YES

Problem:
 active(first(0(),X)) -> mark(nil())
 active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
 active(from(X)) -> mark(cons(X,from(s(X))))
 active(first(X1,X2)) -> first(active(X1),X2)
 active(first(X1,X2)) -> first(X1,active(X2))
 active(s(X)) -> s(active(X))
 active(cons(X1,X2)) -> cons(active(X1),X2)
 active(from(X)) -> from(active(X))
 first(mark(X1),X2) -> mark(first(X1,X2))
 first(X1,mark(X2)) -> mark(first(X1,X2))
 s(mark(X)) -> mark(s(X))
 cons(mark(X1),X2) -> mark(cons(X1,X2))
 from(mark(X)) -> mark(from(X))
 proper(first(X1,X2)) -> first(proper(X1),proper(X2))
 proper(0()) -> ok(0())
 proper(nil()) -> ok(nil())
 proper(s(X)) -> s(proper(X))
 proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
 proper(from(X)) -> from(proper(X))
 first(ok(X1),ok(X2)) -> ok(first(X1,X2))
 s(ok(X)) -> ok(s(X))
 cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
 from(ok(X)) -> ok(from(X))
 top(mark(X)) -> top(proper(X))
 top(ok(X)) -> top(active(X))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                     [0]
   [mark](x0) = x0 + [0]
                     [1],
   
                    
   [proper](x0) = x0
                    ,
   
                               [0]
   [first](x0, x1) = x0 + x1 + [0]
                               [1],
   
           [0]
   [nil] = [0]
           [0],
   
               
   [s](x0) = x0
               ,
   
               [1 0 1]  
   [top](x0) = [1 1 1]x0
               [1 0 0]  ,
   
         [0]
   [0] = [0]
         [0],
   
                [1 1 0]     [0]
   [from](x0) = [1 1 0]x0 + [0]
                [0 0 1]     [1],
   
              [1 1 0]  
   [ok](x0) = [1 1 0]x0
              [0 0 1]  ,
   
                         [1 0 0]  
   [cons](x0, x1) = x0 + [0 1 0]x1
                         [0 0 0]  ,
   
                  [1 1 0]  
   [active](x0) = [1 1 0]x0
                  [0 0 1]  
  orientation:
                          [1 1 0]    [0]    [0]              
   active(first(0(),X)) = [1 1 0]X + [0] >= [0] = mark(nil())
                          [0 0 1]    [1]    [1]              
   
                                   [1 1 0]    [1 1 0]    [1 1 0]    [0]    [1 0 0]        [1 0 0]    [0]                           
   active(first(s(X),cons(Y,Z))) = [1 1 0]X + [1 1 0]Y + [1 1 0]Z + [0] >= [0 1 0]X + Y + [0 1 0]Z + [0] = mark(cons(Y,first(X,Z)))
                                   [0 0 1]    [0 0 1]    [0 0 0]    [1]    [0 0 0]        [0 0 0]    [1]                           
   
                     [2 2 0]    [0]    [2 1 0]    [0]                           
   active(from(X)) = [2 2 0]X + [0] >= [1 2 0]X + [0] = mark(cons(X,from(s(X))))
                     [0 0 1]    [1]    [0 0 1]    [1]                           
   
                          [1 1 0]     [1 1 0]     [0]    [1 1 0]          [0]                       
   active(first(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 + [0] >= [1 1 0]X1 + X2 + [0] = first(active(X1),X2)
                          [0 0 1]     [0 0 1]     [1]    [0 0 1]          [1]                       
   
                          [1 1 0]     [1 1 0]     [0]         [1 1 0]     [0]                       
   active(first(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 + [0] >= X1 + [1 1 0]X2 + [0] = first(X1,active(X2))
                          [0 0 1]     [0 0 1]     [1]         [0 0 1]     [1]                       
   
                  [1 1 0]     [1 1 0]                
   active(s(X)) = [1 1 0]X >= [1 1 0]X = s(active(X))
                  [0 0 1]     [0 0 1]                
   
                         [1 1 0]     [1 1 0]      [1 1 0]     [1 0 0]                        
   active(cons(X1,X2)) = [1 1 0]X1 + [1 1 0]X2 >= [1 1 0]X1 + [0 1 0]X2 = cons(active(X1),X2)
                         [0 0 1]     [0 0 0]      [0 0 1]     [0 0 0]                        
   
                     [2 2 0]    [0]    [2 2 0]    [0]                  
   active(from(X)) = [2 2 0]X + [0] >= [2 2 0]X + [0] = from(active(X))
                     [0 0 1]    [1]    [0 0 1]    [1]                  
   
                                  [0]              [0]                     
   first(mark(X1),X2) = X1 + X2 + [0] >= X1 + X2 + [0] = mark(first(X1,X2))
                                  [2]              [2]                     
   
                                  [0]              [0]                     
   first(X1,mark(X2)) = X1 + X2 + [0] >= X1 + X2 + [0] = mark(first(X1,X2))
                                  [2]              [2]                     
   
                    [0]        [0]             
   s(mark(X)) = X + [0] >= X + [0] = mark(s(X))
                    [1]        [1]             
   
                            [1 0 0]     [0]         [1 0 0]     [0]                    
   cons(mark(X1),X2) = X1 + [0 1 0]X2 + [0] >= X1 + [0 1 0]X2 + [0] = mark(cons(X1,X2))
                            [0 0 0]     [1]         [0 0 0]     [1]                    
   
                   [1 1 0]    [0]    [1 1 0]    [0]                
   from(mark(X)) = [1 1 0]X + [0] >= [1 1 0]X + [0] = mark(from(X))
                   [0 0 1]    [2]    [0 0 1]    [2]                
   
                                    [0]              [0]                               
   proper(first(X1,X2)) = X1 + X2 + [0] >= X1 + X2 + [0] = first(proper(X1),proper(X2))
                                    [1]              [1]                               
   
                 [0]    [0]          
   proper(0()) = [0] >= [0] = ok(0())
                 [0]    [0]          
   
                   [0]    [0]            
   proper(nil()) = [0] >= [0] = ok(nil())
                   [0]    [0]            
   
                                       
   proper(s(X)) = X >= X = s(proper(X))
                                       
   
                              [1 0 0]           [1 0 0]                                
   proper(cons(X1,X2)) = X1 + [0 1 0]X2 >= X1 + [0 1 0]X2 = cons(proper(X1),proper(X2))
                              [0 0 0]           [0 0 0]                                
   
                     [1 1 0]    [0]    [1 1 0]    [0]                  
   proper(from(X)) = [1 1 0]X + [0] >= [1 1 0]X + [0] = from(proper(X))
                     [0 0 1]    [1]    [0 0 1]    [1]                  
   
                          [1 1 0]     [1 1 0]     [0]    [1 1 0]     [1 1 0]     [0]                   
   first(ok(X1),ok(X2)) = [1 1 0]X1 + [1 1 0]X2 + [0] >= [1 1 0]X1 + [1 1 0]X2 + [0] = ok(first(X1,X2))
                          [0 0 1]     [0 0 1]     [1]    [0 0 1]     [0 0 1]     [1]                   
   
              [1 1 0]     [1 1 0]            
   s(ok(X)) = [1 1 0]X >= [1 1 0]X = ok(s(X))
              [0 0 1]     [0 0 1]            
   
                         [1 1 0]     [1 1 0]      [1 1 0]     [1 1 0]                    
   cons(ok(X1),ok(X2)) = [1 1 0]X1 + [1 1 0]X2 >= [1 1 0]X1 + [1 1 0]X2 = ok(cons(X1,X2))
                         [0 0 1]     [0 0 0]      [0 0 1]     [0 0 0]                    
   
                 [2 2 0]    [0]    [2 2 0]    [0]              
   from(ok(X)) = [2 2 0]X + [0] >= [2 2 0]X + [0] = ok(from(X))
                 [0 0 1]    [1]    [0 0 1]    [1]              
   
                  [1 0 1]    [1]    [1 0 1]                  
   top(mark(X)) = [1 1 1]X + [1] >= [1 1 1]X = top(proper(X))
                  [1 0 0]    [0]    [1 0 0]                  
   
                [1 1 1]     [1 1 1]                  
   top(ok(X)) = [2 2 1]X >= [2 2 1]X = top(active(X))
                [1 1 0]     [1 1 0]                  
  problem:
   active(first(0(),X)) -> mark(nil())
   active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
   active(from(X)) -> mark(cons(X,from(s(X))))
   active(first(X1,X2)) -> first(active(X1),X2)
   active(first(X1,X2)) -> first(X1,active(X2))
   active(s(X)) -> s(active(X))
   active(cons(X1,X2)) -> cons(active(X1),X2)
   active(from(X)) -> from(active(X))
   first(mark(X1),X2) -> mark(first(X1,X2))
   first(X1,mark(X2)) -> mark(first(X1,X2))
   s(mark(X)) -> mark(s(X))
   cons(mark(X1),X2) -> mark(cons(X1,X2))
   from(mark(X)) -> mark(from(X))
   proper(first(X1,X2)) -> first(proper(X1),proper(X2))
   proper(0()) -> ok(0())
   proper(nil()) -> ok(nil())
   proper(s(X)) -> s(proper(X))
   proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
   proper(from(X)) -> from(proper(X))
   first(ok(X1),ok(X2)) -> ok(first(X1,X2))
   s(ok(X)) -> ok(s(X))
   cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
   from(ok(X)) -> ok(from(X))
   top(ok(X)) -> top(active(X))
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [mark](x0) = x0,
    
    [proper](x0) = 4x0,
    
    [first](x0, x1) = x0 + 4x1,
    
    [nil] = 1,
    
    [s](x0) = x0,
    
    [top](x0) = 2x0,
    
    [0] = 6,
    
    [from](x0) = x0,
    
    [ok](x0) = 4x0,
    
    [cons](x0, x1) = x0 + x1,
    
    [active](x0) = 2x0
   orientation:
    active(first(0(),X)) = 8X + 12 >= 1 = mark(nil())
    
    active(first(s(X),cons(Y,Z))) = 2X + 8Y + 8Z >= X + Y + 4Z = mark(cons(Y,first(X,Z)))
    
    active(from(X)) = 2X >= 2X = mark(cons(X,from(s(X))))
    
    active(first(X1,X2)) = 2X1 + 8X2 >= 2X1 + 4X2 = first(active(X1),X2)
    
    active(first(X1,X2)) = 2X1 + 8X2 >= X1 + 8X2 = first(X1,active(X2))
    
    active(s(X)) = 2X >= 2X = s(active(X))
    
    active(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + X2 = cons(active(X1),X2)
    
    active(from(X)) = 2X >= 2X = from(active(X))
    
    first(mark(X1),X2) = X1 + 4X2 >= X1 + 4X2 = mark(first(X1,X2))
    
    first(X1,mark(X2)) = X1 + 4X2 >= X1 + 4X2 = mark(first(X1,X2))
    
    s(mark(X)) = X >= X = mark(s(X))
    
    cons(mark(X1),X2) = X1 + X2 >= X1 + X2 = mark(cons(X1,X2))
    
    from(mark(X)) = X >= X = mark(from(X))
    
    proper(first(X1,X2)) = 4X1 + 16X2 >= 4X1 + 16X2 = first(proper(X1),proper(X2))
    
    proper(0()) = 24 >= 24 = ok(0())
    
    proper(nil()) = 4 >= 4 = ok(nil())
    
    proper(s(X)) = 4X >= 4X = s(proper(X))
    
    proper(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(proper(X1),proper(X2))
    
    proper(from(X)) = 4X >= 4X = from(proper(X))
    
    first(ok(X1),ok(X2)) = 4X1 + 16X2 >= 4X1 + 16X2 = ok(first(X1,X2))
    
    s(ok(X)) = 4X >= 4X = ok(s(X))
    
    cons(ok(X1),ok(X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = ok(cons(X1,X2))
    
    from(ok(X)) = 4X >= 4X = ok(from(X))
    
    top(ok(X)) = 8X >= 4X = top(active(X))
   problem:
    active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z)))
    active(from(X)) -> mark(cons(X,from(s(X))))
    active(first(X1,X2)) -> first(active(X1),X2)
    active(first(X1,X2)) -> first(X1,active(X2))
    active(s(X)) -> s(active(X))
    active(cons(X1,X2)) -> cons(active(X1),X2)
    active(from(X)) -> from(active(X))
    first(mark(X1),X2) -> mark(first(X1,X2))
    first(X1,mark(X2)) -> mark(first(X1,X2))
    s(mark(X)) -> mark(s(X))
    cons(mark(X1),X2) -> mark(cons(X1,X2))
    from(mark(X)) -> mark(from(X))
    proper(first(X1,X2)) -> first(proper(X1),proper(X2))
    proper(0()) -> ok(0())
    proper(nil()) -> ok(nil())
    proper(s(X)) -> s(proper(X))
    proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
    proper(from(X)) -> from(proper(X))
    first(ok(X1),ok(X2)) -> ok(first(X1,X2))
    s(ok(X)) -> ok(s(X))
    cons(ok(X1),ok(X2)) -> ok(cons(X1,X2))
    from(ok(X)) -> ok(from(X))
    top(ok(X)) -> top(active(X))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [mark](x0) = x0 + 3,
     
     [proper](x0) = 5x0,
     
     [first](x0, x1) = 2x0 + x1 + 2,
     
     [nil] = 6,
     
     [s](x0) = x0,
     
     [top](x0) = x0 + 6,
     
     [0] = 4,
     
     [from](x0) = 4x0 + 2,
     
     [ok](x0) = 4x0 + 4,
     
     [cons](x0, x1) = 4x0 + 2x1,
     
     [active](x0) = 4x0
    orientation:
     active(first(s(X),cons(Y,Z))) = 8X + 16Y + 8Z + 8 >= 4X + 4Y + 2Z + 7 = mark(cons(Y,first(X,Z)))
     
     active(from(X)) = 16X + 8 >= 12X + 7 = mark(cons(X,from(s(X))))
     
     active(first(X1,X2)) = 8X1 + 4X2 + 8 >= 8X1 + X2 + 2 = first(active(X1),X2)
     
     active(first(X1,X2)) = 8X1 + 4X2 + 8 >= 2X1 + 4X2 + 2 = first(X1,active(X2))
     
     active(s(X)) = 4X >= 4X = s(active(X))
     
     active(cons(X1,X2)) = 16X1 + 8X2 >= 16X1 + 2X2 = cons(active(X1),X2)
     
     active(from(X)) = 16X + 8 >= 16X + 2 = from(active(X))
     
     first(mark(X1),X2) = 2X1 + X2 + 8 >= 2X1 + X2 + 5 = mark(first(X1,X2))
     
     first(X1,mark(X2)) = 2X1 + X2 + 5 >= 2X1 + X2 + 5 = mark(first(X1,X2))
     
     s(mark(X)) = X + 3 >= X + 3 = mark(s(X))
     
     cons(mark(X1),X2) = 4X1 + 2X2 + 12 >= 4X1 + 2X2 + 3 = mark(cons(X1,X2))
     
     from(mark(X)) = 4X + 14 >= 4X + 5 = mark(from(X))
     
     proper(first(X1,X2)) = 10X1 + 5X2 + 10 >= 10X1 + 5X2 + 2 = first(proper(X1),proper(X2))
     
     proper(0()) = 20 >= 20 = ok(0())
     
     proper(nil()) = 30 >= 28 = ok(nil())
     
     proper(s(X)) = 5X >= 5X = s(proper(X))
     
     proper(cons(X1,X2)) = 20X1 + 10X2 >= 20X1 + 10X2 = cons(proper(X1),proper(X2))
     
     proper(from(X)) = 20X + 10 >= 20X + 2 = from(proper(X))
     
     first(ok(X1),ok(X2)) = 8X1 + 4X2 + 14 >= 8X1 + 4X2 + 12 = ok(first(X1,X2))
     
     s(ok(X)) = 4X + 4 >= 4X + 4 = ok(s(X))
     
     cons(ok(X1),ok(X2)) = 16X1 + 8X2 + 24 >= 16X1 + 8X2 + 4 = ok(cons(X1,X2))
     
     from(ok(X)) = 16X + 18 >= 16X + 12 = ok(from(X))
     
     top(ok(X)) = 4X + 10 >= 4X + 6 = top(active(X))
    problem:
     active(s(X)) -> s(active(X))
     active(cons(X1,X2)) -> cons(active(X1),X2)
     first(X1,mark(X2)) -> mark(first(X1,X2))
     s(mark(X)) -> mark(s(X))
     proper(0()) -> ok(0())
     proper(s(X)) -> s(proper(X))
     proper(cons(X1,X2)) -> cons(proper(X1),proper(X2))
     s(ok(X)) -> ok(s(X))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [mark](x0) = x0,
      
      [proper](x0) = 2x0,
      
      [first](x0, x1) = 4x0 + 2x1,
      
      [s](x0) = x0 + 4,
      
      [0] = 4,
      
      [ok](x0) = x0 + 2,
      
      [cons](x0, x1) = 4x0 + x1 + 4,
      
      [active](x0) = x0
     orientation:
      active(s(X)) = X + 4 >= X + 4 = s(active(X))
      
      active(cons(X1,X2)) = 4X1 + X2 + 4 >= 4X1 + X2 + 4 = cons(active(X1),X2)
      
      first(X1,mark(X2)) = 4X1 + 2X2 >= 4X1 + 2X2 = mark(first(X1,X2))
      
      s(mark(X)) = X + 4 >= X + 4 = mark(s(X))
      
      proper(0()) = 8 >= 6 = ok(0())
      
      proper(s(X)) = 2X + 8 >= 2X + 4 = s(proper(X))
      
      proper(cons(X1,X2)) = 8X1 + 2X2 + 8 >= 8X1 + 2X2 + 4 = cons(proper(X1),proper(X2))
      
      s(ok(X)) = X + 6 >= X + 6 = ok(s(X))
     problem:
      active(s(X)) -> s(active(X))
      active(cons(X1,X2)) -> cons(active(X1),X2)
      first(X1,mark(X2)) -> mark(first(X1,X2))
      s(mark(X)) -> mark(s(X))
      s(ok(X)) -> ok(s(X))
     Matrix Interpretation Processor: dim=1
      
      interpretation:
       [mark](x0) = x0,
       
       [first](x0, x1) = 4x0 + x1,
       
       [s](x0) = 2x0 + 3,
       
       [ok](x0) = 2x0 + 4,
       
       [cons](x0, x1) = 4x0 + 4x1,
       
       [active](x0) = x0
      orientation:
       active(s(X)) = 2X + 3 >= 2X + 3 = s(active(X))
       
       active(cons(X1,X2)) = 4X1 + 4X2 >= 4X1 + 4X2 = cons(active(X1),X2)
       
       first(X1,mark(X2)) = 4X1 + X2 >= 4X1 + X2 = mark(first(X1,X2))
       
       s(mark(X)) = 2X + 3 >= 2X + 3 = mark(s(X))
       
       s(ok(X)) = 4X + 11 >= 4X + 10 = ok(s(X))
      problem:
       active(s(X)) -> s(active(X))
       active(cons(X1,X2)) -> cons(active(X1),X2)
       first(X1,mark(X2)) -> mark(first(X1,X2))
       s(mark(X)) -> mark(s(X))
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [mark](x0) = x0 + 3,
        
        [first](x0, x1) = x0 + x1 + 4,
        
        [s](x0) = 6x0,
        
        [cons](x0, x1) = 2x0 + 4x1,
        
        [active](x0) = x0
       orientation:
        active(s(X)) = 6X >= 6X = s(active(X))
        
        active(cons(X1,X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = cons(active(X1),X2)
        
        first(X1,mark(X2)) = X1 + X2 + 7 >= X1 + X2 + 7 = mark(first(X1,X2))
        
        s(mark(X)) = 6X + 18 >= 6X + 3 = mark(s(X))
       problem:
        active(s(X)) -> s(active(X))
        active(cons(X1,X2)) -> cons(active(X1),X2)
        first(X1,mark(X2)) -> mark(first(X1,X2))
       Matrix Interpretation Processor: dim=1
        
        interpretation:
         [mark](x0) = x0 + 5,
         
         [first](x0, x1) = x0 + x1 + 2,
         
         [s](x0) = 4x0,
         
         [cons](x0, x1) = 4x0 + x1 + 6,
         
         [active](x0) = 2x0
        orientation:
         active(s(X)) = 8X >= 8X = s(active(X))
         
         active(cons(X1,X2)) = 8X1 + 2X2 + 12 >= 8X1 + X2 + 6 = cons(active(X1),X2)
         
         first(X1,mark(X2)) = X1 + X2 + 7 >= X1 + X2 + 7 = mark(first(X1,X2))
        problem:
         active(s(X)) -> s(active(X))
         first(X1,mark(X2)) -> mark(first(X1,X2))
        Matrix Interpretation Processor: dim=1
         
         interpretation:
          [mark](x0) = x0 + 5,
          
          [first](x0, x1) = x0 + 2x1 + 3,
          
          [s](x0) = x0 + 4,
          
          [active](x0) = x0
         orientation:
          active(s(X)) = X + 4 >= X + 4 = s(active(X))
          
          first(X1,mark(X2)) = X1 + 2X2 + 13 >= X1 + 2X2 + 8 = mark(first(X1,X2))
         problem:
          active(s(X)) -> s(active(X))
         Matrix Interpretation Processor: dim=3
          
          interpretation:
                     [1 0 0]     [0]
           [s](x0) = [0 1 0]x0 + [1]
                     [0 0 0]     [0],
           
                          [1 1 0]  
           [active](x0) = [0 1 0]x0
                          [0 0 0]  
          orientation:
                          [1 1 0]    [1]    [1 1 0]    [0]               
           active(s(X)) = [0 1 0]X + [1] >= [0 1 0]X + [1] = s(active(X))
                          [0 0 0]    [0]    [0 0 0]    [0]               
          problem:
           
          Qed