```
Problem:
f(0()) -> 0()
f(s(x)) -> f(id(x))
id(0()) -> 0()
id(s(x)) -> s(id(x))

Proof:
Complexity Transformation Processor:
strict:
f(0()) -> 0()
f(s(x)) -> f(id(x))
id(0()) -> 0()
id(s(x)) -> s(id(x))
weak:

Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[id](x0) = x0 + 194,

[s](x0) = x0 + 82,

[f](x0) = x0 + 167,

 = 116
orientation:
f(0()) = 283 >= 116 = 0()

f(s(x)) = x + 249 >= x + 361 = f(id(x))

id(0()) = 310 >= 116 = 0()

id(s(x)) = x + 276 >= x + 276 = s(id(x))
problem:
strict:
f(s(x)) -> f(id(x))
id(s(x)) -> s(id(x))
weak:
f(0()) -> 0()
id(0()) -> 0()
Matrix Interpretation Processor:
dimension: 1
max_matrix:
1
interpretation:
[id](x0) = x0,

[s](x0) = x0 + 1,

[f](x0) = x0,

 = 40
orientation:
f(s(x)) = x + 1 >= x = f(id(x))

id(s(x)) = x + 1 >= x + 1 = s(id(x))

f(0()) = 40 >= 40 = 0()

id(0()) = 40 >= 40 = 0()
problem:
strict:
id(s(x)) -> s(id(x))
weak:
f(s(x)) -> f(id(x))
f(0()) -> 0()
id(0()) -> 0()
Matrix Interpretation Processor:
dimension: 4
max_matrix:
[1 1 1 1]
[0 0 0 0]
[0 0 0 1]
[0 0 0 1]
interpretation:
[1 0 1 0]     
[0 0 0 0]     
[id](x0) = [0 0 0 1]x0 + 
[0 0 0 1]     ,

[1 1 1 1]     
[0 0 0 0]     
[s](x0) = [0 0 0 1]x0 + 
[0 0 0 1]     ,

[1 0 0 1]     
[0 0 0 0]     
[f](x0) = [0 0 0 0]x0 + 
[0 0 0 0]     ,



 = 

orientation:
[1 1 1 2]        [1 0 1 2]    
[0 0 0 0]        [0 0 0 0]    
id(s(x)) = [0 0 0 1]x +  >= [0 0 0 1]x +  = s(id(x))
[0 0 0 1]        [0 0 0 1]    

[1 1 1 2]        [1 0 1 1]    
[0 0 0 0]        [0 0 0 0]    
f(s(x)) = [0 0 0 0]x +  >= [0 0 0 0]x +  = f(id(x))
[0 0 0 0]        [0 0 0 0]    

    
    
f(0()) =  >=  = 0()
    

    
    
id(0()) =  >=  = 0()
    
problem:
strict:

weak:
id(s(x)) -> s(id(x))
f(s(x)) -> f(id(x))
f(0()) -> 0()
id(0()) -> 0()
Qed
```