We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ f(0()) -> 1()
, f(s(x)) -> g(x, s(x))
, g(0(), y) -> y
, g(s(x), y) -> g(x, s(+(y, x)))
, g(s(x), y) -> g(x, +(y, s(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(n^2))
We use the processor 'matrix interpretation of dimension 1' to
orient following rules strictly.
Trs:
{ f(0()) -> 1()
, f(s(x)) -> g(x, s(x))
, g(0(), y) -> y }
The induced complexity on above rules (modulo remaining rules) is
YES(?,O(n^1)) . These rules are moved into the corresponding weak
component(s).
Sub-proof:
----------
The following argument positions are usable:
Uargs(s) = {1}, Uargs(g) = {2}, Uargs(+) = {1}
TcT has computed the following constructor-based matrix
interpretation satisfying not(EDA).
[f](x1) = [1] x1 + [7]
[0] = [1]
[1] = [3]
[s](x1) = [1] x1 + [0]
[g](x1, x2) = [1] x2 + [1]
[+](x1, x2) = [1] x1 + [0]
The order satisfies the following ordering constraints:
[f(0())] = [8]
> [3]
= [1()]
[f(s(x))] = [1] x + [7]
> [1] x + [1]
= [g(x, s(x))]
[g(0(), y)] = [1] y + [1]
> [1] y + [0]
= [y]
[g(s(x), y)] = [1] y + [1]
>= [1] y + [1]
= [g(x, s(+(y, x)))]
[g(s(x), y)] = [1] y + [1]
>= [1] y + [1]
= [g(x, +(y, s(x)))]
[+(x, 0())] = [1] x + [0]
>= [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [0]
>= [1] x + [0]
= [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(n^2)).
Strict Trs:
{ g(s(x), y) -> g(x, s(+(y, x)))
, g(s(x), y) -> g(x, +(y, s(x)))
, +(x, 0()) -> x
, +(x, s(y)) -> s(+(x, y)) }
Weak Trs:
{ f(0()) -> 1()
, f(s(x)) -> g(x, s(x))
, g(0(), y) -> y }
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(g) = {2}, Uargs(+) = {1}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[f](x1) = [1] x1 + [4]
[0] = [4]
[1] = [0]
[s](x1) = [1] x1 + [4]
[g](x1, x2) = [1] x2 + [0]
[+](x1, x2) = [1] x1 + [4]
The order satisfies the following ordering constraints:
[f(0())] = [8]
> [0]
= [1()]
[f(s(x))] = [1] x + [8]
> [1] x + [4]
= [g(x, s(x))]
[g(0(), y)] = [1] y + [0]
>= [1] y + [0]
= [y]
[g(s(x), y)] = [1] y + [0]
? [1] y + [8]
= [g(x, s(+(y, x)))]
[g(s(x), y)] = [1] y + [0]
? [1] y + [4]
= [g(x, +(y, s(x)))]
[+(x, 0())] = [1] x + [4]
> [1] x + [0]
= [x]
[+(x, s(y))] = [1] x + [4]
? [1] x + [8]
= [s(+(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:
{ g(s(x), y) -> g(x, s(+(y, x)))
, g(s(x), y) -> g(x, +(y, s(x)))
, +(x, s(y)) -> s(+(x, y)) }
Weak Trs:
{ f(0()) -> 1()
, f(s(x)) -> g(x, s(x))
, g(0(), y) -> y
, +(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:
{ g(s(x), y) -> g(x, s(+(y, x)))
, g(s(x), y) -> g(x, +(y, s(x))) }
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(g) = {2}, Uargs(+) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[f](x1) = 3*x1 + x1^2
[0]() = 0
[1]() = 0
[s](x1) = 1 + x1
[g](x1, x2) = 3*x1 + x1^2 + x2
[+](x1, x2) = 2 + x1 + x2
This order satisfies the following ordering constraints.
[f(0())] =
>=
= [1()]
[f(s(x))] = 4 + 5*x + x^2
> 4*x + x^2 + 1
= [g(x, s(x))]
[g(0(), y)] = y
>= y
= [y]
[g(s(x), y)] = 4 + 5*x + x^2 + y
> 4*x + x^2 + 3 + y
= [g(x, s(+(y, x)))]
[g(s(x), y)] = 4 + 5*x + x^2 + y
> 4*x + x^2 + 3 + y
= [g(x, +(y, s(x)))]
[+(x, 0())] = 2 + x
> x
= [x]
[+(x, s(y))] = 3 + x + y
>= 3 + x + 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(n^2)).
Strict Trs: { +(x, s(y)) -> s(+(x, y)) }
Weak Trs:
{ f(0()) -> 1()
, f(s(x)) -> g(x, s(x))
, g(0(), y) -> y
, g(s(x), y) -> g(x, s(+(y, x)))
, g(s(x), y) -> g(x, +(y, 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(g) = {2}, Uargs(+) = {1}
TcT has computed the following constructor-restricted polynomial
interpretation.
[f](x1) = 3*x1 + x1^2
[0]() = 0
[1]() = 0
[s](x1) = 1 + x1
[g](x1, x2) = 2*x1 + x1^2 + x2
[+](x1, x2) = 1 + x1 + 2*x2
This order satisfies the following ordering constraints.
[f(0())] =
>=
= [1()]
[f(s(x))] = 4 + 5*x + x^2
> 3*x + x^2 + 1
= [g(x, s(x))]
[g(0(), y)] = y
>= y
= [y]
[g(s(x), y)] = 3 + 4*x + x^2 + y
> 4*x + x^2 + 2 + y
= [g(x, s(+(y, x)))]
[g(s(x), y)] = 3 + 4*x + x^2 + y
>= 4*x + x^2 + 3 + y
= [g(x, +(y, s(x)))]
[+(x, 0())] = 1 + x
> x
= [x]
[+(x, s(y))] = 3 + x + 2*y
> 2 + 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 Trs:
{ f(0()) -> 1()
, f(s(x)) -> g(x, s(x))
, g(0(), y) -> y
, g(s(x), y) -> g(x, s(+(y, x)))
, g(s(x), y) -> g(x, +(y, 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))