'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, f(f(g(x1))) -> g(f(x1))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ g^#(c(x1)) -> c_0(g^#(f(c(x1))))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))
, g^#(g(x1)) -> c_2(g^#(f(g(x1))))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
The usable rules are:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, f(f(g(x1))) -> g(f(x1))}
The estimated dependency graph contains the following edges:
{g^#(c(x1)) -> c_0(g^#(f(c(x1))))}
==> {g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
{g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
==> {g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
==> {g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
==> {g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
We consider the following path(s):
1) { f^#(f(g(x1))) -> c_3(g^#(f(x1)))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
The usable rules for this path are the following:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(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:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [8]
c_2(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [9]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(f(g(x1))) -> g(f(x1))}
and weakly orienting the rules
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(f(g(x1))) -> g(f(x1))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
f(x1) = [1] x1 + [8]
g^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_3(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
Weak Rules:
{ f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
Weak Rules:
{ f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, g^#_0(2) -> 4
, f^#_0(2) -> 8}
2) { g^#(c(x1)) -> c_0(g^#(f(c(x1))))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
The usable rules for this path are the following:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(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:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(c(x1)) -> c_0(g^#(f(c(x1))))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f(f(g(x1))) -> g(f(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(f(g(x1))) -> g(f(x1))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [15]
f(x1) = [1] x1 + [15]
g^#(x1) = [1] x1 + [3]
c_0(x1) = [1] x1 + [7]
c_1(x1) = [1] x1 + [2]
c_2(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_3(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(c(x1)) -> c_0(g^#(f(c(x1))))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
Weak Rules: {f(f(g(x1))) -> g(f(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(c(x1)) -> c_0(g^#(f(c(x1))))
, g^#(f(c(x1))) -> c_1(g^#(f(f(c(x1)))))}
Weak Rules: {f(f(g(x1))) -> g(f(x1))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, c_1(2) -> 5
, c_2(2) -> 9
, f_1(5) -> 4
, f_2(8) -> 7
, f_2(9) -> 8
, g^#_0(2) -> 1
, g^#_1(4) -> 3
, g^#_2(7) -> 6
, c_0_1(3) -> 1
, c_1_2(6) -> 3}
3) { f^#(f(g(x1))) -> c_3(g^#(f(x1)))
, g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
The usable rules for this path are the following:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, f(f(g(x1))) -> g(f(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))
, g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [2]
f^#(x1) = [1] x1 + [4]
c_3(x1) = [1] x1 + [3]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(f(g(x1))) -> g(f(x1))}
and weakly orienting the rules
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(f(g(x1))) -> g(f(x1))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [1]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [8]
g^#(x1) = [1] x1 + [7]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
f^#(x1) = [1] x1 + [6]
c_3(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
Weak Rules:
{ f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(g(x1)) -> c_2(g^#(f(g(x1))))}
Weak Rules:
{ f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, g^#_0(2) -> 4
, f^#_0(2) -> 8}
4) {f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
The usable rules for this path are the following:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(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:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f(f(g(x1))) -> g(f(x1))}
and weakly orienting the rules
{f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(f(g(x1))) -> g(f(x1))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [8]
f(x1) = [1] x1 + [8]
g^#(x1) = [1] x1 + [1]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [9]
c_3(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))}
Weak Rules:
{ f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))}
Weak Rules:
{ f(f(g(x1))) -> g(f(x1))
, f^#(f(g(x1))) -> c_3(g^#(f(x1)))}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, g^#_0(2) -> 4
, f^#_0(2) -> 8}
5) {g^#(c(x1)) -> c_0(g^#(f(c(x1))))}
The usable rules for this path are the following:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(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:
{ f(f(g(x1))) -> g(f(x1))
, g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(c(x1)) -> c_0(g^#(f(c(x1))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f(f(g(x1))) -> g(f(x1))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(f(g(x1))) -> g(f(x1))}
Details:
Interpretation Functions:
g(x1) = [1] x1 + [0]
c(x1) = [1] x1 + [2]
f(x1) = [1] x1 + [4]
g^#(x1) = [1] x1 + [2]
c_0(x1) = [1] x1 + [2]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_3(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(c(x1)) -> c_0(g^#(f(c(x1))))}
Weak Rules: {f(f(g(x1))) -> g(f(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:
{ g(c(x1)) -> g(f(c(x1)))
, g(f(c(x1))) -> g(f(f(c(x1))))
, g(g(x1)) -> g(f(g(x1)))
, g^#(c(x1)) -> c_0(g^#(f(c(x1))))}
Weak Rules: {f(f(g(x1))) -> g(f(x1))}
Details:
The problem is Match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ c_0(2) -> 2
, c_1(2) -> 7
, f_1(7) -> 6
, g^#_0(2) -> 4
, g^#_1(6) -> 5
, c_0_1(5) -> 4}