```* Step 1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
- Signature:
{f/2} / {0/0,1/0,2/0,g/2,i/1}
- Obligation:
runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
+ Applied Processor:
NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
+ Details:
We apply a polynomial interpretation of kind constructor-based(mixed(2)):
The following argument positions are considered usable:
uargs(f) = {2},
uargs(g) = {1,2}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 0
p(1) = 0
p(2) = 0
p(f) = 1 + x1 + x1*x2 + 2*x1^2 + x2
p(g) = 1 + x1 + x2
p(i) = 0

Following rules are strictly oriented:
f(x,0()) = 1 + x + 2*x^2
> x
= x

f(0(),y) = 1 + y
> y
= y

f(1(),g(x,y)) = 2 + x + y
> x
= x

f(2(),g(x,y)) = 2 + x + y
> y
= y

f(f(x,y),z) = 4 + 5*x + 9*x*y + x*y*z + 4*x*y^2 + x*z + 12*x^2 + 12*x^2*y + 2*x^2*y^2 + 2*x^2*z + 8*x^3 + 8*x^3*y + 8*x^4 + 5*y + y*z + 2*y^2 + 2*z
> 2 + 2*x + x*y + x*y*z + 2*x*y^2 + x*z + 2*x^2 + y + y*z + 2*y^2 + z
= f(x,f(y,z))

f(g(x,y),z) = 4 + 5*x + 4*x*y + x*z + 2*x^2 + 5*y + y*z + 2*y^2 + 2*z
> 3 + x + x*z + 2*x^2 + y + y*z + 2*y^2 + 2*z
= g(f(x,z),f(y,z))

f(i(x),y) = 1 + y
> 0
= i(x)

Following rules are (at-least) weakly oriented:

* Step 2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,0()) -> x
f(0(),y) -> y
f(1(),g(x,y)) -> x
f(2(),g(x,y)) -> y
f(f(x,y),z) -> f(x,f(y,z))
f(g(x,y),z) -> g(f(x,z),f(y,z))
f(i(x),y) -> i(x)
- Signature:
{f/2} / {0/0,1/0,2/0,g/2,i/1}
- Obligation:
runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))
```