'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(0()) -> s(0())
, f(s(0())) -> s(s(0()))
, f(s(0())) -> *(s(s(0())), f(0()))
, f(+(x, s(0()))) -> +(s(s(0())), f(x))
, f(+(x, y)) -> *(f(x), f(y))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(0()) -> c_0()
, f^#(s(0())) -> c_1()
, f^#(s(0())) -> c_2(f^#(0()))
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{f^#(s(0())) -> c_2(f^#(0()))}
==> {f^#(0()) -> c_0()}
{f^#(+(x, s(0()))) -> c_3(f^#(x))}
==> {f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
{f^#(+(x, s(0()))) -> c_3(f^#(x))}
==> {f^#(+(x, s(0()))) -> c_3(f^#(x))}
{f^#(+(x, s(0()))) -> c_3(f^#(x))}
==> {f^#(s(0())) -> c_2(f^#(0()))}
{f^#(+(x, s(0()))) -> c_3(f^#(x))}
==> {f^#(s(0())) -> c_1()}
{f^#(+(x, s(0()))) -> c_3(f^#(x))}
==> {f^#(0()) -> c_0()}
{f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
==> {f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
{f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
==> {f^#(+(x, s(0()))) -> c_3(f^#(x))}
{f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
==> {f^#(s(0())) -> c_2(f^#(0()))}
{f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
==> {f^#(s(0())) -> c_1()}
{f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
==> {f^#(0()) -> c_0()}
We consider the following path(s):
1) { f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
, f^#(s(0())) -> c_2(f^#(0()))
, f^#(0()) -> 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]
0() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [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^#(0()) -> c_0()}
Weak Rules:
{ f^#(s(0())) -> c_2(f^#(0()))
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(0()) -> c_0()}
and weakly orienting the rules
{ f^#(s(0())) -> c_2(f^#(0()))
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(0()) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [1] x1 + [1] x2 + [3]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [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:
{ f^#(0()) -> c_0()
, f^#(s(0())) -> c_2(f^#(0()))
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
The given problem does not contain any strict rules
2) { f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
, f^#(s(0())) -> c_2(f^#(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]
0() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [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^#(s(0())) -> c_2(f^#(0()))}
Weak Rules:
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(s(0())) -> c_2(f^#(0()))}
and weakly orienting the rules
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(0())) -> c_2(f^#(0()))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [8]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [1] x1 + [1] x2 + [12]
f^#(x1) = [1] x1 + [9]
c_0() = [0]
c_1() = [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [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:
{ f^#(s(0())) -> c_2(f^#(0()))
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
The given problem does not contain any strict rules
3) { f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
, f^#(s(0())) -> c_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]
0() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [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^#(s(0())) -> c_1()}
Weak Rules:
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(s(0())) -> c_1()}
and weakly orienting the rules
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(0())) -> c_1()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [1] x1 + [1] x2 + [8]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [7]
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^#(s(0())) -> c_1()
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
The given problem does not contain any strict rules
4) { f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))
, f^#(0()) -> 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]
0() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [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^#(0()) -> c_0()}
Weak Rules:
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(0()) -> c_0()}
and weakly orienting the rules
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(0()) -> c_0()}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [1] x1 + [1] x2 + [8]
f^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [7]
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^#(0()) -> c_0()
, f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
The given problem does not contain any strict rules
5) { f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
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]
0() = [0]
s(x1) = [0] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
f^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
c_4(x1, x2) = [0] x1 + [0] x2 + [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, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
Interpretation Functions:
f(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
*(x1, x2) = [0] x1 + [0] x2 + [0]
+(x1, x2) = [1] x1 + [1] x2 + [8]
f^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [1] x1 + [0]
c_4(x1, x2) = [1] x1 + [1] x2 + [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:
{ f^#(+(x, s(0()))) -> c_3(f^#(x))
, f^#(+(x, y)) -> c_4(f^#(x), f^#(y))}
Details:
The given problem does not contain any strict rules