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)) -> +(sum(x), s(x))
, sum1(0()) -> 0()
, sum1(s(x)) -> s(+(sum1(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(sum^#(x), x)
, sum1^#(0()) -> c_3()
, sum1^#(s(x)) -> c_4(sum1^#(x), 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(sum^#(x), x)
, sum1^#(0()) -> c_3()
, sum1^#(s(x)) -> c_4(sum1^#(x), x, x) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, sum1(0()) -> 0()
, sum1(s(x)) -> s(+(sum1(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(sum^#(x), x)
, sum1^#(0()) -> c_3()
, sum1^#(s(x)) -> c_4(sum1^#(x), 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}, Uargs(c_4) = {1}
TcT has computed the following constructor-restricted matrix
interpretation.
[0] = [0]
[0]
[s](x1) = [1 0] x1 + [0]
[0 0] [0]
[sum^#](x1) = [0]
[0]
[c_1] = [0]
[0]
[c_2](x1, x2) = [1 0] x1 + [0]
[0 1] [2]
[sum1^#](x1) = [1]
[0]
[c_3] = [0]
[0]
[c_4](x1, x2, x3) = [1 0] x1 + [0]
[0 1] [0]
The order satisfies the following ordering constraints:
[sum^#(0())] = [0]
[0]
>= [0]
[0]
= [c_1()]
[sum^#(s(x))] = [0]
[0]
? [0]
[2]
= [c_2(sum^#(x), x)]
[sum1^#(0())] = [1]
[0]
> [0]
[0]
= [c_3()]
[sum1^#(s(x))] = [1]
[0]
>= [1]
[0]
= [c_4(sum1^#(x), 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(sum^#(x), x)
, sum1^#(s(x)) -> c_4(sum1^#(x), x, x) }
Weak DPs: { sum1^#(0()) -> c_3() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^1))
We estimate the number of application of {1} by applications of
Pre({1}) = {2,3}. Here rules are labeled as follows:
DPs:
{ 1: sum^#(0()) -> c_1()
, 2: sum^#(s(x)) -> c_2(sum^#(x), x)
, 3: sum1^#(s(x)) -> c_4(sum1^#(x), x, x)
, 4: sum1^#(0()) -> c_3() }
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(sum^#(x), x)
, sum1^#(s(x)) -> c_4(sum1^#(x), x, x) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum1^#(0()) -> c_3() }
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()
, sum1^#(0()) -> c_3() }
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(sum^#(x), x)
, sum1^#(s(x)) -> c_4(sum1^#(x), 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(sum^#(x), x) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[s](x1) = [1] x1 + [4]
[sum^#](x1) = [2] x1 + [0]
[c_2](x1, x2) = [1] x1 + [7]
[sum1^#](x1) = [6]
[c_4](x1, x2, x3) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum^#(s(x))] = [2] x + [8]
> [2] x + [7]
= [c_2(sum^#(x), x)]
[sum1^#(s(x))] = [6]
>= [6]
= [c_4(sum1^#(x), 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: { sum1^#(s(x)) -> c_4(sum1^#(x), x, x) }
Weak DPs: { sum^#(s(x)) -> c_2(sum^#(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: sum1^#(s(x)) -> c_4(sum1^#(x), x, x) }
Sub-proof:
----------
The following argument positions are usable:
Uargs(c_2) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[s](x1) = [1] x1 + [4]
[sum^#](x1) = [0]
[c_2](x1, x2) = [1] x1 + [0]
[sum1^#](x1) = [2] x1 + [0]
[c_4](x1, x2, x3) = [1] x1 + [1]
The order satisfies the following ordering constraints:
[sum^#(s(x))] = [0]
>= [0]
= [c_2(sum^#(x), x)]
[sum1^#(s(x))] = [2] x + [8]
> [2] x + [1]
= [c_4(sum1^#(x), 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(sum^#(x), x)
, sum1^#(s(x)) -> c_4(sum1^#(x), 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(sum^#(x), x)
, sum1^#(s(x)) -> c_4(sum1^#(x), 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))