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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [1]
p(1) = [0]
p(f) = [9]
p(g) = [2] x_1 + [2] x_2 + [5]
p(s) = [1] x_1 + [0]

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

g(x,y) = [2] x + [2] y + [5]
> [1] y + [0]
= y

Following rules are (at-least) weakly oriented:
f(x,y,s(z)) =  [9]
>= [9]
=  s(f(0(),1(),z))

f(0(),1(),x) =  [9]
>= [9]
=  f(s(x),x,x)

* Step 2: WeightGap WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
- Weak TRS:
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {f,g} and constructors {0,1,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(0) = [1]
p(1) = [1]
p(f) = [8] x3 + [0]
p(g) = [1] x1 + [2] x2 + [8]
p(s) = [1] x1 + [2]

Following rules are strictly oriented:
f(x,y,s(z)) = [8] z + [16]
> [8] z + [2]
= s(f(0(),1(),z))

Following rules are (at-least) weakly oriented:
f(0(),1(),x) =  [8] x + [0]
>= [8] x + [0]
=  f(s(x),x,x)

g(x,y) =  [1] x + [2] y + [8]
>= [1] x + [0]
=  x

g(x,y) =  [1] x + [2] y + [8]
>= [1] y + [0]
=  y

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:
f(0(),1(),x) -> f(s(x),x,x)
- Weak TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
MI {miKind = MaximalMatrix (UpperTriangular (Multiplicity (Just 1))), miDimension = 2, miUArgs = UArgs, miURules = URules, miSelector = Just any strict-rules}
+ Details:
We apply a matrix interpretation of kind MaximalMatrix (UpperTriangular (Multiplicity (Just 1))):

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

Following symbols are considered usable:
all
TcT has computed the following interpretation:
p(0) = [4]
[1]
p(1) = [0]
[0]
p(f) = [0 2] x_1 + [5 9] x_3 + [1]
[0 0]       [0 2]       [4]
p(g) = [1 4] x_1 + [2 1] x_2 + [0]
[1 2]       [8 8]       [0]
p(s) = [1 3] x_1 + [4]
[0 0]       [0]

Following rules are strictly oriented:
f(0(),1(),x) = [5 9] x + [3]
[0 2]     [4]
> [5 9] x + [1]
[0 2]     [4]
= f(s(x),x,x)

Following rules are (at-least) weakly oriented:
f(x,y,s(z)) =  [0 2] x + [5 15] z + [21]
[0 0]     [0  0]     [4]
>= [5 15] z + [19]
[0  0]     [0]
=  s(f(0(),1(),z))

g(x,y) =  [1 4] x + [2 1] y + [0]
[1 2]     [8 8]     [0]
>= [1 0] x + [0]
[0 1]     [0]
=  x

g(x,y) =  [1 4] x + [2 1] y + [0]
[1 2]     [8 8]     [0]
>= [1 0] y + [0]
[0 1]     [0]
=  y

* Step 4: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
f(x,y,s(z)) -> s(f(0(),1(),z))
f(0(),1(),x) -> f(s(x),x,x)
g(x,y) -> x
g(x,y) -> y
- Signature:
{f/3,g/2} / {0/0,1/0,s/1}
- Obligation:
runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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