'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ g(c(x, s(y))) -> g(c(s(x), y))
, f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f(x) -> x}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))
, f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))
, f^#(f(x)) -> c_2(f^#(d(f(x))))
, f^#(x) -> c_3()}
The usable rules are:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f(x) -> x}
The estimated dependency graph contains the following edges:
{g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
==> {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
{f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
==> {f^#(x) -> c_3()}
{f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
==> {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
{f^#(f(x)) -> c_2(f^#(d(f(x))))}
==> {f^#(x) -> c_3()}
We consider the following path(s):
1) { f^#(f(x)) -> c_2(f^#(d(f(x))))
, f^#(x) -> c_3()}
The usable rules for this path are the following:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f(x) -> 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(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f(x) -> x
, f^#(f(x)) -> c_2(f^#(d(f(x))))
, f^#(x) -> c_3()}
Details:
We apply the weight gap principle, strictly orienting the rules
{f(x) -> x}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(x) -> x}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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
{f(x) -> x}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(x) -> c_3()}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [8]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_2(f^#(d(f(x))))}
Weak Rules:
{ f^#(x) -> c_3()
, f(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:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_2(f^#(d(f(x))))}
Weak Rules:
{ f^#(x) -> c_3()
, f(x) -> x}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2, 2) -> 2
, s_0(2) -> 2
, d_0(2) -> 2
, f^#_0(2) -> 1
, c_3_0() -> 1}
2) {f^#(f(x)) -> c_2(f^#(d(f(x))))}
The usable rules for this path are the following:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f(x) -> 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(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f(x) -> x
, f^#(f(x)) -> c_2(f^#(d(f(x))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f(x) -> x}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f(x) -> x}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [1] x1 + [1]
d(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3() = [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:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_2(f^#(d(f(x))))}
Weak Rules: {f(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:
{ f(c(s(x), y)) -> f(c(x, s(y)))
, f(f(x)) -> f(d(f(x)))
, f^#(f(x)) -> c_2(f^#(d(f(x))))}
Weak Rules: {f(x) -> x}
Details:
The problem is Match-bounded by 0.
The enriched problem is compatible with the following automaton:
{ c_0(2, 2) -> 2
, s_0(2) -> 2
, d_0(2) -> 2
, f^#_0(2) -> 1}
3) {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
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: {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
Details:
a) We first check the conditional [Success]:
We are considering a strict trs contains single rule TRS.
b) We continue with the then-branch:
The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
--------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(c(x, s(y))) -> c_0(g^#(c(s(x), y)))}
Weak Rules: {}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [1] x2 + [7]
s(x1) = [1] x1 + [4]
f(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g^#(x1) = [4] x1 + [0]
c_0(x1) = [1] x1 + [5]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
4) {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
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^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
Details:
a) We first check the conditional [Success]:
We are considering a strict trs contains single rule TRS.
b) We continue with the then-branch:
The problem was solved by processor 'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation'':
'fastest of 'Matrix Interpretation', 'Matrix Interpretation', 'Matrix Interpretation''
--------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Weak Rules: {}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [0] x2 + [7]
s(x1) = [1] x1 + [4]
f(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [4] x1 + [0]
c_1(x1) = [1] x1 + [5]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
5) { f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))
, 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:
g(x1) = [0] x1 + [0]
c(x1, x2) = [0] x1 + [0] x2 + [0]
s(x1) = [0] x1 + [0]
f(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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: {f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(x) -> c_3()}
and weakly orienting the rules
{f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(x) -> c_3()}
Details:
Interpretation Functions:
g(x1) = [0] x1 + [0]
c(x1, x2) = [1] x1 + [1] x2 + [0]
s(x1) = [1] x1 + [0]
f(x1) = [0] x1 + [0]
d(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_0(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [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()
, f^#(c(s(x), y)) -> c_1(f^#(c(x, s(y))))}
Details:
The given problem does not contain any strict rules