```* Step 1: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ 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(cons) = {1,2},
uargs(copy) = {2,3},
uargs(f) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 0
p(cons) = 1 + x1 + x2
p(copy) = 2 + 4*x1 + x1*x2 + 2*x1^2 + x2 + x3
p(f) = x1
p(n) = 0
p(nil) = 0
p(s) = 1 + x1

Following rules are strictly oriented:
copy(0(),y,z) = 2 + y + z
> z
= f(z)

copy(s(x),y,z) = 8 + 8*x + x*y + 2*x^2 + 2*y + z
> 3 + 4*x + x*y + 2*x^2 + 2*y + z
= copy(x,y,cons(f(y),z))

f(cons(nil(),y)) = 1 + y
> y
= y

Following rules are (at-least) weakly oriented:
f(cons(f(cons(nil(),y)),z)) =  2 + y + z
>= 2 + y + z
=  copy(n(),y,z)

* Step 2: NaturalPI WORST_CASE(?,O(n^2))
+ Considered Problem:
- Strict TRS:
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
- Weak TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ 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(cons) = {1,2},
uargs(copy) = {2,3},
uargs(f) = {1}

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = 1
p(cons) = x1 + x2
p(copy) = 2*x1 + 6*x1*x2 + 6*x1^2 + 2*x2 + 2*x3
p(f) = 2 + 2*x1
p(n) = 0
p(nil) = 0
p(s) = 1 + x1

Following rules are strictly oriented:
f(cons(f(cons(nil(),y)),z)) = 6 + 4*y + 2*z
> 2*y + 2*z
= copy(n(),y,z)

Following rules are (at-least) weakly oriented:
copy(0(),y,z) =  8 + 8*y + 2*z
>= 2 + 2*z
=  f(z)

copy(s(x),y,z) =  8 + 14*x + 6*x*y + 6*x^2 + 8*y + 2*z
>= 4 + 2*x + 6*x*y + 6*x^2 + 6*y + 2*z
=  copy(x,y,cons(f(y),z))

f(cons(nil(),y)) =  2 + 2*y
>= y
=  y

* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
copy(0(),y,z) -> f(z)
copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
f(cons(nil(),y)) -> y
- Signature:
{copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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