'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(0()) -> cons(0(), n__f(s(0())))
, f(s(0())) -> f(p(s(0())))
, p(s(0())) -> 0()
, f(X) -> n__f(X)
, activate(n__f(X)) -> f(X)
, activate(X) -> X}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(0()) -> c_0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, p^#(s(0())) -> c_2()
, f^#(X) -> c_3()
, activate^#(n__f(X)) -> c_4(f^#(X))
, activate^#(X) -> c_5()}
The usable rules are:
{p(s(0())) -> 0()}
The estimated dependency graph contains the following edges:
{f^#(s(0())) -> c_1(f^#(p(s(0()))))}
==> {f^#(X) -> c_3()}
{f^#(s(0())) -> c_1(f^#(p(s(0()))))}
==> {f^#(s(0())) -> c_1(f^#(p(s(0()))))}
{f^#(s(0())) -> c_1(f^#(p(s(0()))))}
==> {f^#(0()) -> c_0()}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(X) -> c_3()}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(s(0())) -> c_1(f^#(p(s(0()))))}
{activate^#(n__f(X)) -> c_4(f^#(X))}
==> {f^#(0()) -> c_0()}
We consider the following path(s):
1) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, f^#(0()) -> c_0()}
The usable rules for this path are the following:
{p(s(0())) -> 0()}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [1]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(0()) -> c_0()}
Weak Rules:
{ p(s(0())) -> 0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(0()) -> c_0()}
and weakly orienting the rules
{ p(s(0())) -> 0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(0()) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ f^#(0()) -> c_0()
, p(s(0())) -> 0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
2) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, f^#(X) -> c_3()}
The usable rules for this path are the following:
{p(s(0())) -> 0()}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [1]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(X) -> c_3()}
Weak Rules:
{ p(s(0())) -> 0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_3()}
and weakly orienting the rules
{ p(s(0())) -> 0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [1] x1 + [0]
s(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [9]
c_4(x1) = [1] x1 + [1]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ f^#(X) -> c_3()
, p(s(0())) -> 0()
, f^#(s(0())) -> c_1(f^#(p(s(0()))))
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
3) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(s(0())) -> c_1(f^#(p(s(0()))))}
The usable rules for this path are the following:
{p(s(0())) -> 0()}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
p(x1) = [1] x1 + [1]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(s(0())) -> c_1(f^#(p(s(0()))))}
Weak Rules:
{ p(s(0())) -> 0()
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(s(0())) -> c_1(f^#(p(s(0()))))}
Weak Rules:
{ p(s(0())) -> 0()
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(s(0())) -> c_1(f^#(p(s(0()))))}
Weak Rules:
{ p(s(0())) -> 0()
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0() -> 2
, 0_1() -> 4
, 0_1() -> 6
, n__f_0(2) -> 2
, s_0(2) -> 2
, s_1(6) -> 5
, p_1(5) -> 4
, f^#_0(2) -> 1
, f^#_1(4) -> 3
, c_1_1(3) -> 1
, activate^#_0(2) -> 1
, c_4_0(1) -> 1}
4) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(X) -> c_3()}
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]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(X) -> c_3()}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(X) -> c_3()}
and weakly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(X) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [1] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ f^#(X) -> c_3()
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
5) { activate^#(n__f(X)) -> c_4(f^#(X))
, f^#(0()) -> c_0()}
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]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(0()) -> c_0()}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(0()) -> c_0()}
and weakly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(X))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(0()) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [1] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules:
{ f^#(0()) -> c_0()
, activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
6) {activate^#(n__f(X)) -> c_4(f^#(X))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {activate^#(n__f(X)) -> c_4(f^#(X))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(n__f(X)) -> c_4(f^#(X))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [1] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {activate^#(n__f(X)) -> c_4(f^#(X))}
Details:
The given problem does not contain any strict rules
7) {p^#(s(0())) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {p^#(s(0())) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{p^#(s(0())) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{p^#(s(0())) -> c_2()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {p^#(s(0())) -> c_2()}
Details:
The given problem does not contain any strict rules
8) {activate^#(X) -> c_5()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {activate^#(X) -> c_5()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{activate^#(X) -> c_5()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{activate^#(X) -> c_5()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
cons(x1, x2) = [0] x1 + [0] x2 + [0]
n__f(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
p(x1) = [0] x1 + [0]
activate(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
p^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
activate^#(x1) = [1] x1 + [4]
c_4(x1) = [0] x1 + [0]
c_5() = [0]
Finally we apply the subprocessor
'Empty TRS'
-----------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {}
Weak Rules: {activate^#(X) -> c_5()}
Details:
The given problem does not contain any strict rules