interpretations
YES(?,O(n^3))
We are left with following problem, upon which TcT provides the
certificate YES(?,O(n^3)).
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(x, c(y)) -> g(x, g(s(c(y)), y))
, g(s(x), s(y)) -> if(f(x), s(x), s(y)) }
Obligation:
innermost runtime complexity
Answer:
YES(?,O(n^3))
The following argument positions are usable:
Uargs(f) = {}, Uargs(s) = {}, Uargs(if) = {1}, Uargs(g) = {2},
Uargs(c) = {}
TcT has computed following constructor-based matrix interpretation
satisfying not(EDA).
[0 1 0] [0]
[f](x1) = [0 0 0] x1 + [0]
[0 3 0] [0]
[0]
[0] = [3]
[1]
[1]
[true] = [0]
[0]
[0]
[1] = [3]
[2]
[2]
[false] = [0]
[0]
[0 1 0] [0]
[s](x1) = [0 1 0] x1 + [1]
[0 0 0] [1]
[1 0 0] [1 0 0] [1 0 0] [0]
[if](x1, x2, x3) = [1 1 0] x1 + [0 1 1] x2 + [0 1 0] x3 + [1]
[0 0 0] [0 0 1] [0 0 1] [0]
[1 1 0] [1 0 2] [0]
[g](x1, x2) = [1 1 0] x1 + [0 1 2] x2 + [0]
[0 0 1] [0 0 0] [1]
[1 0 0] [0]
[c](x1) = [0 1 0] x1 + [0]
[1 1 1] [3]
This order satisfies following ordering constraints
[f(0())] = [3]
[0]
[9]
> [1]
[0]
[0]
= [true()]
[f(1())] = [3]
[0]
[9]
> [2]
[0]
[0]
= [false()]
[f(s(x))] = [0 1 0] [1]
[0 0 0] x + [0]
[0 3 0] [3]
> [0 1 0] [0]
[0 0 0] x + [0]
[0 3 0] [0]
= [f(x)]
[if(true(), x, y)] = [1 0 0] [1 0 0] [1]
[0 1 1] x + [0 1 0] y + [2]
[0 0 1] [0 0 1] [0]
> [1 0 0] [0]
[0 1 0] x + [0]
[0 0 1] [0]
= [x]
[if(false(), x, y)] = [1 0 0] [1 0 0] [2]
[0 1 1] x + [0 1 0] y + [3]
[0 0 1] [0 0 1] [0]
> [1 0 0] [0]
[0 1 0] y + [0]
[0 0 1] [0]
= [y]
[g(x, c(y))] = [1 1 0] [3 2 2] [6]
[1 1 0] x + [2 3 2] y + [6]
[0 0 1] [0 0 0] [1]
> [1 1 0] [1 2 2] [5]
[1 1 0] x + [0 3 2] y + [5]
[0 0 1] [0 0 0] [1]
= [g(x, g(s(c(y)), y))]
[g(s(x), s(y))] = [0 2 0] [0 1 0] [3]
[0 2 0] x + [0 1 0] y + [4]
[0 0 0] [0 0 0] [2]
> [0 2 0] [0 1 0] [0]
[0 2 0] x + [0 1 0] y + [4]
[0 0 0] [0 0 0] [2]
= [if(f(x), s(x), s(y))]
Hurray, we answered YES(?,O(n^3))
lmpo
MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
The input cannot be shown compatible
Arrrr..
mpo
MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
The input cannot be shown compatible
Arrrr..
popstar
MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
The input cannot be shown compatible
Arrrr..
popstar-ps
MAYBE
We are left with following problem, upon which TcT provides the
certificate MAYBE.
Strict Trs:
{ f(0()) -> true()
, f(1()) -> false()
, f(s(x)) -> f(x)
, if(true(), x, y) -> x
, if(false(), x, y) -> y
, g(s(x), s(y)) -> if(f(x), s(x), s(y))
, g(x, c(y)) -> g(x, g(s(c(y)), y)) }
Obligation:
innermost runtime complexity
Answer:
MAYBE
The input cannot be shown compatible
Arrrr..