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

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

Following rules are strictly oriented:
+(s(x),y) =  x + 
>  x + 
= s(+(x,y))

double(x) =  x + 
>  x + 
= +(x,x)

double(0()) = 
> 
= 0()

double(s(x)) =  x + 
>  x + 
= s(s(double(x)))

Following rules are (at-least) weakly oriented:
+(x,0()) =   x + 
>=  x + 
=  x

+(x,s(y)) =   x + 
>=  x + 
=  s(+(x,y))

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:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
- Weak TRS:
+(s(x),y) -> s(+(x,y))
double(x) -> +(x,x)
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
- Signature:
{+/2,double/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+,double} 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(s) = {1}

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

Following rules are strictly oriented:
+(x,0()) =  x + 
>  x + 
= x

+(x,s(y)) =  x +  y + 
>  x +  y + 
= s(+(x,y))

Following rules are (at-least) weakly oriented:
+(s(x),y) =   x +  y + 
>=  x +  y + 
=  s(+(x,y))

double(x) =   x + 
>=  x + 
=  +(x,x)

double(0()) =  
>= 
=  0()

double(s(x)) =   x + 
>=  x + 
=  s(s(double(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:
+(x,0()) -> x
+(x,s(y)) -> s(+(x,y))
+(s(x),y) -> s(+(x,y))
double(x) -> +(x,x)
double(0()) -> 0()
double(s(x)) -> s(s(double(x)))
- Signature:
{+/2,double/1} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+,double} and constructors {0,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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