We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
We add the following weak dependency pairs:
Strict DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x)
, +^#(x, s(y)) -> c_4(+^#(x, y)) }
and mark the set of starting terms.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x)
, +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
We estimate the number of application of {1} by applications of
Pre({1}) = {3}. Here rules are labeled as follows:
DPs:
{ 1: sum^#(0()) -> c_1()
, 2: sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, 3: +^#(x, 0()) -> c_3(x)
, 4: +^#(x, s(y)) -> c_4(+^#(x, y)) }
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict DPs:
{ sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x)
, +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Weak DPs: { sum^#(0()) -> c_1() }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[sum](x1) = [0]
[0] = [0]
[s](x1) = [1] x1 + [0]
[+](x1, x2) = [1] x1 + [0]
[sum^#](x1) = [1] x1 + [6]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[+^#](x1, x2) = [1] x1 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum(0())] = [0]
>= [0]
= [0()]
[sum(s(x))] = [0]
>= [0]
= [+(sum(x), s(x))]
[+(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [0]
>= [1] x + [0]
= [s(+(x, y))]
[sum^#(0())] = [6]
> [0]
= [c_1()]
[sum^#(s(x))] = [1] x + [6]
> [0]
= [c_2(+^#(sum(x), s(x)))]
[+^#(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [c_3(x)]
[+^#(x, s(y))] = [1] x + [0]
>= [1] x + [0]
= [c_4(+^#(x, y))]
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^2)).
Strict DPs:
{ +^#(x, 0()) -> c_3(x)
, +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x))) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[sum](x1) = [0]
[0] = [1]
[s](x1) = [1] x1 + [0]
[+](x1, x2) = [1] x1 + [0]
[sum^#](x1) = [1] x1 + [4]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[+^#](x1, x2) = [1] x1 + [1] x2 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum(0())] = [0]
? [1]
= [0()]
[sum(s(x))] = [0]
>= [0]
= [+(sum(x), s(x))]
[+(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [0]
>= [1] x + [0]
= [s(+(x, y))]
[sum^#(0())] = [5]
> [0]
= [c_1()]
[sum^#(s(x))] = [1] x + [4]
> [1] x + [0]
= [c_2(+^#(sum(x), s(x)))]
[+^#(x, 0())] = [1] x + [1]
> [1] x + [0]
= [c_3(x)]
[+^#(x, s(y))] = [1] x + [1] y + [0]
>= [1] x + [1] y + [0]
= [c_4(+^#(x, y))]
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^2)).
Strict DPs: { +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[sum](x1) = [0]
[0] = [0]
[s](x1) = [1] x1 + [7]
[+](x1, x2) = [1] x1 + [4]
[sum^#](x1) = [1] x1 + [6]
[c_1] = [0]
[c_2](x1) = [1] x1 + [3]
[+^#](x1, x2) = [1] x1 + [1]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum(0())] = [0]
>= [0]
= [0()]
[sum(s(x))] = [0]
? [4]
= [+(sum(x), s(x))]
[+(x, 0())] = [1] x + [4]
> [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [4]
? [1] x + [11]
= [s(+(x, y))]
[sum^#(0())] = [6]
> [0]
= [c_1()]
[sum^#(s(x))] = [1] x + [13]
> [4]
= [c_2(+^#(sum(x), s(x)))]
[+^#(x, 0())] = [1] x + [1]
> [1] x + [0]
= [c_3(x)]
[+^#(x, s(y))] = [1] x + [1]
>= [1] x + [1]
= [c_4(+^#(x, y))]
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^2)).
Strict DPs: { +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, s(y)) -> s(+(x, y)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x) }
Weak Trs: { +(x, 0()) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[sum](x1) = [4]
[0] = [0]
[s](x1) = [1] x1 + [0]
[+](x1, x2) = [1] x1 + [4]
[sum^#](x1) = [1] x1 + [6]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[+^#](x1, x2) = [1] x1 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum(0())] = [4]
> [0]
= [0()]
[sum(s(x))] = [4]
? [8]
= [+(sum(x), s(x))]
[+(x, 0())] = [1] x + [4]
> [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [4]
>= [1] x + [4]
= [s(+(x, y))]
[sum^#(0())] = [6]
> [0]
= [c_1()]
[sum^#(s(x))] = [1] x + [6]
> [4]
= [c_2(+^#(sum(x), s(x)))]
[+^#(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [c_3(x)]
[+^#(x, s(y))] = [1] x + [0]
>= [1] x + [0]
= [c_4(+^#(x, y))]
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^2)).
Strict DPs: { +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs:
{ sum(s(x)) -> +(sum(x), s(x))
, +(x, s(y)) -> s(+(x, y)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x) }
Weak Trs:
{ sum(0()) -> 0()
, +(x, 0()) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[sum](x1) = [1] x1 + [0]
[0] = [0]
[s](x1) = [1] x1 + [4]
[+](x1, x2) = [1] x1 + [0]
[sum^#](x1) = [1] x1 + [4]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[+^#](x1, x2) = [1] x1 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum(0())] = [0]
>= [0]
= [0()]
[sum(s(x))] = [1] x + [4]
> [1] x + [0]
= [+(sum(x), s(x))]
[+(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [0]
? [1] x + [4]
= [s(+(x, y))]
[sum^#(0())] = [4]
> [0]
= [c_1()]
[sum^#(s(x))] = [1] x + [8]
> [1] x + [0]
= [c_2(+^#(sum(x), s(x)))]
[+^#(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [c_3(x)]
[+^#(x, s(y))] = [1] x + [0]
>= [1] x + [0]
= [c_4(+^#(x, y))]
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^2)).
Strict DPs: { +^#(x, s(y)) -> c_4(+^#(x, y)) }
Strict Trs: { +(x, s(y)) -> s(+(x, y)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x) }
Weak Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
The weightgap principle applies (using the following nonconstant
growth matrix-interpretation)
The following argument positions are usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[sum](x1) = [0]
[0] = [0]
[s](x1) = [1] x1 + [4]
[+](x1, x2) = [1] x1 + [0]
[sum^#](x1) = [1] x1 + [0]
[c_1] = [0]
[c_2](x1) = [1] x1 + [0]
[+^#](x1, x2) = [1] x1 + [1] x2 + [0]
[c_3](x1) = [1] x1 + [0]
[c_4](x1) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[sum(0())] = [0]
>= [0]
= [0()]
[sum(s(x))] = [0]
>= [0]
= [+(sum(x), s(x))]
[+(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [0]
? [1] x + [4]
= [s(+(x, y))]
[sum^#(0())] = [0]
>= [0]
= [c_1()]
[sum^#(s(x))] = [1] x + [4]
>= [1] x + [4]
= [c_2(+^#(sum(x), s(x)))]
[+^#(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [c_3(x)]
[+^#(x, s(y))] = [1] x + [1] y + [4]
> [1] x + [1] y + [0]
= [c_4(+^#(x, y))]
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^2)).
Strict Trs: { +(x, s(y)) -> s(+(x, y)) }
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x)
, +^#(x, s(y)) -> c_4(+^#(x, y)) }
Weak Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
We use the processor 'custom shape polynomial interpretation' to
orient following rules strictly.
Trs: { +(x, s(y)) -> s(+(x, y)) }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^2)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are considered usable:
Uargs(s) = {1}, Uargs(+) = {1}, Uargs(c_2) = {1}, Uargs(+^#) = {1},
Uargs(c_3) = {1}, Uargs(c_4) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[sum](x1) = x1 + x1^2
[0]() = 0
[s](x1) = 1 + x1
[+](x1, x2) = x1 + 2*x2
[sum^#](x1) = 3*x1 + x1^2
[c_1]() = 0
[c_2](x1) = x1
[+^#](x1, x2) = 2 + x1
[c_3](x1) = x1
[c_4](x1) = x1
This order satisfies the following ordering constraints.
[sum(0())] =
>=
= [0()]
[sum(s(x))] = 2 + 3*x + x^2
>= 3*x + x^2 + 2
= [+(sum(x), s(x))]
[+(x, 0())] = x
>= x
= [x]
[+(x, s(y))] = x + 2 + 2*y
> 1 + x + 2*y
= [s(+(x, y))]
We return to the main proof.
We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(1)).
Weak DPs:
{ sum^#(0()) -> c_1()
, sum^#(s(x)) -> c_2(+^#(sum(x), s(x)))
, +^#(x, 0()) -> c_3(x)
, +^#(x, s(y)) -> c_4(+^#(x, y)) }
Weak Trs:
{ sum(0()) -> 0()
, sum(s(x)) -> +(sum(x), s(x))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^2))