We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ prime(0()) -> false()
, prime(s(0())) -> false()
, prime(s(s(x))) -> prime1(s(s(x)), s(x))
, prime1(x, 0()) -> false()
, prime1(x, s(0())) -> true()
, prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
, divp(x, y) -> =(rem(x, y), 0()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7() }
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:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7() }
Strict Trs:
{ prime(0()) -> false()
, prime(s(0())) -> false()
, prime(s(s(x))) -> prime1(s(s(x)), s(x))
, prime1(x, 0()) -> false()
, prime1(x, s(0())) -> true()
, prime1(x, s(s(y))) -> and(not(divp(s(s(y)), x)), prime1(x, s(y)))
, divp(x, y) -> =(rem(x, y), 0()) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^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(n^1)).
Strict DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7() }
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(c_3) = {1}, Uargs(c_6) = {1, 2}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[s](x1) = [0]
[1]
[prime^#](x1) = [1]
[0]
[c_1] = [0]
[0]
[c_2] = [0]
[0]
[c_3](x1) = [1 0] x1 + [0]
[0 1] [0]
[prime1^#](x1, x2) = [0 1] x2 + [0]
[0 0] [0]
[c_4] = [0]
[0]
[c_5] = [0]
[0]
[c_6](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[divp^#](x1, x2) = [0]
[0]
[c_7] = [0]
[0]
The order satisfies the following ordering constraints:
[prime^#(0())] = [1]
[0]
> [0]
[0]
= [c_1()]
[prime^#(s(0()))] = [1]
[0]
> [0]
[0]
= [c_2()]
[prime^#(s(s(x)))] = [1]
[0]
>= [1]
[0]
= [c_3(prime1^#(s(s(x)), s(x)))]
[prime1^#(x, 0())] = [0]
[0]
>= [0]
[0]
= [c_4()]
[prime1^#(x, s(0()))] = [1]
[0]
> [0]
[0]
= [c_5()]
[prime1^#(x, s(s(y)))] = [1]
[0]
>= [1]
[0]
= [c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))]
[divp^#(x, y)] = [0]
[0]
>= [0]
[0]
= [c_7()]
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(n^1)).
Strict DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, divp^#(x, y) -> c_7() }
Weak DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime1^#(x, s(0())) -> c_5() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {2,4} by applications of
Pre({2,4}) = {3}. Here rules are labeled as follows:
DPs:
{ 1: prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, 2: prime1^#(x, 0()) -> c_4()
, 3: prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y)))
, 4: divp^#(x, y) -> c_7()
, 5: prime^#(0()) -> c_1()
, 6: prime^#(s(0())) -> c_2()
, 7: prime1^#(x, s(0())) -> c_5() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Weak DPs:
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, divp^#(x, y) -> c_7() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
The following weak DPs constitute a sub-graph of the DG that is
closed under successors. The DPs are removed.
{ prime^#(0()) -> c_1()
, prime^#(s(0())) -> c_2()
, prime1^#(x, 0()) -> c_4()
, prime1^#(x, s(0())) -> c_5()
, divp^#(x, y) -> c_7() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime^#(s(s(x))) -> c_3(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Due to missing edges in the dependency-graph, the right-hand sides
of following rules could be simplified:
{ prime1^#(x, s(s(y))) ->
c_6(divp^#(s(s(y)), x), prime1^#(x, s(y))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x)))
, prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
Consider the dependency graph
1: prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x)))
-->_1 prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) :2
2: prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y)))
-->_1 prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) :2
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).
{ prime^#(s(s(x))) -> c_1(prime1^#(s(s(x)), s(x))) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs: { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'Small Polynomial Path Order (PS,1-bounded)'
to orient following rules strictly.
DPs:
{ 1: prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
Sub-proof:
----------
The input was oriented with the instance of 'Small Polynomial Path
Order (PS,1-bounded)' as induced by the safe mapping
safe(s) = {1}, safe(prime1^#) = {}, safe(c_2) = {}
and precedence
empty .
Following symbols are considered recursive:
{prime1^#}
The recursion depth is 1.
Further, following argument filtering is employed:
pi(s) = [1], pi(prime1^#) = [1, 2], pi(c_2) = [1]
Usable defined function symbols are a subset of:
{prime1^#}
For your convenience, here are the satisfied ordering constraints:
pi(prime1^#(x, s(s(y)))) = prime1^#(x, s(; s(; y));)
> c_2(prime1^#(x, s(; y););)
= pi(c_2(prime1^#(x, s(y))))
The strictly oriented rules are moved into the weak component.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs: { prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
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.
{ prime1^#(x, s(s(y))) -> c_2(prime1^#(x, s(y))) }
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))