YES

Problem:
 active(f(x)) -> mark(x)
 top(active(c())) -> top(mark(c()))
 top(mark(x)) -> top(check(x))
 check(f(x)) -> f(check(x))
 check(x) -> start(match(f(X()),x))
 match(f(x),f(y)) -> f(match(x,y))
 match(X(),x) -> proper(x)
 proper(c()) -> ok(c())
 proper(f(x)) -> f(proper(x))
 f(ok(x)) -> ok(f(x))
 start(ok(x)) -> found(x)
 f(found(x)) -> found(f(x))
 top(found(x)) -> top(active(x))
 active(f(x)) -> f(active(x))
 f(mark(x)) -> mark(f(x))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
               [1 0 1]     [0]
   [top](x0) = [0 0 0]x0 + [0]
               [0 0 0]     [1],
   
                 [1 0 1]  
   [start](x0) = [1 0 1]x0
                 [0 0 0]  ,
   
                  [1 0 0]     [0]
   [active](x0) = [1 0 0]x0 + [1]
                  [0 0 1]     [0],
   
         [0]
   [c] = [0]
         [1],
   
                 [1 0 0]  
   [check](x0) = [1 0 0]x0
                 [0 0 0]  ,
   
              [1 0 0]  
   [ok](x0) = [0 0 0]x0
              [0 0 1]  ,
   
             [1 0 0]  
   [f](x0) = [0 0 0]x0
             [0 0 0]  ,
   
                     [1 0 0]     [1 0 0]     [0]
   [match](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                     [0 0 1]     [0 0 0]     [0],
   
                 [1 0 1]  
   [found](x0) = [0 0 1]x0
                 [0 0 0]  ,
   
                  [1 0 0]     [0]
   [proper](x0) = [0 0 0]x0 + [0]
                  [0 0 0]     [1],
   
         [0]
   [X] = [0]
         [1],
   
                [1 0 0]  
   [mark](x0) = [0 0 0]x0
                [0 0 0]  
  orientation:
                  [1 0 0]    [0]    [1 0 0]           
   active(f(x)) = [1 0 0]x + [1] >= [0 0 0]x = mark(x)
                  [0 0 0]    [0]    [0 0 0]           
   
                      [1]    [0]                 
   top(active(c())) = [0] >= [0] = top(mark(c()))
                      [1]    [1]                 
   
                  [1 0 0]    [0]    [1 0 0]    [0]                
   top(mark(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = top(check(x))
                  [0 0 0]    [1]    [0 0 0]    [1]                
   
                 [1 0 0]     [1 0 0]               
   check(f(x)) = [1 0 0]x >= [0 0 0]x = f(check(x))
                 [0 0 0]     [0 0 0]               
   
              [1 0 0]     [1 0 0]                          
   check(x) = [1 0 0]x >= [1 0 0]x = start(match(f(X()),x))
              [0 0 0]     [0 0 0]                          
   
                      [1 0 0]    [1 0 0]    [0]    [1 0 0]    [1 0 0]                 
   match(f(x),f(y)) = [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]x + [0 0 0]y = f(match(x,y))
                      [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]                 
   
                  [1 0 0]    [0]    [1 0 0]    [0]            
   match(X(),x) = [0 0 0]x + [1] >= [0 0 0]x + [0] = proper(x)
                  [0 0 0]    [1]    [0 0 0]    [1]            
   
                 [0]    [0]          
   proper(c()) = [0] >= [0] = ok(c())
                 [1]    [1]          
   
                  [1 0 0]    [0]    [1 0 0]                
   proper(f(x)) = [0 0 0]x + [0] >= [0 0 0]x = f(proper(x))
                  [0 0 0]    [1]    [0 0 0]                
   
              [1 0 0]     [1 0 0]            
   f(ok(x)) = [0 0 0]x >= [0 0 0]x = ok(f(x))
              [0 0 0]     [0 0 0]            
   
                  [1 0 1]     [1 0 1]            
   start(ok(x)) = [1 0 1]x >= [0 0 1]x = found(x)
                  [0 0 0]     [0 0 0]            
   
                 [1 0 1]     [1 0 0]               
   f(found(x)) = [0 0 0]x >= [0 0 0]x = found(f(x))
                 [0 0 0]     [0 0 0]               
   
                   [1 0 1]    [0]    [1 0 1]    [0]                 
   top(found(x)) = [0 0 0]x + [0] >= [0 0 0]x + [0] = top(active(x))
                   [0 0 0]    [1]    [0 0 0]    [1]                 
   
                  [1 0 0]    [0]    [1 0 0]                
   active(f(x)) = [1 0 0]x + [1] >= [0 0 0]x = f(active(x))
                  [0 0 0]    [0]    [0 0 0]                
   
                [1 0 0]     [1 0 0]              
   f(mark(x)) = [0 0 0]x >= [0 0 0]x = mark(f(x))
                [0 0 0]     [0 0 0]              
  problem:
   active(f(x)) -> mark(x)
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X()),x))
   match(f(x),f(y)) -> f(match(x,y))
   match(X(),x) -> proper(x)
   proper(c()) -> ok(c())
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                [1 0 0]     [0]
    [top](x0) = [0 1 0]x0 + [0]
                [1 0 0]     [1],
    
                    
    [start](x0) = x0
                    ,
    
                     
    [active](x0) = x0
                     ,
    
          [0]
    [c] = [1]
          [1],
    
                  [1 1 0]  
    [check](x0) = [0 0 1]x0
                  [1 1 0]  ,
    
                    [1]
    [ok](x0) = x0 + [0]
                    [0],
    
              [1 1 0]  
    [f](x0) = [0 0 1]x0
              [1 1 0]  ,
    
                           [1 1 0]  
    [match](x0, x1) = x0 + [0 0 1]x1
                           [1 1 0]  ,
    
                    
    [found](x0) = x0
                    ,
    
                   [1 1 0]  
    [proper](x0) = [0 0 1]x0
                   [1 1 0]  ,
    
          [0]
    [X] = [0]
          [0],
    
                 [1 1 0]  
    [mark](x0) = [0 0 1]x0
                 [1 1 0]  
   orientation:
                   [1 1 0]     [1 1 0]           
    active(f(x)) = [0 0 1]x >= [0 0 1]x = mark(x)
                   [1 1 0]     [1 1 0]           
    
                   [1 1 0]    [0]    [1 1 0]    [0]                
    top(mark(x)) = [0 0 1]x + [0] >= [0 0 1]x + [0] = top(check(x))
                   [1 1 0]    [1]    [1 1 0]    [1]                
    
                  [1 1 1]     [1 1 1]               
    check(f(x)) = [1 1 0]x >= [1 1 0]x = f(check(x))
                  [1 1 1]     [1 1 1]               
    
               [1 1 0]     [1 1 0]                          
    check(x) = [0 0 1]x >= [0 0 1]x = start(match(f(X()),x))
               [1 1 0]     [1 1 0]                          
    
                       [1 1 0]    [1 1 1]     [1 1 0]    [1 1 1]                 
    match(f(x),f(y)) = [0 0 1]x + [1 1 0]y >= [0 0 1]x + [1 1 0]y = f(match(x,y))
                       [1 1 0]    [1 1 1]     [1 1 0]    [1 1 1]                 
    
                   [1 1 0]     [1 1 0]             
    match(X(),x) = [0 0 1]x >= [0 0 1]x = proper(x)
                   [1 1 0]     [1 1 0]             
    
                  [1]    [1]          
    proper(c()) = [1] >= [1] = ok(c())
                  [1]    [1]          
    
                   [1 1 1]     [1 1 1]                
    proper(f(x)) = [1 1 0]x >= [1 1 0]x = f(proper(x))
                   [1 1 1]     [1 1 1]                
    
               [1 1 0]    [1]    [1 1 0]    [1]           
    f(ok(x)) = [0 0 1]x + [0] >= [0 0 1]x + [0] = ok(f(x))
               [1 1 0]    [1]    [1 1 0]    [0]           
    
                       [1]                
    start(ok(x)) = x + [0] >= x = found(x)
                       [0]                
    
                  [1 1 0]     [1 1 0]               
    f(found(x)) = [0 0 1]x >= [0 0 1]x = found(f(x))
                  [1 1 0]     [1 1 0]               
    
                    [1 0 0]    [0]    [1 0 0]    [0]                 
    top(found(x)) = [0 1 0]x + [0] >= [0 1 0]x + [0] = top(active(x))
                    [1 0 0]    [1]    [1 0 0]    [1]                 
    
                   [1 1 0]     [1 1 0]                
    active(f(x)) = [0 0 1]x >= [0 0 1]x = f(active(x))
                   [1 1 0]     [1 1 0]                
    
                 [1 1 1]     [1 1 1]              
    f(mark(x)) = [1 1 0]x >= [1 1 0]x = mark(f(x))
                 [1 1 1]     [1 1 1]              
   problem:
    active(f(x)) -> mark(x)
    top(mark(x)) -> top(check(x))
    check(f(x)) -> f(check(x))
    check(x) -> start(match(f(X()),x))
    match(f(x),f(y)) -> f(match(x,y))
    match(X(),x) -> proper(x)
    proper(c()) -> ok(c())
    proper(f(x)) -> f(proper(x))
    f(ok(x)) -> ok(f(x))
    f(found(x)) -> found(f(x))
    top(found(x)) -> top(active(x))
    active(f(x)) -> f(active(x))
    f(mark(x)) -> mark(f(x))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [top](x0) = 4x0,
     
     [start](x0) = x0,
     
     [active](x0) = x0,
     
     [c] = 4,
     
     [check](x0) = 2x0,
     
     [ok](x0) = x0,
     
     [f](x0) = 2x0,
     
     [match](x0, x1) = x0 + 2x1,
     
     [found](x0) = x0 + 1,
     
     [proper](x0) = x0,
     
     [X] = 0,
     
     [mark](x0) = 2x0
    orientation:
     active(f(x)) = 2x >= 2x = mark(x)
     
     top(mark(x)) = 8x >= 8x = top(check(x))
     
     check(f(x)) = 4x >= 4x = f(check(x))
     
     check(x) = 2x >= 2x = start(match(f(X()),x))
     
     match(f(x),f(y)) = 2x + 4y >= 2x + 4y = f(match(x,y))
     
     match(X(),x) = 2x >= x = proper(x)
     
     proper(c()) = 4 >= 4 = ok(c())
     
     proper(f(x)) = 2x >= 2x = f(proper(x))
     
     f(ok(x)) = 2x >= 2x = ok(f(x))
     
     f(found(x)) = 2x + 2 >= 2x + 1 = found(f(x))
     
     top(found(x)) = 4x + 4 >= 4x = top(active(x))
     
     active(f(x)) = 2x >= 2x = f(active(x))
     
     f(mark(x)) = 4x >= 4x = mark(f(x))
    problem:
     active(f(x)) -> mark(x)
     top(mark(x)) -> top(check(x))
     check(f(x)) -> f(check(x))
     check(x) -> start(match(f(X()),x))
     match(f(x),f(y)) -> f(match(x,y))
     match(X(),x) -> proper(x)
     proper(c()) -> ok(c())
     proper(f(x)) -> f(proper(x))
     f(ok(x)) -> ok(f(x))
     active(f(x)) -> f(active(x))
     f(mark(x)) -> mark(f(x))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [top](x0) = x0,
      
      [start](x0) = 4x0,
      
      [active](x0) = 5x0 + 5,
      
      [c] = 4,
      
      [check](x0) = 4x0,
      
      [ok](x0) = x0,
      
      [f](x0) = x0,
      
      [match](x0, x1) = 2x0 + x1,
      
      [proper](x0) = x0,
      
      [X] = 0,
      
      [mark](x0) = 4x0 + 5
     orientation:
      active(f(x)) = 5x + 5 >= 4x + 5 = mark(x)
      
      top(mark(x)) = 4x + 5 >= 4x = top(check(x))
      
      check(f(x)) = 4x >= 4x = f(check(x))
      
      check(x) = 4x >= 4x = start(match(f(X()),x))
      
      match(f(x),f(y)) = 2x + y >= 2x + y = f(match(x,y))
      
      match(X(),x) = x >= x = proper(x)
      
      proper(c()) = 4 >= 4 = ok(c())
      
      proper(f(x)) = x >= x = f(proper(x))
      
      f(ok(x)) = x >= x = ok(f(x))
      
      active(f(x)) = 5x + 5 >= 5x + 5 = f(active(x))
      
      f(mark(x)) = 4x + 5 >= 4x + 5 = mark(f(x))
     problem:
      active(f(x)) -> mark(x)
      check(f(x)) -> f(check(x))
      check(x) -> start(match(f(X()),x))
      match(f(x),f(y)) -> f(match(x,y))
      match(X(),x) -> proper(x)
      proper(c()) -> ok(c())
      proper(f(x)) -> f(proper(x))
      f(ok(x)) -> ok(f(x))
      active(f(x)) -> f(active(x))
      f(mark(x)) -> mark(f(x))
     Matrix Interpretation Processor: dim=3
      
      interpretation:
                     [1 0 0]  
       [start](x0) = [0 0 0]x0
                     [0 0 0]  ,
       
                           [1]
       [active](x0) = x0 + [0]
                           [0],
       
             [0]
       [c] = [0]
             [0],
       
                     [1 1 1]     [0]
       [check](x0) = [0 1 1]x0 + [0]
                     [0 0 0]     [1],
       
                  [1 0 0]  
       [ok](x0) = [0 0 0]x0
                  [0 0 0]  ,
       
                 [1 0 0]     [0]
       [f](x0) = [0 1 1]x0 + [0]
                 [0 0 0]     [1],
       
                         [1 1 0]     [1 1 1]     [0]
       [match](x0, x1) = [0 0 0]x0 + [0 1 1]x1 + [0]
                         [0 0 0]     [0 0 0]     [1],
       
                      [1 1 1]     [0]
       [proper](x0) = [0 1 1]x0 + [0]
                      [0 0 0]     [1],
       
             [0]
       [X] = [0]
             [0],
       
                    [1 0 0]     [0]
       [mark](x0) = [0 1 1]x0 + [0]
                    [0 0 0]     [1]
      orientation:
                      [1 0 0]    [1]    [1 0 0]    [0]          
       active(f(x)) = [0 1 1]x + [0] >= [0 1 1]x + [0] = mark(x)
                      [0 0 0]    [1]    [0 0 0]    [1]          
       
                     [1 1 1]    [1]    [1 1 1]    [0]              
       check(f(x)) = [0 1 1]x + [1] >= [0 1 1]x + [1] = f(check(x))
                     [0 0 0]    [1]    [0 0 0]    [1]              
       
                  [1 1 1]    [0]    [1 1 1]                          
       check(x) = [0 1 1]x + [0] >= [0 0 0]x = start(match(f(X()),x))
                  [0 0 0]    [1]    [0 0 0]                          
       
                          [1 1 1]    [1 1 1]    [1]    [1 1 0]    [1 1 1]    [0]                
       match(f(x),f(y)) = [0 0 0]x + [0 1 1]y + [1] >= [0 0 0]x + [0 1 1]y + [1] = f(match(x,y))
                          [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                
       
                      [1 1 1]    [0]    [1 1 1]    [0]            
       match(X(),x) = [0 1 1]x + [0] >= [0 1 1]x + [0] = proper(x)
                      [0 0 0]    [1]    [0 0 0]    [1]            
       
                     [0]    [0]          
       proper(c()) = [0] >= [0] = ok(c())
                     [1]    [0]          
       
                      [1 1 1]    [1]    [1 1 1]    [0]               
       proper(f(x)) = [0 1 1]x + [1] >= [0 1 1]x + [1] = f(proper(x))
                      [0 0 0]    [1]    [0 0 0]    [1]               
       
                  [1 0 0]    [0]    [1 0 0]            
       f(ok(x)) = [0 0 0]x + [0] >= [0 0 0]x = ok(f(x))
                  [0 0 0]    [1]    [0 0 0]            
       
                      [1 0 0]    [1]    [1 0 0]    [1]               
       active(f(x)) = [0 1 1]x + [0] >= [0 1 1]x + [0] = f(active(x))
                      [0 0 0]    [1]    [0 0 0]    [1]               
       
                    [1 0 0]    [0]    [1 0 0]    [0]             
       f(mark(x)) = [0 1 1]x + [1] >= [0 1 1]x + [1] = mark(f(x))
                    [0 0 0]    [1]    [0 0 0]    [1]             
      problem:
       check(x) -> start(match(f(X()),x))
       match(X(),x) -> proper(x)
       proper(c()) -> ok(c())
       f(ok(x)) -> ok(f(x))
       active(f(x)) -> f(active(x))
       f(mark(x)) -> mark(f(x))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [start](x0) = x0,
        
        [active](x0) = 4x0,
        
        [c] = 0,
        
        [check](x0) = 4x0 + 2,
        
        [ok](x0) = 2x0 + 1,
        
        [f](x0) = 5x0,
        
        [match](x0, x1) = x0 + x1 + 2,
        
        [proper](x0) = x0 + 2,
        
        [X] = 0,
        
        [mark](x0) = 4x0 + 1
       orientation:
        check(x) = 4x + 2 >= x + 2 = start(match(f(X()),x))
        
        match(X(),x) = x + 2 >= x + 2 = proper(x)
        
        proper(c()) = 2 >= 1 = ok(c())
        
        f(ok(x)) = 10x + 5 >= 10x + 1 = ok(f(x))
        
        active(f(x)) = 20x >= 20x = f(active(x))
        
        f(mark(x)) = 20x + 5 >= 20x + 1 = mark(f(x))
       problem:
        check(x) -> start(match(f(X()),x))
        match(X(),x) -> proper(x)
        active(f(x)) -> f(active(x))
       Matrix Interpretation Processor: dim=3
        
        interpretation:
                       [1 0 0]     [0]
         [start](x0) = [0 1 0]x0 + [0]
                       [1 0 0]     [1],
         
                        [1 0 0]     [0]
         [active](x0) = [1 0 0]x0 + [1]
                        [0 0 1]     [0],
         
                       [1 0 1]     [0]
         [check](x0) = [0 0 0]x0 + [1]
                       [1 0 0]     [1],
         
                   [1 0 1]     [0]
         [f](x0) = [0 0 0]x0 + [0]
                   [0 0 0]     [1],
         
                           [1 1 0]     [1 0 0]     [0]
         [match](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                           [0 0 0]     [0 0 0]     [0],
         
                        [1 0 0]  
         [proper](x0) = [0 0 0]x0
                        [0 0 0]  ,
         
               [0]
         [X] = [1]
               [0]
        orientation:
                    [1 0 1]    [0]    [1 0 0]    [0]                         
         check(x) = [0 0 0]x + [1] >= [0 0 0]x + [1] = start(match(f(X()),x))
                    [1 0 0]    [1]    [1 0 0]    [1]                         
         
                        [1 0 0]    [1]    [1 0 0]             
         match(X(),x) = [0 0 0]x + [1] >= [0 0 0]x = proper(x)
                        [0 0 0]    [0]    [0 0 0]             
         
                        [1 0 1]    [0]    [1 0 1]    [0]               
         active(f(x)) = [1 0 1]x + [1] >= [0 0 0]x + [0] = f(active(x))
                        [0 0 0]    [1]    [0 0 0]    [1]               
        problem:
         check(x) -> start(match(f(X()),x))
         active(f(x)) -> f(active(x))
        Matrix Interpretation Processor: dim=3
         
         interpretation:
                        [1 0 0]     [0]
          [start](x0) = [0 0 0]x0 + [1]
                        [0 0 0]     [0],
          
                         [1 1 0]  
          [active](x0) = [0 1 0]x0
                         [0 0 0]  ,
          
                        [1 0 0]     [0]
          [check](x0) = [0 0 0]x0 + [1]
                        [0 0 0]     [0],
          
                    [1 0 0]     [0]
          [f](x0) = [0 1 0]x0 + [1]
                    [0 0 0]     [0],
          
                            [1 0 0]     [1 0 0]  
          [match](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                            [0 0 0]     [0 0 0]  ,
          
                [0]
          [X] = [0]
                [0]
         orientation:
                     [1 0 0]    [0]    [1 0 0]    [0]                         
          check(x) = [0 0 0]x + [1] >= [0 0 0]x + [1] = start(match(f(X()),x))
                     [0 0 0]    [0]    [0 0 0]    [0]                         
          
                         [1 1 0]    [1]    [1 1 0]    [0]               
          active(f(x)) = [0 1 0]x + [1] >= [0 1 0]x + [1] = f(active(x))
                         [0 0 0]    [0]    [0 0 0]    [0]               
         problem:
          check(x) -> start(match(f(X()),x))
         Matrix Interpretation Processor: dim=3
          
          interpretation:
                         [1 0 0]  
           [start](x0) = [0 0 0]x0
                         [0 0 0]  ,
           
                         [1 0 0]     [1]
           [check](x0) = [0 0 0]x0 + [0]
                         [0 0 0]     [0],
           
                     [1 0 0]  
           [f](x0) = [0 0 0]x0
                     [0 0 0]  ,
           
                             [1 0 0]     [1 0 0]  
           [match](x0, x1) = [0 0 0]x0 + [0 0 0]x1
                             [0 0 0]     [0 0 0]  ,
           
                 [0]
           [X] = [0]
                 [0]
          orientation:
                      [1 0 0]    [1]    [1 0 0]                          
           check(x) = [0 0 0]x + [0] >= [0 0 0]x = start(match(f(X()),x))
                      [0 0 0]    [0]    [0 0 0]                          
          problem:
           
          Qed