We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sqr(s(x)), sum(x))
, sum(s(x)) -> +(*(s(x), s(x)), sum(x))
, sqr(x) -> *(x, x) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We add the following weak dependency pairs:
Strict DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x))
, sqr^#(x) -> c_4(x, x) }
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:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x))
, sqr^#(x) -> c_4(x, x) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sqr(s(x)), sum(x))
, sum(s(x)) -> +(*(s(x), s(x)), sum(x))
, sqr(x) -> *(x, x) }
Obligation:
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:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x))
, sqr^#(x) -> c_4(x, x) }
Obligation:
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_2) = {1, 2}, Uargs(c_3) = {3}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[s](x1) = [0]
[0]
[sum^#](x1) = [0]
[0]
[c_1] = [0]
[0]
[c_2](x1, x2) = [1 0] x1 + [1 0] x2 + [0]
[0 1] [0 1] [0]
[sqr^#](x1) = [1]
[0]
[c_3](x1, x2, x3) = [1 0] x3 + [0]
[0 1] [0]
[c_4](x1, x2) = [0]
[0]
The order satisfies the following ordering constraints:
[sum^#(0())] = [0]
[0]
>= [0]
[0]
= [c_1()]
[sum^#(s(x))] = [0]
[0]
? [1]
[0]
= [c_2(sqr^#(s(x)), sum^#(x))]
[sum^#(s(x))] = [0]
[0]
>= [0]
[0]
= [c_3(x, x, sum^#(x))]
[sqr^#(x)] = [1]
[0]
> [0]
[0]
= [c_4(x, x)]
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:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x)) }
Weak DPs: { sqr^#(x) -> c_4(x, x) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 1: sum^#(0()) -> c_1() }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1, 2}, Uargs(c_3) = {3}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [4]
[s](x1) = [1] x1 + [0]
[sum^#](x1) = [2] x1 + [0]
[c_1] = [1]
[c_2](x1, x2) = [2] x1 + [1] x2 + [0]
[sqr^#](x1) = [0]
[c_3](x1, x2, x3) = [1] x3 + [0]
[c_4](x1, x2) = [0]
The order satisfies the following ordering constraints:
[sum^#(0())] = [8]
> [1]
= [c_1()]
[sum^#(s(x))] = [2] x + [0]
>= [2] x + [0]
= [c_2(sqr^#(s(x)), sum^#(x))]
[sum^#(s(x))] = [2] x + [0]
>= [2] x + [0]
= [c_3(x, x, sum^#(x))]
[sqr^#(x)] = [0]
>= [0]
= [c_4(x, x)]
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(n^1)).
Strict DPs:
{ sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sqr^#(x) -> c_4(x, x) }
Obligation:
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.
{ sum^#(0()) -> c_1() }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^1)).
Strict DPs:
{ sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x)) }
Weak DPs: { sqr^#(x) -> c_4(x, x) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
DPs:
{ 1: sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, 2: sum^#(s(x)) -> c_3(x, x, sum^#(x)) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1, 2}, Uargs(c_3) = {3}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[0] = [0]
[s](x1) = [1] x1 + [1]
[sum^#](x1) = [1] x1 + [0]
[c_1] = [0]
[c_2](x1, x2) = [1] x1 + [1] x2 + [0]
[sqr^#](x1) = [0]
[c_3](x1, x2, x3) = [1] x3 + [0]
[c_4](x1, x2) = [0]
The order satisfies the following ordering constraints:
[sum^#(s(x))] = [1] x + [1]
> [1] x + [0]
= [c_2(sqr^#(s(x)), sum^#(x))]
[sum^#(s(x))] = [1] x + [1]
> [1] x + [0]
= [c_3(x, x, sum^#(x))]
[sqr^#(x)] = [0]
>= [0]
= [c_4(x, x)]
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:
{ sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x))
, sqr^#(x) -> c_4(x, x) }
Obligation:
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.
{ sum^#(s(x)) -> c_2(sqr^#(s(x)), sum^#(x))
, sum^#(s(x)) -> c_3(x, x, sum^#(x))
, sqr^#(x) -> c_4(x, x) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Rules: Empty
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^1))