We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ 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()) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
The input is overlay and right-linear. Switching to innermost
rewriting.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ 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()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ not^#(true()) -> c_1()
, not^#(false()) -> c_2()
, evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
, evenodd^#(0(), s(0())) -> c_4()
, evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ not^#(true()) -> c_1()
, not^#(false()) -> c_2()
, evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
, evenodd^#(0(), s(0())) -> c_4()
, evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Strict Trs:
{ 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()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(not) = {1}, Uargs(not^#) = {1}, Uargs(c_3) = {1},
Uargs(c_5) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[not](x1) = [1 1] x1 + [0]
[0 0] [1]
[true] = [0]
[1]
[false] = [0]
[1]
[evenodd](x1, x2) = [0 2] x1 + [2 0] x2 + [1]
[0 0] [0 0] [1]
[0] = [2]
[0]
[s](x1) = [0 0] x1 + [1]
[0 1] [2]
[not^#](x1) = [1 0] x1 + [0]
[0 0] [0]
[c_1] = [0]
[0]
[c_2] = [0]
[0]
[evenodd^#](x1, x2) = [0 2] x1 + [0]
[0 0] [0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[c_4] = [0]
[0]
[c_5](x1) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[not(true())] = [1]
[1]
> [0]
[1]
= [false()]
[not(false())] = [1]
[1]
> [0]
[1]
= [true()]
[evenodd(x, 0())] = [0 2] x + [5]
[0 0] [1]
> [0 2] x + [4]
[0 0] [1]
= [not(evenodd(x, s(0())))]
[evenodd(0(), s(0()))] = [3]
[1]
> [0]
[1]
= [false()]
[evenodd(s(x), s(0()))] = [0 2] x + [7]
[0 0] [1]
> [0 2] x + [5]
[0 0] [1]
= [evenodd(x, 0())]
[not^#(true())] = [0]
[0]
>= [0]
[0]
= [c_1()]
[not^#(false())] = [0]
[0]
>= [0]
[0]
= [c_2()]
[evenodd^#(x, 0())] = [0 2] x + [0]
[0 0] [0]
? [0 2] x + [3]
[0 0] [0]
= [c_3(not^#(evenodd(x, s(0()))))]
[evenodd^#(0(), s(0()))] = [0]
[0]
>= [0]
[0]
= [c_4()]
[evenodd^#(s(x), s(0()))] = [0 2] x + [4]
[0 0] [0]
> [0 2] x + [0]
[0 0] [0]
= [c_5(evenodd^#(x, 0()))]
Further, it can be verified that all rules not oriented are covered by the weightgap condition.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ not^#(true()) -> c_1()
, not^#(false()) -> c_2()
, evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
, evenodd^#(0(), s(0())) -> c_4() }
Weak DPs: { evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak Trs:
{ 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()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We estimate the number of application of {1,2,4} by applications of
Pre({1,2,4}) = {3}. Here rules are labeled as follows:
DPs:
{ 1: not^#(true()) -> c_1()
, 2: not^#(false()) -> c_2()
, 3: evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0()))))
, 4: evenodd^#(0(), s(0())) -> c_4()
, 5: evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs: { evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0())))) }
Weak DPs:
{ not^#(true()) -> c_1()
, not^#(false()) -> c_2()
, evenodd^#(0(), s(0())) -> c_4()
, evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak Trs:
{ 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()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ not^#(true()) -> c_1()
, not^#(false()) -> c_2()
, evenodd^#(0(), s(0())) -> c_4() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs: { evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0())))) }
Weak DPs: { evenodd^#(s(x), s(0())) -> c_5(evenodd^#(x, 0())) }
Weak Trs:
{ 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()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ evenodd^#(x, 0()) -> c_3(not^#(evenodd(x, s(0())))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs: { evenodd^#(x, 0()) -> c_1() }
Weak DPs: { evenodd^#(s(x), s(0())) -> c_2(evenodd^#(x, 0())) }
Weak Trs:
{ 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()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
No rule is usable, rules are removed from the input problem.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs: { evenodd^#(x, 0()) -> c_1() }
Weak DPs: { evenodd^#(s(x), s(0())) -> c_2(evenodd^#(x, 0())) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Consider the dependency graph
1: evenodd^#(x, 0()) -> c_1()
2: evenodd^#(s(x), s(0())) -> c_2(evenodd^#(x, 0()))
-->_1 evenodd^#(x, 0()) -> c_1() :1
Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).
{ evenodd^#(s(x), s(0())) -> c_2(evenodd^#(x, 0())) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs: { evenodd^#(x, 0()) -> c_1() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Consider the dependency graph
1: evenodd^#(x, 0()) -> c_1()
Following roots of the dependency graph are removed, as the
considered set of starting terms is closed under reduction with
respect to these rules (modulo compound contexts).
{ evenodd^#(x, 0()) -> c_1() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))