Problem:
 f(X) -> if(X,c(),n__f(true()))
 if(true(),X,Y) -> X
 if(false(),X,Y) -> activate(Y)
 f(X) -> n__f(X)
 activate(n__f(X)) -> f(X)
 activate(X) -> X

Proof:
 Complexity Transformation Processor:
  strict:
   f(X) -> if(X,c(),n__f(true()))
   if(true(),X,Y) -> X
   if(false(),X,Y) -> activate(Y)
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(X)
   activate(X) -> X
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   max_matrix:
    1
    interpretation:
     [activate](x0) = x0 + 1,
     
     [false] = 0,
     
     [if](x0, x1, x2) = x0 + x1 + x2 + 15,
     
     [n__f](x0) = x0 + 7,
     
     [true] = 1,
     
     [c] = 2,
     
     [f](x0) = x0
    orientation:
     f(X) = X >= X + 25 = if(X,c(),n__f(true()))
     
     if(true(),X,Y) = X + Y + 16 >= X = X
     
     if(false(),X,Y) = X + Y + 15 >= Y + 1 = activate(Y)
     
     f(X) = X >= X + 7 = n__f(X)
     
     activate(n__f(X)) = X + 8 >= X = f(X)
     
     activate(X) = X + 1 >= X = X
    problem:
     strict:
      f(X) -> if(X,c(),n__f(true()))
      f(X) -> n__f(X)
     weak:
      if(true(),X,Y) -> X
      if(false(),X,Y) -> activate(Y)
      activate(n__f(X)) -> f(X)
      activate(X) -> X
    Matrix Interpretation Processor:
     dimension: 1
     max_matrix:
      1
      interpretation:
       [activate](x0) = x0 + 2,
       
       [false] = 8,
       
       [if](x0, x1, x2) = x0 + x1 + x2,
       
       [n__f](x0) = x0 + 1,
       
       [true] = 0,
       
       [c] = 8,
       
       [f](x0) = x0 + 3
      orientation:
       f(X) = X + 3 >= X + 9 = if(X,c(),n__f(true()))
       
       f(X) = X + 3 >= X + 1 = n__f(X)
       
       if(true(),X,Y) = X + Y >= X = X
       
       if(false(),X,Y) = X + Y + 8 >= Y + 2 = activate(Y)
       
       activate(n__f(X)) = X + 3 >= X + 3 = f(X)
       
       activate(X) = X + 2 >= X = X
      problem:
       strict:
        f(X) -> if(X,c(),n__f(true()))
       weak:
        f(X) -> n__f(X)
        if(true(),X,Y) -> X
        if(false(),X,Y) -> activate(Y)
        activate(n__f(X)) -> f(X)
        activate(X) -> X
      Matrix Interpretation Processor:
       dimension: 1
       max_matrix:
        1
        interpretation:
         [activate](x0) = x0 + 1,
         
         [false] = 1,
         
         [if](x0, x1, x2) = x0 + x1 + x2,
         
         [n__f](x0) = x0 + 4,
         
         [true] = 0,
         
         [c] = 0,
         
         [f](x0) = x0 + 5
        orientation:
         f(X) = X + 5 >= X + 4 = if(X,c(),n__f(true()))
         
         f(X) = X + 5 >= X + 4 = n__f(X)
         
         if(true(),X,Y) = X + Y >= X = X
         
         if(false(),X,Y) = X + Y + 1 >= Y + 1 = activate(Y)
         
         activate(n__f(X)) = X + 5 >= X + 5 = f(X)
         
         activate(X) = X + 1 >= X = X
        problem:
         strict:
          
         weak:
          f(X) -> if(X,c(),n__f(true()))
          f(X) -> n__f(X)
          if(true(),X,Y) -> X
          if(false(),X,Y) -> activate(Y)
          activate(n__f(X)) -> f(X)
          activate(X) -> X
        Qed