YES

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

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                      [1 0 0]          [1 1 0]  
   [if](x0, x1, x2) = [0 0 0]x0 + x1 + [0 1 0]x2
                      [0 1 0]          [1 0 1]  ,
   
         [0]
   [c] = [0]
         [0],
   
                [1 0 0]  
   [n__f](x0) = [0 0 0]x0
                [0 1 0]  ,
   
            [1]
   [true] = [1]
            [1],
   
             [1 0 0]     [0]
   [f](x0) = [0 0 0]x0 + [0]
             [0 1 0]     [1],
   
                         [1]
   [activate](x0) = x0 + [0]
                         [1],
   
             [1]
   [false] = [1]
             [0],
   
               [0]
   [n__true] = [1]
               [0]
  orientation:
          [1 0 0]    [0]    [1 0 0]    [0]                            
   f(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = if(X,c(),n__f(n__true()))
          [0 1 0]    [1]    [0 1 0]    [1]                            
   
                        [1 1 0]    [1]         
   if(true(),X,Y) = X + [0 1 0]Y + [0] >= X = X
                        [1 0 1]    [1]         
   
                         [1 1 0]    [1]        [1]              
   if(false(),X,Y) = X + [0 1 0]Y + [0] >= Y + [0] = activate(Y)
                         [1 0 1]    [1]        [1]              
   
          [1 0 0]    [0]    [1 0 0]           
   f(X) = [0 0 0]X + [0] >= [0 0 0]X = n__f(X)
          [0 1 0]    [1]    [0 1 0]           
   
            [1]    [0]            
   true() = [1] >= [1] = n__true()
            [1]    [0]            
   
                       [1 0 0]    [1]    [1 0 0]    [1]                 
   activate(n__f(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = f(activate(X))
                       [0 1 0]    [1]    [0 1 0]    [1]                 
   
                         [1]    [1]         
   activate(n__true()) = [1] >= [1] = true()
                         [1]    [1]         
   
                     [1]         
   activate(X) = X + [0] >= X = X
                     [1]         
  problem:
   f(X) -> if(X,c(),n__f(n__true()))
   if(false(),X,Y) -> activate(Y)
   f(X) -> n__f(X)
   activate(n__f(X)) -> f(activate(X))
   activate(n__true()) -> true()
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                       [1 0 0]     [1 0 0]     [1 1 0]  
    [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 1 1]x2
                       [0 0 1]     [0 0 0]     [0 0 1]  ,
    
          [0]
    [c] = [0]
          [0],
    
                 [1 0 0]  
    [n__f](x0) = [0 0 0]x0
                 [1 0 1]  ,
    
             [0]
    [true] = [0]
             [0],
    
              [1 0 0]  
    [f](x0) = [0 0 0]x0
              [1 0 1]  ,
    
                     [1 0 0]     [0]
    [activate](x0) = [0 0 1]x0 + [0]
                     [0 0 1]     [1],
    
              [1]
    [false] = [0]
              [1],
    
                [0]
    [n__true] = [0]
                [0]
   orientation:
           [1 0 0]     [1 0 0]                             
    f(X) = [0 0 0]X >= [0 0 0]X = if(X,c(),n__f(n__true()))
           [1 0 1]     [0 0 1]                             
    
                      [1 0 0]    [1 1 0]    [1]    [1 0 0]    [0]              
    if(false(),X,Y) = [0 0 0]X + [0 1 1]Y + [0] >= [0 0 1]Y + [0] = activate(Y)
                      [0 0 0]    [0 0 1]    [1]    [0 0 1]    [1]              
    
           [1 0 0]     [1 0 0]           
    f(X) = [0 0 0]X >= [0 0 0]X = n__f(X)
           [1 0 1]     [1 0 1]           
    
                        [1 0 0]    [0]    [1 0 0]    [0]                 
    activate(n__f(X)) = [1 0 1]X + [0] >= [0 0 0]X + [0] = f(activate(X))
                        [1 0 1]    [1]    [1 0 1]    [1]                 
    
                          [0]    [0]         
    activate(n__true()) = [0] >= [0] = true()
                          [1]    [0]         
   problem:
    f(X) -> if(X,c(),n__f(n__true()))
    f(X) -> n__f(X)
    activate(n__f(X)) -> f(activate(X))
    activate(n__true()) -> true()
   Matrix Interpretation Processor: dim=3
    
    interpretation:
                        [1 0 0]     [1 0 0]     [1 0 0]  
     [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2
                        [0 0 0]     [0 1 0]     [0 0 0]  ,
     
           [0]
     [c] = [1]
           [0],
     
                  [1 0 0]     [0]
     [n__f](x0) = [0 1 1]x0 + [0]
                  [0 0 0]     [1],
     
              [0]
     [true] = [0]
              [0],
     
               [1 0 0]     [1]
     [f](x0) = [0 1 1]x0 + [0]
               [0 0 0]     [1],
     
                      [1 1 1]     [0]
     [activate](x0) = [1 1 1]x0 + [0]
                      [0 0 0]     [1],
     
                 [0]
     [n__true] = [0]
                 [0]
    orientation:
            [1 0 0]    [1]    [1 0 0]    [0]                            
     f(X) = [0 1 1]X + [0] >= [0 0 0]X + [0] = if(X,c(),n__f(n__true()))
            [0 0 0]    [1]    [0 0 0]    [1]                            
     
            [1 0 0]    [1]    [1 0 0]    [0]          
     f(X) = [0 1 1]X + [0] >= [0 1 1]X + [0] = n__f(X)
            [0 0 0]    [1]    [0 0 0]    [1]          
     
                         [1 1 1]    [1]    [1 1 1]    [1]                 
     activate(n__f(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = f(activate(X))
                         [0 0 0]    [1]    [0 0 0]    [1]                 
     
                           [0]    [0]         
     activate(n__true()) = [0] >= [0] = true()
                           [1]    [0]         
    problem:
     activate(n__f(X)) -> f(activate(X))
     activate(n__true()) -> true()
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                   [1 0 0]     [1]
      [n__f](x0) = [0 0 0]x0 + [0]
                   [0 0 0]     [0],
      
               [0]
      [true] = [0]
               [0],
      
                [1 0 0]  
      [f](x0) = [0 0 0]x0
                [0 0 0]  ,
      
                       [1 0 0]  
      [activate](x0) = [0 0 0]x0
                       [0 0 0]  ,
      
                  [1]
      [n__true] = [0]
                  [0]
     orientation:
                          [1 0 0]    [1]    [1 0 0]                  
      activate(n__f(X)) = [0 0 0]X + [0] >= [0 0 0]X = f(activate(X))
                          [0 0 0]    [0]    [0 0 0]                  
      
                            [1]    [0]         
      activate(n__true()) = [0] >= [0] = true()
                            [0]    [0]         
     problem:
      
     Qed