'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()
, evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ not^#(true()) -> c_0()
, not^#(false()) -> c_1()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd^#(0(), s(0())) -> c_3()
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
The usable rules are:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()}
The estimated dependency graph contains the following edges:
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
==> {not^#(false()) -> c_1()}
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
==> {not^#(true()) -> c_0()}
{evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
==> {evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
We consider the following path(s):
1) { evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, not^#(true()) -> c_0()}
The usable rules for this path are the following:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()}
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:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, not^#(true()) -> c_0()}
Details:
We apply the weight gap principle, strictly orienting the rules
{not(false()) -> true()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not(false()) -> true()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [2]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{evenodd(0(), s(0())) -> false()}
and weakly orienting the rules
{not(false()) -> true()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd(0(), s(0())) -> false()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{not^#(true()) -> c_0()}
and weakly orienting the rules
{ evenodd(0(), s(0())) -> false()
, not(false()) -> true()}
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]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [8]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [3]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
and weakly orienting the rules
{ not^#(true()) -> c_0()
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [2]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
and weakly orienting the rules
{ evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, not^#(true()) -> c_0()
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [1]
true() = [0]
false() = [2]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [2]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{not(true()) -> false()}
and weakly orienting the rules
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, not^#(true()) -> c_0()
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not(true()) -> false()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [8]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(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: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
Weak Rules:
{ not(true()) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, not^#(true()) -> c_0()
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()}
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: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
Weak Rules:
{ not(true()) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, not^#(true()) -> c_0()
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ not_1(5) -> 4
, not_2(9) -> 5
, not_2(9) -> 9
, true_0() -> 2
, true_1() -> 4
, true_2() -> 4
, true_2() -> 5
, true_2() -> 9
, false_0() -> 2
, false_0() -> 4
, false_1() -> 5
, false_1() -> 9
, false_2() -> 4
, false_2() -> 5
, false_2() -> 9
, evenodd_0(2, 2) -> 4
, evenodd_1(2, 6) -> 5
, evenodd_1(2, 7) -> 5
, evenodd_1(2, 7) -> 9
, evenodd_2(2, 10) -> 9
, 0_0() -> 2
, 0_1() -> 7
, 0_2() -> 11
, s_0(2) -> 2
, s_1(7) -> 6
, s_2(11) -> 10
, not^#_0(2) -> 1
, not^#_0(4) -> 3
, not^#_1(5) -> 8
, c_0_0() -> 1
, c_0_1() -> 3
, c_0_2() -> 8
, evenodd^#_0(2, 2) -> 1
, c_2_0(3) -> 1
, c_2_1(8) -> 1
, c_4_0(1) -> 1}
2) { evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, not^#(false()) -> c_1()}
The usable rules for this path are the following:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()}
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:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, not^#(false()) -> c_1()}
Details:
We apply the weight gap principle, strictly orienting the rules
{ not(false()) -> true()
, not^#(false()) -> c_1()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ not(false()) -> true()
, not^#(false()) -> c_1()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [2]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{evenodd(0(), s(0())) -> false()}
and weakly orienting the rules
{ not(false()) -> true()
, not^#(false()) -> c_1()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd(0(), s(0())) -> false()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
and weakly orienting the rules
{ evenodd(0(), s(0())) -> false()
, not(false()) -> true()
, not^#(false()) -> c_1()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [2]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [9]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [8]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{not(true()) -> false()}
and weakly orienting the rules
{ evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()
, not^#(false()) -> c_1()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not(true()) -> false()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [13]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_2(x1) = [1] x1 + [4]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
and weakly orienting the rules
{ not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()
, not^#(false()) -> c_1()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [2]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(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: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
Weak Rules:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()
, not^#(false()) -> c_1()}
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: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
Weak Rules:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()
, not(false()) -> true()
, not^#(false()) -> c_1()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ not_1(5) -> 4
, not_2(9) -> 5
, not_2(9) -> 9
, true_0() -> 2
, true_1() -> 4
, true_2() -> 4
, true_2() -> 5
, true_2() -> 9
, false_0() -> 2
, false_0() -> 4
, false_1() -> 5
, false_1() -> 9
, false_2() -> 4
, false_2() -> 5
, false_2() -> 9
, evenodd_0(2, 2) -> 4
, evenodd_1(2, 6) -> 5
, evenodd_1(2, 7) -> 5
, evenodd_1(2, 7) -> 9
, evenodd_2(2, 10) -> 9
, 0_0() -> 2
, 0_1() -> 7
, 0_2() -> 11
, s_0(2) -> 2
, s_1(7) -> 6
, s_2(11) -> 10
, not^#_0(2) -> 1
, not^#_0(4) -> 3
, not^#_1(5) -> 8
, c_1_0() -> 1
, c_1_0() -> 3
, c_1_1() -> 3
, c_1_1() -> 8
, c_1_2() -> 8
, evenodd^#_0(2, 2) -> 1
, c_2_0(3) -> 1
, c_2_1(8) -> 1
, c_4_0(1) -> 1}
3) { evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
The usable rules for this path are the following:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()}
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:
{ evenodd(x, 0()) -> not(evenodd(x, s(0())))
, evenodd(0(), s(0())) -> false()
, evenodd(s(x), s(0())) -> evenodd(x, 0())
, not(true()) -> false()
, not(false()) -> true()
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
Details:
We apply the weight gap principle, strictly orienting the rules
{evenodd(0(), s(0())) -> false()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd(0(), s(0())) -> false()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [7]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
and weakly orienting the rules
{evenodd(0(), s(0())) -> false()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [5]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{not(true()) -> false()}
and weakly orienting the rules
{ evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not(true()) -> false()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [8]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [4]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [1]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{not(false()) -> true()}
and weakly orienting the rules
{ not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{not(false()) -> true()}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [1]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [1] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
and weakly orienting the rules
{ not(false()) -> true()
, not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
Details:
Interpretation Functions:
not(x1) = [1] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [1] x1 + [1] x2 + [1]
0() = [0]
s(x1) = [1] x1 + [6]
not^#(x1) = [1] x1 + [1]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [8]
c_2(x1) = [1] x1 + [0]
c_3() = [0]
c_4(x1) = [1] x1 + [9]
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: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
Weak Rules:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, not(false()) -> true()
, not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()}
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: {evenodd(x, 0()) -> not(evenodd(x, s(0())))}
Weak Rules:
{ evenodd(s(x), s(0())) -> evenodd(x, 0())
, evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))
, not(false()) -> true()
, not(true()) -> false()
, evenodd^#(x, 0()) -> c_2(not^#(evenodd(x, s(0()))))
, evenodd(0(), s(0())) -> false()}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ not_1(5) -> 4
, not_2(9) -> 5
, not_2(9) -> 9
, true_0() -> 2
, true_1() -> 4
, true_2() -> 4
, true_2() -> 5
, true_2() -> 9
, false_0() -> 2
, false_0() -> 4
, false_1() -> 5
, false_1() -> 9
, false_2() -> 4
, false_2() -> 5
, false_2() -> 9
, evenodd_0(2, 2) -> 4
, evenodd_1(2, 6) -> 5
, evenodd_1(2, 7) -> 5
, evenodd_1(2, 7) -> 9
, evenodd_2(2, 10) -> 9
, 0_0() -> 2
, 0_1() -> 7
, 0_2() -> 11
, s_0(2) -> 2
, s_1(7) -> 6
, s_2(11) -> 10
, not^#_0(2) -> 1
, not^#_0(4) -> 3
, not^#_1(5) -> 8
, evenodd^#_0(2, 2) -> 1
, c_2_0(3) -> 1
, c_2_1(8) -> 1
, c_4_0(1) -> 1}
4) {evenodd^#(0(), s(0())) -> c_3()}
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]
evenodd(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {evenodd^#(0(), s(0())) -> c_3()}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{evenodd^#(0(), s(0())) -> c_3()}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd^#(0(), s(0())) -> c_3()}
Details:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {evenodd^#(0(), s(0())) -> c_3()}
Details:
The given problem does not contain any strict rules
5) {evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
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]
evenodd(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [0] x1 + [0]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
Details:
Interpretation Functions:
not(x1) = [0] x1 + [0]
true() = [0]
false() = [0]
evenodd(x1, x2) = [0] x1 + [0] x2 + [0]
0() = [0]
s(x1) = [1] x1 + [8]
not^#(x1) = [0] x1 + [0]
c_0() = [0]
c_1() = [0]
evenodd^#(x1, x2) = [1] x1 + [1] x2 + [0]
c_2(x1) = [0] x1 + [0]
c_3() = [0]
c_4(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: {evenodd^#(s(x), s(0())) -> c_4(evenodd^#(x, 0()))}
Details:
The given problem does not contain any strict rules