```* Step 1: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
sqr(x) -> *(x,x)
sum(0()) -> 0()
sum(s(x)) -> +(*(s(x),s(x)),sum(x))
sum(s(x)) -> +(sqr(s(x)),sum(x))
- Signature:
{sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(+) = {1,2}

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

Following rules are strictly oriented:
sum(0()) = 
> 
= 0()

sum(s(x)) =  x + 
>  x + 
= +(*(s(x),s(x)),sum(x))

sum(s(x)) =  x + 
>  x + 
= +(sqr(s(x)),sum(x))

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

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
sqr(x) -> *(x,x)
- Weak TRS:
sum(0()) -> 0()
sum(s(x)) -> +(*(s(x),s(x)),sum(x))
sum(s(x)) -> +(sqr(s(x)),sum(x))
- Signature:
{sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
+ Applied Processor:
WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
+ Details:
The weightgap principle applies using the following nonconstant growth matrix-interpretation:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(+) = {1,2}

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

Following rules are strictly oriented:
sqr(x) = 
> 
= *(x,x)

Following rules are (at-least) weakly oriented:
sum(0()) =  
>= 
=  0()

sum(s(x)) =   x + 
>=  x + 
=  +(*(s(x),s(x)),sum(x))

sum(s(x)) =   x + 
>=  x + 
=  +(sqr(s(x)),sum(x))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
sqr(x) -> *(x,x)
sum(0()) -> 0()
sum(s(x)) -> +(*(s(x),s(x)),sum(x))
sum(s(x)) -> +(sqr(s(x)),sum(x))
- Signature:
{sqr/1,sum/1} / {*/2,+/2,0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {sqr,sum} and constructors {*,+,0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```