'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(nil()) -> nil()
, f(.(nil(), y)) -> .(nil(), f(y))
, f(.(.(x, y), z)) -> f(.(x, .(y, z)))
, g(nil()) -> nil()
, g(.(x, nil())) -> .(g(x), nil())
, g(.(x, .(y, z))) -> g(.(.(x, y), z))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(nil()) -> c_0()
, f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))
, g^#(nil()) -> c_3()
, g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{f^#(.(nil(), y)) -> c_1(f^#(y))}
==> {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
{f^#(.(nil(), y)) -> c_1(f^#(y))}
==> {f^#(.(nil(), y)) -> c_1(f^#(y))}
{f^#(.(nil(), y)) -> c_1(f^#(y))}
==> {f^#(nil()) -> c_0()}
{f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
==> {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
{f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
==> {f^#(.(nil(), y)) -> c_1(f^#(y))}
{g^#(.(x, nil())) -> c_4(g^#(x))}
==> {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
{g^#(.(x, nil())) -> c_4(g^#(x))}
==> {g^#(.(x, nil())) -> c_4(g^#(x))}
{g^#(.(x, nil())) -> c_4(g^#(x))}
==> {g^#(nil()) -> c_3()}
{g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
==> {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
{g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
==> {g^#(.(x, nil())) -> c_4(g^#(x))}
We consider the following path(s):
1) { g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
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]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(.(x, nil())) -> c_4(g^#(x))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(.(x, nil())) -> c_4(g^#(x))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [8]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
c_5(x1) = [1] x1 + [2]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {g^#(.(x, nil())) -> c_4(g^#(x))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {g^#(.(x, nil())) -> c_4(g^#(x))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {g^#(.(x, nil())) -> c_4(g^#(x))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {g^#(.(x, nil())) -> c_4(g^#(x))}
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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {g^#(.(x, nil())) -> c_4(g^#(x))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Weak Rules: {g^#(.(x, nil())) -> c_4(g^#(x))}
Details:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[1]
[0]
.(x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0]
[0 1 0] [0 1 1] [0]
[0 0 1] [0 0 0] [1]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_0() = [0]
[0]
[0]
c_1(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_2(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
g^#(x1) = [0 1 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [1]
c_3() = [0]
[0]
[0]
c_4(x1) = [1 0 0] x1 + [1]
[0 0 0] [0]
[0 0 0] [1]
c_5(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 1] [0]
2) { f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
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]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(.(nil(), y)) -> c_1(f^#(y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(.(nil(), y)) -> c_1(f^#(y))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [8]
g(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1(x1) = [1] x1 + [1]
c_2(x1) = [1] x1 + [2]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(x1) = [0] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {f^#(.(nil(), y)) -> c_1(f^#(y))}
Details:
The problem was solved by processor 'combine':
'combine'
---------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {f^#(.(nil(), y)) -> c_1(f^#(y))}
Details:
'sequentially if-then-else, sequentially'
-----------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {f^#(.(nil(), y)) -> c_1(f^#(y))}
Details:
'if Check whether the TRS is strict trs contains single rule then fastest else fastest'
---------------------------------------------------------------------------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {f^#(.(nil(), y)) -> c_1(f^#(y))}
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^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {f^#(.(nil(), y)) -> c_1(f^#(y))}
Details:
The problem was solved by processor 'Matrix Interpretation':
'Matrix Interpretation'
-----------------------
Answer: YES(?,O(n^2))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Weak Rules: {f^#(.(nil(), y)) -> c_1(f^#(y))}
Details:
Interpretation Functions:
f(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
nil() = [0]
[0]
[1]
.(x1, x2) = [0 0 0] x1 + [0 0 1] x2 + [0]
[0 0 1] [0 1 0] [0]
[0 0 1] [0 0 1] [1]
g(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
f^#(x1) = [0 1 0] x1 + [1]
[0 1 1] [1]
[1 0 0] [1]
c_0() = [0]
[0]
[0]
c_1(x1) = [1 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [1]
c_2(x1) = [1 0 0] x1 + [0]
[0 0 1] [1]
[0 0 0] [0]
g^#(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_3() = [0]
[0]
[0]
c_4(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
c_5(x1) = [0 0 0] x1 + [0]
[0 0 0] [0]
[0 0 0] [0]
3) { f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))
, f^#(nil()) -> 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]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(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: {f^#(nil()) -> c_0()}
Weak Rules:
{ f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(nil()) -> c_0()}
and weakly orienting the rules
{ f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(nil()) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(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:
{ f^#(nil()) -> c_0()
, f^#(.(nil(), y)) -> c_1(f^#(y))
, f^#(.(.(x, y), z)) -> c_2(f^#(.(x, .(y, z))))}
Details:
The given problem does not contain any strict rules
4) { g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))
, g^#(nil()) -> 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]
nil() = [0]
.(x1, x2) = [0] x1 + [0] x2 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [0] x1 + [0]
c_3() = [0]
c_4(x1) = [0] x1 + [0]
c_5(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^#(nil()) -> c_3()}
Weak Rules:
{ g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{g^#(nil()) -> c_3()}
and weakly orienting the rules
{ g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{g^#(nil()) -> c_3()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
nil() = [0]
.(x1, x2) = [1] x1 + [1] x2 + [0]
g(x1) = [0] x1 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
g^#(x1) = [1] x1 + [1]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
c_5(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^#(nil()) -> c_3()
, g^#(.(x, nil())) -> c_4(g^#(x))
, g^#(.(x, .(y, z))) -> c_5(g^#(.(.(x, y), z)))}
Details:
The given problem does not contain any strict rules