We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)))
, sqr(0()) -> 0()
, sqr(s()) -> s()
, dbl(0()) -> 0()
, dbl(s()) -> s()
, add(0(), X) -> X
, add(s(), Y) -> s()
, first(0(), X) -> nil()
, first(s(), cons(Y)) -> cons(Y) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We add the following weak dependency pairs:
Strict DPs:
{ terms^#(N) -> c_1(sqr^#(N))
, sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3()
, dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8()
, first^#(s(), cons(Y)) -> c_9() }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ terms^#(N) -> c_1(sqr^#(N))
, sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3()
, dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8()
, first^#(s(), cons(Y)) -> c_9() }
Strict Trs:
{ terms(N) -> cons(recip(sqr(N)))
, sqr(0()) -> 0()
, sqr(s()) -> s()
, dbl(0()) -> 0()
, dbl(s()) -> s()
, add(0(), X) -> X
, add(s(), Y) -> s()
, first(0(), X) -> nil()
, first(s(), cons(Y)) -> cons(Y) }
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:
{ terms^#(N) -> c_1(sqr^#(N))
, sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3()
, dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8()
, first^#(s(), cons(Y)) -> c_9() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
The weightgap principle applies (using the following constant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(c_1) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[cons](x1) = [0]
[0]
[0] = [0]
[0]
[s] = [1]
[1]
[terms^#](x1) = [1 2] x1 + [2]
[2 1] [2]
[c_1](x1) = [1 0] x1 + [0]
[0 1] [0]
[sqr^#](x1) = [0]
[0]
[c_2] = [0]
[0]
[c_3] = [0]
[0]
[dbl^#](x1) = [0]
[0]
[c_4] = [0]
[0]
[c_5] = [0]
[0]
[add^#](x1, x2) = [0]
[0]
[c_6] = [0]
[0]
[c_7] = [0]
[0]
[first^#](x1, x2) = [1 0] x1 + [0]
[0 0] [0]
[c_8] = [0]
[0]
[c_9] = [0]
[0]
The order satisfies the following ordering constraints:
[terms^#(N)] = [1 2] N + [2]
[2 1] [2]
> [0]
[0]
= [c_1(sqr^#(N))]
[sqr^#(0())] = [0]
[0]
>= [0]
[0]
= [c_2()]
[sqr^#(s())] = [0]
[0]
>= [0]
[0]
= [c_3()]
[dbl^#(0())] = [0]
[0]
>= [0]
[0]
= [c_4()]
[dbl^#(s())] = [0]
[0]
>= [0]
[0]
= [c_5()]
[add^#(0(), X)] = [0]
[0]
>= [0]
[0]
= [c_6()]
[add^#(s(), Y)] = [0]
[0]
>= [0]
[0]
= [c_7()]
[first^#(0(), X)] = [0]
[0]
>= [0]
[0]
= [c_8()]
[first^#(s(), cons(Y))] = [1]
[0]
> [0]
[0]
= [c_9()]
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:
{ sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3()
, dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8() }
Weak DPs:
{ terms^#(N) -> c_1(sqr^#(N))
, first^#(s(), cons(Y)) -> c_9() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
We estimate the number of application of {3,4,5,6,7} by
applications of Pre({3,4,5,6,7}) = {}. Here rules are labeled as
follows:
DPs:
{ 1: sqr^#(0()) -> c_2()
, 2: sqr^#(s()) -> c_3()
, 3: dbl^#(0()) -> c_4()
, 4: dbl^#(s()) -> c_5()
, 5: add^#(0(), X) -> c_6()
, 6: add^#(s(), Y) -> c_7()
, 7: first^#(0(), X) -> c_8()
, 8: terms^#(N) -> c_1(sqr^#(N))
, 9: first^#(s(), cons(Y)) -> c_9() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3() }
Weak DPs:
{ terms^#(N) -> c_1(sqr^#(N))
, dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8()
, first^#(s(), cons(Y)) -> c_9() }
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.
{ dbl^#(0()) -> c_4()
, dbl^#(s()) -> c_5()
, add^#(0(), X) -> c_6()
, add^#(s(), Y) -> c_7()
, first^#(0(), X) -> c_8()
, first^#(s(), cons(Y)) -> c_9() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3() }
Weak DPs: { terms^#(N) -> c_1(sqr^#(N)) }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Consider the dependency graph
1: sqr^#(0()) -> c_2()
2: sqr^#(s()) -> c_3()
3: terms^#(N) -> c_1(sqr^#(N))
-->_1 sqr^#(s()) -> c_3() :2
-->_1 sqr^#(0()) -> c_2() :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).
{ terms^#(N) -> c_1(sqr^#(N)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Strict DPs:
{ sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3() }
Obligation:
innermost runtime complexity
Answer:
YES(O(1),O(1))
Consider the dependency graph
1: sqr^#(0()) -> c_2()
2: sqr^#(s()) -> c_3()
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).
{ sqr^#(0()) -> c_2()
, sqr^#(s()) -> c_3() }
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(1))