'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(x1) -> n(c(c(x1)))
, c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(x1) -> c_0(n^#(c(c(x1))))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, c^#(c(x1)) -> c_2(c^#(x1))
, n^#(s(x1)) -> c_3(f^#(s(s(x1))))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
The usable rules are:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))}
The estimated dependency graph contains the following edges:
{f^#(x1) -> c_0(n^#(c(c(x1))))}
==> {n^#(f(x1)) -> c_4(f^#(n(x1)))}
{c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
==> {f^#(x1) -> c_0(n^#(c(c(x1))))}
{c^#(c(x1)) -> c_2(c^#(x1))}
==> {c^#(c(x1)) -> c_2(c^#(x1))}
{c^#(c(x1)) -> c_2(c^#(x1))}
==> {c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
{n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
==> {f^#(x1) -> c_0(n^#(c(c(x1))))}
{n^#(f(x1)) -> c_4(f^#(n(x1)))}
==> {f^#(x1) -> c_0(n^#(c(c(x1))))}
We consider the following path(s):
1) { n^#(s(x1)) -> c_3(f^#(s(s(x1))))
, f^#(x1) -> c_0(n^#(c(c(x1))))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
The usable rules for this path are the following:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))}
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:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))
, n^#(s(x1)) -> c_3(f^#(s(s(x1))))
, f^#(x1) -> c_0(n^#(c(c(x1))))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [1]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [1]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
and weakly orienting the rules
{ c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [0]
n^#(x1) = [1] x1 + [1]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{n^#(f(x1)) -> c_4(f^#(n(x1)))}
and weakly orienting the rules
{ n^#(s(x1)) -> c_3(f^#(s(s(x1))))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n^#(f(x1)) -> c_4(f^#(n(x1)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [1] x1 + [7]
n^#(x1) = [1] x1 + [8]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [1] x1 + [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:
{ c(f(x1)) -> f(c(c(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
Weak Rules:
{ n^#(f(x1)) -> c_4(f^#(n(x1)))
, n^#(s(x1)) -> c_3(f^#(s(s(x1))))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))}
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:
{ c(f(x1)) -> f(c(c(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
Weak Rules:
{ n^#(f(x1)) -> c_4(f^#(n(x1)))
, n^#(s(x1)) -> c_3(f^#(s(s(x1))))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ c_1(4) -> 9
, c_1(4) -> 10
, c_1(10) -> 9
, c_2(12) -> 15
, c_2(12) -> 16
, c_2(16) -> 15
, s_0(4) -> 4
, s_1(4) -> 13
, s_1(13) -> 12
, f^#_0(4) -> 5
, f^#_1(12) -> 11
, c_0_1(8) -> 5
, c_0_2(14) -> 11
, n^#_0(4) -> 7
, n^#_1(9) -> 8
, n^#_2(15) -> 14
, c_3_0(5) -> 7
, c_3_1(11) -> 7}
2) { c^#(c(x1)) -> c_2(c^#(x1))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
The usable rules for this path are the following:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, f(x1) -> n(c(c(x1)))
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))}
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:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, f(x1) -> n(c(c(x1)))
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))
, c^#(c(x1)) -> c_2(c^#(x1))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{c(c(x1)) -> c(x1)}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c(c(x1)) -> c(x1)}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [2]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{n(s(x1)) -> f(s(s(x1)))}
and weakly orienting the rules
{c(c(x1)) -> c(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n(s(x1)) -> f(s(s(x1)))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
and weakly orienting the rules
{ n(s(x1)) -> f(s(s(x1)))
, c(c(x1)) -> c(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [4]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(c(x1)) -> c_2(c^#(x1))}
and weakly orienting the rules
{ c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, n(s(x1)) -> f(s(s(x1)))
, c(c(x1)) -> c(x1)}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(c(x1)) -> c_2(c^#(x1))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [8]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] x1 + [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:
{ c(f(x1)) -> f(c(c(x1)))
, f(x1) -> n(c(c(x1)))
, n(f(x1)) -> f(n(x1))}
Weak Rules:
{ c^#(c(x1)) -> c_2(c^#(x1))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, n(s(x1)) -> f(s(s(x1)))
, c(c(x1)) -> c(x1)}
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:
{ c(f(x1)) -> f(c(c(x1)))
, f(x1) -> n(c(c(x1)))
, n(f(x1)) -> f(n(x1))}
Weak Rules:
{ c^#(c(x1)) -> c_2(c^#(x1))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, n(s(x1)) -> f(s(s(x1)))
, c(c(x1)) -> c(x1)}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ s_0(4) -> 4
, f^#_0(4) -> 5
, c^#_0(4) -> 8}
3) { c^#(c(x1)) -> c_2(c^#(x1))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, f^#(x1) -> c_0(n^#(c(c(x1))))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
The usable rules for this path are the following:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))}
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:
{ c(f(x1)) -> f(c(c(x1)))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))
, c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, c^#(c(x1)) -> c_2(c^#(x1))
, f^#(x1) -> c_0(n^#(c(c(x1))))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{ c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, c^#(c(x1)) -> c_2(c^#(x1))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, c^#(c(x1)) -> c_2(c^#(x1))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [14]
c_0(x1) = [1] x1 + [3]
n^#(x1) = [1] x1 + [0]
c^#(x1) = [1] x1 + [3]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
and weakly orienting the rules
{ c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, c^#(c(x1)) -> c_2(c^#(x1))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(f(x1)) -> c_1(f^#(c(c(x1))))}
Details:
Interpretation Functions:
f(x1) = [1] x1 + [0]
n(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [1]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [6]
c_0(x1) = [1] x1 + [2]
n^#(x1) = [1] x1 + [1]
c^#(x1) = [1] x1 + [10]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [1]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [1] x1 + [12]
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:
{ c(f(x1)) -> f(c(c(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
Weak Rules:
{ c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, c^#(c(x1)) -> c_2(c^#(x1))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
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:
{ c(f(x1)) -> f(c(c(x1)))
, n(f(x1)) -> f(n(x1))
, f(x1) -> n(c(c(x1)))
, n^#(f(x1)) -> c_4(f^#(n(x1)))}
Weak Rules:
{ c^#(f(x1)) -> c_1(f^#(c(c(x1))))
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, c^#(c(x1)) -> c_2(c^#(x1))
, f^#(x1) -> c_0(n^#(c(c(x1))))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(4) -> 10
, c_0(4) -> 11
, c_0(11) -> 10
, s_0(4) -> 4
, f^#_0(4) -> 5
, c_0_0(9) -> 5
, n^#_0(4) -> 7
, n^#_0(10) -> 9
, c^#_0(4) -> 8}
4) {n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(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: {n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [1] x1 + [1]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1) = [0] 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: {n^#(s(x1)) -> c_3(f^#(s(s(x1))))}
Details:
The given problem does not contain any strict rules
5) {c^#(c(x1)) -> c_2(c^#(x1))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n(x1) = [0] x1 + [0]
c(x1) = [0] x1 + [0]
s(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(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: {c^#(c(x1)) -> c_2(c^#(x1))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{c^#(c(x1)) -> c_2(c^#(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{c^#(c(x1)) -> c_2(c^#(x1))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
n(x1) = [0] x1 + [0]
c(x1) = [1] x1 + [8]
s(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
n^#(x1) = [0] x1 + [0]
c^#(x1) = [1] x1 + [1]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [3]
c_3(x1) = [0] x1 + [0]
c_4(x1) = [0] 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: {c^#(c(x1)) -> c_2(c^#(x1))}
Details:
The given problem does not contain any strict rules