(12) -> 14
, n__f_2(12) -> 16
, n__f_3(15) -> 14
, n__f_3(15) -> 16
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(2) -> 5
, n__g_1(4) -> 8
, n__g_1(4) -> 10
, n__g_2(4) -> 10
, n__g_2(13) -> 12
, n__g_2(13) -> 15
, n__g_3(13) -> 15
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(9) -> 4
, d_1(9) -> 5
, d_2(14) -> 4
, d_2(14) -> 5
, d_3(16) -> 4
, d_3(16) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, activate_1(7) -> 9
, activate_2(7) -> 14
, activate_2(8) -> 10
, activate_2(11) -> 14
, activate_3(11) -> 16
, activate_3(12) -> 15
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 5
, n__d_1(2) -> 5
, n__d_1(9) -> 4
, n__d_1(9) -> 5
, n__d_2(14) -> 4
, n__d_2(14) -> 5
, n__d_3(16) -> 4
, n__d_3(16) -> 5
, g_0(2) -> 4
, g_1(2) -> 5
, g_2(4) -> 10
, g_3(13) -> 15
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_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}
7) {activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
The usable rules for this path are the following:
{ activate(n__f(X)) -> f(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(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(activate(X))
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, g(X) -> n__g(X)
, d(X) -> n__d(X)
, c(X) -> d(activate(X))
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> 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]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
and weakly orienting the rules
{ g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_6(f^#(activate(X)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [9]
c_6(x1) = [1] x1 + [1]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{d(X) -> n__d(X)}
and weakly orienting the rules
{ activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d(X) -> n__d(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [8]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [7]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
and weakly orienting the rules
{ d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [4]
c(x1) = [1] x1 + [1]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c(X) -> d(activate(X))}
and weakly orienting the rules
{ f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(X) -> d(activate(X))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [8]
c(x1) = [1] x1 + [8]
n__f(x1) = [1] x1 + [0]
n__g(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [5]
activate(x1) = [1] x1 + [1]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [5]
g(x1) = [1] x1 + [1]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [0]
c_9() = [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__f(X)) -> f(activate(X))}
Weak Rules:
{ c(X) -> d(activate(X))
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(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__f(X)) -> f(activate(X))}
Weak Rules:
{ c(X) -> d(activate(X))
, f(f(X)) -> c(n__f(n__g(n__f(X))))
, f(X) -> n__f(X)
, d(X) -> n__d(X)
, activate^#(n__f(X)) -> c_6(f^#(activate(X)))
, g(X) -> n__g(X)
, activate(n__g(X)) -> g(X)
, activate(n__d(X)) -> d(X)
, activate(X) -> X}
Details:
The problem is Match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_1(5) -> 4
, f_1(5) -> 5
, f_2(10) -> 9
, f_2(10) -> 11
, f_3(15) -> 11
, f_3(15) -> 16
, c_1(7) -> 4
, c_1(7) -> 5
, c_2(12) -> 4
, c_2(12) -> 5
, 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(8) -> 7
, n__f_1(8) -> 9
, n__f_1(8) -> 11
, n__f_2(5) -> 14
, n__f_2(10) -> 9
, n__f_2(10) -> 11
, n__f_2(13) -> 11
, n__f_2(13) -> 12
, n__f_2(13) -> 16
, n__f_3(15) -> 11
, n__f_3(15) -> 16
, n__g_0(2) -> 2
, n__g_0(2) -> 4
, n__g_0(2) -> 5
, n__g_1(2) -> 5
, n__g_1(4) -> 8
, n__g_1(4) -> 10
, n__g_2(4) -> 10
, n__g_2(14) -> 13
, n__g_2(14) -> 15
, n__g_3(14) -> 15
, d_0(2) -> 4
, d_1(2) -> 5
, d_1(9) -> 4
, d_1(9) -> 5
, d_2(11) -> 4
, d_2(11) -> 5
, d_3(16) -> 4
, d_3(16) -> 5
, activate_0(2) -> 4
, activate_1(2) -> 5
, activate_1(7) -> 9
, activate_2(7) -> 11
, activate_2(8) -> 10
, activate_2(12) -> 11
, activate_3(12) -> 16
, activate_3(13) -> 15
, n__d_0(2) -> 2
, n__d_0(2) -> 4
, n__d_0(2) -> 5
, n__d_1(2) -> 5
, n__d_1(9) -> 4
, n__d_1(9) -> 5
, n__d_2(11) -> 4
, n__d_2(11) -> 5
, n__d_3(16) -> 4
, n__d_3(16) -> 5
, g_0(2) -> 4
, g_1(2) -> 5
, g_2(4) -> 10
, g_3(14) -> 15
, f^#_0(2) -> 1
, f^#_0(4) -> 3
, f^#_1(5) -> 6
, activate^#_0(2) -> 1
, c_6_0(3) -> 1
, c_6_1(6) -> 1}
8) { activate^#(n__g(X)) -> c_7(g^#(X))
, 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:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {g^#(X) -> c_4()}
Weak Rules: {activate^#(n__g(X)) -> c_7(g^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(X) -> c_4()}
and weakly orienting the rules
{activate^#(n__g(X)) -> c_7(g^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(X) -> c_4()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [1] x1 + [1]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [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:
{ g^#(X) -> c_4()
, activate^#(n__g(X)) -> c_7(g^#(X))}
Details:
The given problem does not contain any strict rules
9) {activate^#(n__g(X)) -> c_7(g^#(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]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {activate^#(n__g(X)) -> c_7(g^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__g(X)) -> c_7(g^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__g(X)) -> c_7(g^#(X))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [1] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [1] x1 + [0]
c_8(x1) = [0] x1 + [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: {activate^#(n__g(X)) -> c_7(g^#(X))}
Details:
The given problem does not contain any strict rules
10)
{ activate^#(n__d(X)) -> c_8(d^#(X))
, d^#(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]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {d^#(X) -> c_5()}
Weak Rules: {activate^#(n__d(X)) -> c_8(d^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{d^#(X) -> c_5()}
and weakly orienting the rules
{activate^#(n__d(X)) -> c_8(d^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{d^#(X) -> c_5()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [1]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [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:
{ d^#(X) -> c_5()
, activate^#(n__d(X)) -> c_8(d^#(X))}
Details:
The given problem does not contain any strict rules
11)
{h^#(X) -> c_2(c^#(n__d(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]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {h^#(X) -> c_2(c^#(n__d(X)))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{h^#(X) -> c_2(c^#(n__d(X)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{h^#(X) -> c_2(c^#(n__d(X)))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [1]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {h^#(X) -> c_2(c^#(n__d(X)))}
Details:
The given problem does not contain any strict rules
12)
{activate^#(n__d(X)) -> c_8(d^#(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]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {activate^#(n__d(X)) -> c_8(d^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__d(X)) -> c_8(d^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__d(X)) -> c_8(d^#(X))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [1] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [1]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [1] x1 + [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: {activate^#(n__d(X)) -> c_8(d^#(X))}
Details:
The given problem does not contain any strict rules
13)
{activate^#(X) -> c_9()}
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]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {activate^#(X) -> c_9()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_9()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_9()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
n__f(x1) = [0] x1 + [0]
n__g(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
n__d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
d^#(x1) = [0] x1 + [0]
h^#(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
g^#(x1) = [0] x1 + [0]
c_4() = [0]
c_5() = [0]
activate^#(x1) = [1] x1 + [4]
c_6(x1) = [0] x1 + [0]
c_7(x1) = [0] x1 + [0]
c_8(x1) = [0] x1 + [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: {activate^#(X) -> c_9()}
Details:
The given problem does not contain any strict rules