'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, g(c(x)) -> x
, g(d(x)) -> x
, g(c(h(0()))) -> g(d(1()))
, g(c(1())) -> g(d(h(0())))
, g(h(x)) -> g(x)}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(f(x)) -> c_0(f^#(c(f(x))))
, f^#(f(x)) -> c_1(f^#(d(f(x))))
, g^#(c(x)) -> c_2()
, g^#(d(x)) -> c_3()
, g^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(h(x)) -> c_6(g^#(x))}
The usable rules are:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))}
The estimated dependency graph contains the following edges:
{g^#(c(h(0()))) -> c_4(g^#(d(1())))}
==> {g^#(d(x)) -> c_3()}
{g^#(c(1())) -> c_5(g^#(d(h(0()))))}
==> {g^#(d(x)) -> c_3()}
{g^#(h(x)) -> c_6(g^#(x))}
==> {g^#(h(x)) -> c_6(g^#(x))}
{g^#(h(x)) -> c_6(g^#(x))}
==> {g^#(c(1())) -> c_5(g^#(d(h(0()))))}
{g^#(h(x)) -> c_6(g^#(x))}
==> {g^#(c(h(0()))) -> c_4(g^#(d(1())))}
{g^#(h(x)) -> c_6(g^#(x))}
==> {g^#(d(x)) -> c_3()}
{g^#(h(x)) -> c_6(g^#(x))}
==> {g^#(c(x)) -> c_2()}
We consider the following path(s):
1) {f^#(f(x)) -> c_0(f^#(c(f(x))))}
The usable rules for this path are the following:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_0(f^#(c(f(x))))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_0(f^#(c(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 runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_0(f^#(c(f(x))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, c_0(3) -> 2
, d_0(2) -> 3
, d_0(3) -> 3
, f^#_0(2) -> 8
, f^#_0(3) -> 8}
2) {f^#(f(x)) -> c_1(f^#(d(f(x))))}
The usable rules for this path are the following:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_1(f^#(d(f(x))))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_1(f^#(d(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 runtime-complexity with respect to
Rules:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_1(f^#(d(f(x))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, c_0(3) -> 2
, d_0(2) -> 3
, d_0(3) -> 3
, f^#_0(2) -> 8
, f^#_0(3) -> 8}
3) { g^#(h(x)) -> c_6(g^#(x))
, g^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(d(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]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(d(x)) -> c_3()}
Weak Rules:
{ g^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(h(x)) -> c_6(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(d(x)) -> c_3()}
and weakly orienting the rules
{ g^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(h(x)) -> c_6(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(d(x)) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [0]
c_6(x1) = [1] x1 + [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^#(d(x)) -> c_3()
, g^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules
4) { g^#(h(x)) -> c_6(g^#(x))
, g^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(d(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]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(d(x)) -> c_3()}
Weak Rules:
{ g^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(h(x)) -> c_6(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(d(x)) -> c_3()}
and weakly orienting the rules
{ g^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(h(x)) -> c_6(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(d(x)) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [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^#(d(x)) -> c_3()
, g^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules
5) { g^#(h(x)) -> c_6(g^#(x))
, g^#(c(1())) -> c_5(g^#(d(h(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]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(c(1())) -> c_5(g^#(d(h(0()))))}
Weak Rules: {g^#(h(x)) -> c_6(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(c(1())) -> c_5(g^#(d(h(0()))))}
and weakly orienting the rules
{g^#(h(x)) -> c_6(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(c(1())) -> c_5(g^#(d(h(0()))))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [4]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [2]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [10]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [1] x1 + [1]
c_6(x1) = [1] x1 + [1]
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^#(c(1())) -> c_5(g^#(d(h(0()))))
, g^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules
6) { g^#(h(x)) -> c_6(g^#(x))
, g^#(d(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]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(d(x)) -> c_3()}
Weak Rules: {g^#(h(x)) -> c_6(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(d(x)) -> c_3()}
and weakly orienting the rules
{g^#(h(x)) -> c_6(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(d(x)) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [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^#(d(x)) -> c_3()
, g^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules
7) { g^#(h(x)) -> c_6(g^#(x))
, g^#(c(h(0()))) -> c_4(g^#(d(1())))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(c(h(0()))) -> c_4(g^#(d(1())))}
Weak Rules: {g^#(h(x)) -> c_6(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(c(h(0()))) -> c_4(g^#(d(1())))}
and weakly orienting the rules
{g^#(h(x)) -> c_6(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(c(h(0()))) -> c_4(g^#(d(1())))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [8]
d(x1) = [1] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [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^#(c(h(0()))) -> c_4(g^#(d(1())))
, g^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules
8) { g^#(h(x)) -> c_6(g^#(x))
, g^#(c(x)) -> 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]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(c(x)) -> c_2()}
Weak Rules: {g^#(h(x)) -> c_6(g^#(x))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(c(x)) -> c_2()}
and weakly orienting the rules
{g^#(h(x)) -> c_6(g^#(x))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(c(x)) -> c_2()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [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^#(c(x)) -> c_2()
, g^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules
9) {g^#(h(x)) -> c_6(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]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [0] x1 + [0]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [0] x1 + [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^#(h(x)) -> c_6(g^#(x))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(h(x)) -> c_6(g^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(h(x)) -> c_6(g^#(x))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g(x1) = [0] x1 + [0]
h(x1) = [1] x1 + [8]
0() = [0]
1() = [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
c_6(x1) = [1] x1 + [3]
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^#(h(x)) -> c_6(g^#(x))}
Details:
The given problem does not contain any strict rules