'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ not(true()) -> false()
, not(false()) -> true()
, odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y))
, +(s(x), y) -> s(+(x, y))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ not^#(true()) -> c_0()
, not^#(false()) -> c_1()
, odd^#(0()) -> c_2()
, odd^#(s(x)) -> c_3(not^#(odd(x)))
, +^#(x, 0()) -> c_4()
, +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
The usable rules are:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
The estimated dependency graph contains the following edges:
{odd^#(s(x)) -> c_3(not^#(odd(x)))}
==> {not^#(false()) -> c_1()}
{odd^#(s(x)) -> c_3(not^#(odd(x)))}
==> {not^#(true()) -> c_0()}
{+^#(x, s(y)) -> c_5(+^#(x, y))}
==> {+^#(s(x), y) -> c_6(+^#(x, y))}
{+^#(x, s(y)) -> c_5(+^#(x, y))}
==> {+^#(x, s(y)) -> c_5(+^#(x, y))}
{+^#(x, s(y)) -> c_5(+^#(x, y))}
==> {+^#(x, 0()) -> c_4()}
{+^#(s(x), y) -> c_6(+^#(x, y))}
==> {+^#(s(x), y) -> c_6(+^#(x, y))}
{+^#(s(x), y) -> c_6(+^#(x, y))}
==> {+^#(x, s(y)) -> c_5(+^#(x, y))}
{+^#(s(x), y) -> c_6(+^#(x, y))}
==> {+^#(x, 0()) -> c_4()}
We consider the following path(s):
1) { odd^#(s(x)) -> c_3(not^#(odd(x)))
, not^#(false()) -> c_1()}
The usable rules for this path are the following:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
not(x1) = [1] x1 + [7]
true() = [4]
false() = [0]
odd(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [8]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {not^#(false()) -> c_1()}
Weak Rules:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()
, odd^#(s(x)) -> c_3(not^#(odd(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{not^#(false()) -> c_1()}
and weakly orienting the rules
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()
, odd^#(s(x)) -> c_3(not^#(odd(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not^#(false()) -> c_1()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [1] x1 + [9]
c_2() = [0]
c_3(x1) = [1] x1 + [1]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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:
{ not^#(false()) -> c_1()
, odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()
, odd^#(s(x)) -> c_3(not^#(odd(x)))}
Details:
The given problem does not contain any strict rules
2) { odd^#(s(x)) -> c_3(not^#(odd(x)))
, not^#(true()) -> c_0()}
The usable rules for this path are the following:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
not(x1) = [1] x1 + [7]
true() = [4]
false() = [0]
odd(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [8]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {not^#(true()) -> c_0()}
Weak Rules:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()
, odd^#(s(x)) -> c_3(not^#(odd(x)))}
Details:
We apply the weight gap principle, strictly orienting the rules
{not^#(true()) -> c_0()}
and weakly orienting the rules
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()
, odd^#(s(x)) -> c_3(not^#(odd(x)))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not^#(true()) -> c_0()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [1] x1 + [9]
c_2() = [0]
c_3(x1) = [1] x1 + [1]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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:
{ not^#(true()) -> c_0()
, odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()
, odd^#(s(x)) -> c_3(not^#(odd(x)))}
Details:
The given problem does not contain any strict rules
3) {odd^#(s(x)) -> c_3(not^#(odd(x)))}
The usable rules for this path are the following:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
not(x1) = [1] x1 + [7]
true() = [4]
false() = [0]
odd(x1) = [1] x1 + [1]
0() = [0]
s(x1) = [1] x1 + [8]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {odd^#(s(x)) -> c_3(not^#(odd(x)))}
Weak Rules:
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
Details:
We apply the weight gap principle, strictly orienting the rules
{odd^#(s(x)) -> c_3(not^#(odd(x)))}
and weakly orienting the rules
{ odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{odd^#(s(x)) -> c_3(not^#(odd(x)))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [1] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [1] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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:
{ odd^#(s(x)) -> c_3(not^#(odd(x)))
, odd(0()) -> false()
, odd(s(x)) -> not(odd(x))
, not(true()) -> false()
, not(false()) -> true()}
Details:
The given problem does not contain any strict rules
4) { +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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:
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
Details:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [8]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [4]
c_6(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:
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
Details:
The given problem does not contain any strict rules
5) { +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))
, +^#(x, 0()) -> c_4()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {+^#(x, 0()) -> c_4()}
Weak Rules:
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
Details:
We apply the weight gap principle, strictly orienting the rules
{+^#(x, 0()) -> c_4()}
and weakly orienting the rules
{ +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{+^#(x, 0()) -> c_4()}
Details:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_4() = [0]
c_5(x1) = [1] x1 + [0]
c_6(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:
{ +^#(x, 0()) -> c_4()
, +^#(x, s(y)) -> c_5(+^#(x, y))
, +^#(s(x), y) -> c_6(+^#(x, y))}
Details:
The given problem does not contain any strict rules
6) {odd^#(0()) -> c_2()}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [0] x1 + [0]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {odd^#(0()) -> c_2()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{odd^#(0()) -> c_2()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{odd^#(0()) -> c_2()}
Details:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
odd(x1) = [0] x1 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
+(x1, x2) = [0] x1 + [0] x2 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
odd^#(x1) = [1] x1 + [1]
c_2() = [0]
c_3(x1) = [0] x1 + [0]
+^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_4() = [0]
c_5(x1) = [0] x1 + [0]
c_6(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: {odd^#(0()) -> c_2()}
Details:
The given problem does not contain any strict rules