```* Step 1: ToInnermost WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
+(0(),y) -> y
+(s(x),0()) -> s(x)
+(s(x),s(y)) -> s(+(s(x),+(y,0())))
- Signature:
{+/2} / {0/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {+} and constructors {0,s}
+ Applied Processor:
ToInnermost
+ Details:
switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
+(0(),y) -> y
+(s(x),0()) -> s(x)
+(s(x),s(y)) -> s(+(s(x),+(y,0())))
- Signature:
{+/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+} 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(+) = {2},
uargs(s) = {1}

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

Following rules are strictly oriented:
+(0(),y) = [1] y + [1]
> [1] y + [0]
= y

+(s(x),0()) = [1] x + [1]
> [1] x + [0]
= s(x)

Following rules are (at-least) weakly oriented:
+(s(x),s(y)) =  [1] x + [1] y + [0]
>= [1] x + [1] y + [1]
=  s(+(s(x),+(y,0())))

Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: MI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
+(s(x),s(y)) -> s(+(s(x),+(y,0())))
- Weak TRS:
+(0(),y) -> y
+(s(x),0()) -> s(x)
- Signature:
{+/2} / {0/0,s/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {+} and constructors {0,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity Nothing)), miDimension = 1, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity Nothing)):

The following argument positions are considered usable:
uargs(+) = {2},
uargs(s) = {1}

Following symbols are considered usable:
{+}
TcT has computed the following interpretation:
p(+) = [1] x_1 + [4] x_2 + [0]
p(0) = [0]
p(s) = [1] x_1 + [1]

Following rules are strictly oriented:
+(s(x),s(y)) = [1] x + [4] y + [5]
> [1] x + [4] y + [2]
= s(+(s(x),+(y,0())))

Following rules are (at-least) weakly oriented:
+(0(),y) =  [4] y + [0]
>= [1] y + [0]
=  y

+(s(x),0()) =  [1] x + [1]
>= [1] x + [1]
=  s(x)

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

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