We are left with following problem, upon which TcT provides the
certificate YES(O(1),O(n^2)).
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), 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(g) = {1, 2}, Uargs(h) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[f](x1) = [0]
[a] = [0]
[b] = [0]
[c] = [0]
[d] = [0]
[g](x1, x2) = [1] x1 + [1] x2 + [4]
[h](x1, x2) = [1] x2 + [0]
[e] = [0]
The order satisfies the following ordering constraints:
[f(a())] = [0]
>= [0]
= [b()]
[f(c())] = [0]
>= [0]
= [d()]
[f(g(x, y))] = [0]
? [4]
= [g(f(x), f(y))]
[f(h(x, y))] = [0]
? [4]
= [g(h(y, f(x)), h(x, f(y)))]
[g(x, x)] = [2] x + [4]
> [1] x + [0]
= [h(e(), 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^2)).
Strict Trs:
{ f(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y))) }
Weak Trs: { g(x, x) -> h(e(), 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(g) = {1, 2}, Uargs(h) = {2}
TcT has computed the following matrix interpretation satisfying
not(EDA) and not(IDA(1)).
[f](x1) = [4]
[a] = [0]
[b] = [0]
[c] = [0]
[d] = [0]
[g](x1, x2) = [1] x1 + [1] x2 + [4]
[h](x1, x2) = [1] x2 + [0]
[e] = [0]
The order satisfies the following ordering constraints:
[f(a())] = [4]
> [0]
= [b()]
[f(c())] = [4]
> [0]
= [d()]
[f(g(x, y))] = [4]
? [12]
= [g(f(x), f(y))]
[f(h(x, y))] = [4]
? [12]
= [g(h(y, f(x)), h(x, f(y)))]
[g(x, x)] = [2] x + [4]
> [1] x + [0]
= [h(e(), 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^2)).
Strict Trs:
{ f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y))) }
Weak Trs:
{ f(a()) -> b()
, f(c()) -> d()
, g(x, x) -> h(e(), 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:
{ f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(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(g) = {1, 2}, Uargs(h) = {2}
TcT has computed the following constructor-restricted polynomial
interpretation.
[f](x1) = 2*x1 + 2*x1^2
[a]() = 0
[b]() = 0
[c]() = 0
[d]() = 0
[g](x1, x2) = 1 + x1 + x2
[h](x1, x2) = 1 + x1 + x2
[e]() = 0
This order satisfies the following ordering constraints.
[f(a())] =
>=
= [b()]
[f(c())] =
>=
= [d()]
[f(g(x, y))] = 4 + 6*x + 6*y + 2*x^2 + 2*x*y + 2*y*x + 2*y^2
> 1 + 2*x + 2*x^2 + 2*y + 2*y^2
= [g(f(x), f(y))]
[f(h(x, y))] = 4 + 6*x + 6*y + 2*x^2 + 2*x*y + 2*y*x + 2*y^2
> 3 + 3*y + 3*x + 2*x^2 + 2*y^2
= [g(h(y, f(x)), h(x, f(y)))]
[g(x, x)] = 1 + 2*x
>= 1 + x
= [h(e(), x)]
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(a()) -> b()
, f(c()) -> d()
, f(g(x, y)) -> g(f(x), f(y))
, f(h(x, y)) -> g(h(y, f(x)), h(x, f(y)))
, g(x, x) -> h(e(), x) }
Obligation:
runtime complexity
Answer:
YES(O(1),O(1))
Empty rules are trivially bounded
Hurray, we answered YES(O(1),O(n^2))