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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = x1 + 4*x1*x2 + 2*x2
p(+) = 1 + x1 + x2
p(1) = 0

Following rules are strictly oriented:
*(x,+(y,z)) = 2 + 5*x + 4*x*y + 4*x*z + 2*y + 2*z
> 1 + 2*x + 4*x*y + 4*x*z + 2*y + 2*z
= +(*(x,y),*(x,z))

Following rules are (at-least) weakly oriented:
*(x,1()) =  x
>= x
=  x

*(+(x,y),z) =  1 + x + 4*x*z + y + 4*y*z + 6*z
>= 1 + x + 4*x*z + y + 4*y*z + 4*z
=  +(*(x,z),*(y,z))

*(1(),y) =  2*y
>= y
=  y

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

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

Following rules are strictly oriented:
*(+(x,y),z) = 2 + 2*x + 2*x*z + 2*y + 2*y*z + 3*z
> 1 + 2*x + 2*x*z + 2*y + 2*y*z + 2*z
= +(*(x,z),*(y,z))

Following rules are (at-least) weakly oriented:
*(x,+(y,z)) =  1 + 4*x + 2*x*y + 2*x*z + y + z
>= 1 + 4*x + 2*x*y + 2*x*z + y + z
=  +(*(x,y),*(x,z))

*(x,1()) =  2*x
>= x
=  x

*(1(),y) =  y
>= y
=  y

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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(*) = 2 + 7*x1 + 7*x1*x2 + 6*x2
p(+) = 1 + x1 + x2
p(1) = 1

Following rules are strictly oriented:
*(x,1()) = 8 + 14*x
> x
= x

*(1(),y) = 9 + 13*y
> y
= y

Following rules are (at-least) weakly oriented:
*(x,+(y,z)) =  8 + 14*x + 7*x*y + 7*x*z + 6*y + 6*z
>= 5 + 14*x + 7*x*y + 7*x*z + 6*y + 6*z
=  +(*(x,y),*(x,z))

*(+(x,y),z) =  9 + 7*x + 7*x*z + 7*y + 7*y*z + 13*z
>= 5 + 7*x + 7*x*z + 7*y + 7*y*z + 12*z
=  +(*(x,z),*(y,z))

* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
*(x,+(y,z)) -> +(*(x,y),*(x,z))
*(x,1()) -> x
*(+(x,y),z) -> +(*(x,z),*(y,z))
*(1(),y) -> y
- Signature:
{*/2} / {+/2,1/0}
- Obligation:
runtime complexity wrt. defined symbols {*} and constructors {+,1}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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