MAYBE

Problem:
 app(nil(),k) -> k
 app(l,nil()) -> l
 app(cons(x,l),k) -> cons(x,app(l,k))
 sum(cons(x,nil())) -> cons(x,nil())
 sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
 sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
 plus(0(),y) -> y
 plus(s(x),y) -> s(plus(x,y))
 sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
 pred(cons(s(x),nil())) -> cons(x,nil())

Proof:
 RT Transformation Processor:
  strict:
   app(nil(),k) -> k
   app(l,nil()) -> l
   app(cons(x,l),k) -> cons(x,app(l,k))
   sum(cons(x,nil())) -> cons(x,nil())
   sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
   sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
   plus(0(),y) -> y
   plus(s(x),y) -> s(plus(x,y))
   sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
   pred(cons(s(x),nil())) -> cons(x,nil())
  weak:
   
  Matrix Interpretation Processor:
   dimension: 1
   interpretation:
    [pred](x0) = x0 + 9,
    
    [s](x0) = x0 + 24,
    
    [0] = 11,
    
    [plus](x0, x1) = x0 + x1 + 2,
    
    [sum](x0) = x0,
    
    [cons](x0, x1) = x0 + x1 + 2,
    
    [app](x0, x1) = x0 + x1,
    
    [nil] = 2
   orientation:
    app(nil(),k) = k + 2 >= k = k
    
    app(l,nil()) = l + 2 >= l = l
    
    app(cons(x,l),k) = k + l + x + 2 >= k + l + x + 2 = cons(x,app(l,k))
    
    sum(cons(x,nil())) = x + 4 >= x + 4 = cons(x,nil())
    
    sum(cons(x,cons(y,l))) = l + x + y + 4 >= l + x + y + 4 = sum(cons(plus(x,y),l))
    
    sum(app(l,cons(x,cons(y,k)))) = k + l + x + y + 4 >= k + l + x + y + 4 = sum(app(l,sum(cons(x,cons(y,k)))))
    
    plus(0(),y) = y + 13 >= y = y
    
    plus(s(x),y) = x + y + 26 >= x + y + 26 = s(plus(x,y))
    
    sum(plus(cons(0(),x),cons(y,l))) = l + x + y + 17 >= l + x + y + 37 = pred(sum(cons(s(x),cons(y,l))))
    
    pred(cons(s(x),nil())) = x + 37 >= x + 4 = cons(x,nil())
   problem:
    strict:
     app(cons(x,l),k) -> cons(x,app(l,k))
     sum(cons(x,nil())) -> cons(x,nil())
     sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
     sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     plus(s(x),y) -> s(plus(x,y))
     sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
    weak:
     app(nil(),k) -> k
     app(l,nil()) -> l
     plus(0(),y) -> y
     pred(cons(s(x),nil())) -> cons(x,nil())
   Matrix Interpretation Processor:
    dimension: 1
    interpretation:
     [pred](x0) = x0 + 3,
     
     [s](x0) = x0 + 22,
     
     [0] = 17,
     
     [plus](x0, x1) = x0 + x1 + 13,
     
     [sum](x0) = x0 + 5,
     
     [cons](x0, x1) = x0 + x1 + 1,
     
     [app](x0, x1) = x0 + x1 + 2,
     
     [nil] = 11
    orientation:
     app(cons(x,l),k) = k + l + x + 3 >= k + l + x + 3 = cons(x,app(l,k))
     
     sum(cons(x,nil())) = x + 17 >= x + 12 = cons(x,nil())
     
     sum(cons(x,cons(y,l))) = l + x + y + 7 >= l + x + y + 19 = sum(cons(plus(x,y),l))
     
     sum(app(l,cons(x,cons(y,k)))) = k + l + x + y + 9 >= k + l + x + y + 14 = sum(app(l,sum(cons(x,cons(y,k)))))
     
     plus(s(x),y) = x + y + 35 >= x + y + 35 = s(plus(x,y))
     
     sum(plus(cons(0(),x),cons(y,l))) = l + x + y + 37 >= l + x + y + 32 = pred(sum(cons(s(x),cons(y,l))))
     
     app(nil(),k) = k + 13 >= k = k
     
     app(l,nil()) = l + 13 >= l = l
     
     plus(0(),y) = y + 30 >= y = y
     
     pred(cons(s(x),nil())) = x + 37 >= x + 12 = cons(x,nil())
    problem:
     strict:
      app(cons(x,l),k) -> cons(x,app(l,k))
      sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
      sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
      plus(s(x),y) -> s(plus(x,y))
     weak:
      sum(cons(x,nil())) -> cons(x,nil())
      sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
      app(nil(),k) -> k
      app(l,nil()) -> l
      plus(0(),y) -> y
      pred(cons(s(x),nil())) -> cons(x,nil())
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [pred](x0) = x0,
      
      [s](x0) = x0 + 17,
      
      [0] = 17,
      
      [plus](x0, x1) = x0 + x1,
      
      [sum](x0) = x0,
      
      [cons](x0, x1) = x0 + x1 + 3,
      
      [app](x0, x1) = x0 + x1 + 10,
      
      [nil] = 14
     orientation:
      app(cons(x,l),k) = k + l + x + 13 >= k + l + x + 13 = cons(x,app(l,k))
      
      sum(cons(x,cons(y,l))) = l + x + y + 6 >= l + x + y + 3 = sum(cons(plus(x,y),l))
      
      sum(app(l,cons(x,cons(y,k)))) = k + l + x + y + 16 >= k + l + x + y + 16 = sum(app(l,sum(cons(x,cons(y,k)))))
      
      plus(s(x),y) = x + y + 17 >= x + y + 17 = s(plus(x,y))
      
      sum(cons(x,nil())) = x + 17 >= x + 17 = cons(x,nil())
      
      sum(plus(cons(0(),x),cons(y,l))) = l + x + y + 23 >= l + x + y + 23 = pred(sum(cons(s(x),cons(y,l))))
      
      app(nil(),k) = k + 24 >= k = k
      
      app(l,nil()) = l + 24 >= l = l
      
      plus(0(),y) = y + 17 >= y = y
      
      pred(cons(s(x),nil())) = x + 34 >= x + 17 = cons(x,nil())
     problem:
      strict:
       app(cons(x,l),k) -> cons(x,app(l,k))
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
       plus(s(x),y) -> s(plus(x,y))
      weak:
       sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
       sum(cons(x,nil())) -> cons(x,nil())
       sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
       app(nil(),k) -> k
       app(l,nil()) -> l
       plus(0(),y) -> y
       pred(cons(s(x),nil())) -> cons(x,nil())
     Matrix Interpretation Processor:
      dimension: 2
      interpretation:
                    [1 4]     [0]
       [pred](x0) = [0 0]x0 + [4],
       
                   
       [s](x0) = x0,
       
             [10]
       [0] = [4 ],
       
                        [1 2]       
       [plus](x0, x1) = [0 0]x0 + x1,
       
                   [1 0]     [0]
       [sum](x0) = [0 0]x0 + [4],
       
                        [1 2]          [0]
       [cons](x0, x1) = [0 0]x0 + x1 + [4],
       
                       [1 4]          [1]
       [app](x0, x1) = [0 1]x0 + x1 + [8],
       
               [6]
       [nil] = [0]
      orientation:
                              [1 4]    [1 2]    [17]        [1 4]    [1 2]    [1 ]                   
       app(cons(x,l),k) = k + [0 1]l + [0 0]x + [12] >= k + [0 1]l + [0 0]x + [12] = cons(x,app(l,k))
       
                                       [1 0]    [1 4]    [1 2]    [1 2]    [1]    [1 0]    [1 4]    [1 2]    [1 2]    [1]                                     
       sum(app(l,cons(x,cons(y,k)))) = [0 0]k + [0 0]l + [0 0]x + [0 0]y + [4] >= [0 0]k + [0 0]l + [0 0]x + [0 0]y + [4] = sum(app(l,sum(cons(x,cons(y,k)))))
       
                      [1 2]         [1 2]                    
       plus(s(x),y) = [0 0]x + y >= [0 0]x + y = s(plus(x,y))
       
                                [1 0]    [1 2]    [1 2]    [0]    [1 0]    [1 2]    [1 2]    [0]                         
       sum(cons(x,cons(y,l))) = [0 0]l + [0 0]x + [0 0]y + [4] >= [0 0]l + [0 0]x + [0 0]y + [4] = sum(cons(plus(x,y),l))
       
                            [1 2]    [6]    [1 2]    [6]                
       sum(cons(x,nil())) = [0 0]x + [4] >= [0 0]x + [4] = cons(x,nil())
       
                                          [1 0]    [1 2]    [1 2]    [26]    [1 0]    [1 2]    [1 2]    [16]                                  
       sum(plus(cons(0(),x),cons(y,l))) = [0 0]l + [0 0]x + [0 0]y + [4 ] >= [0 0]l + [0 0]x + [0 0]y + [4 ] = pred(sum(cons(s(x),cons(y,l))))
       
                          [7]         
       app(nil(),k) = k + [8] >= k = k
       
                      [1 4]    [7]         
       app(l,nil()) = [0 1]l + [8] >= l = l
       
                         [18]         
       plus(0(),y) = y + [0 ] >= y = y
       
                                [1 2]    [22]    [1 2]    [6]                
       pred(cons(s(x),nil())) = [0 0]x + [4 ] >= [0 0]x + [4] = cons(x,nil())
      problem:
       strict:
        sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
        plus(s(x),y) -> s(plus(x,y))
       weak:
        app(cons(x,l),k) -> cons(x,app(l,k))
        sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
        sum(cons(x,nil())) -> cons(x,nil())
        sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
        app(nil(),k) -> k
        app(l,nil()) -> l
        plus(0(),y) -> y
        pred(cons(s(x),nil())) -> cons(x,nil())
      Matrix Interpretation Processor:
       dimension: 2
       interpretation:
                     [1 0]     [10]
        [pred](x0) = [0 0]x0 + [9 ],
        
                  [1 0]  
        [s](x0) = [0 0]x0,
        
              [10]
        [0] = [0 ],
        
                         [1 0]       
        [plus](x0, x1) = [0 0]x0 + x1,
        
                    [1 0]     [0]
        [sum](x0) = [0 0]x0 + [9],
        
                         [1 0]          [0]
        [cons](x0, x1) = [0 0]x0 + x1 + [8],
        
                             [1 1]     [15]
        [app](x0, x1) = x0 + [0 1]x1 + [5 ],
        
                [4]
        [nil] = [1]
       orientation:
                                        [1 1]    [1 0]    [1 0]    [1 0]    [31]    [1 0]    [1 0]    [1 0]    [1 0]    [24]                                     
        sum(app(l,cons(x,cons(y,k)))) = [0 0]k + [0 0]l + [0 0]x + [0 0]y + [9 ] >= [0 0]k + [0 0]l + [0 0]x + [0 0]y + [9 ] = sum(app(l,sum(cons(x,cons(y,k)))))
        
                       [1 0]         [1 0]    [1 0]                
        plus(s(x),y) = [0 0]x + y >= [0 0]x + [0 0]y = s(plus(x,y))
        
                           [1 1]        [1 0]    [15]    [1 1]        [1 0]    [15]                   
        app(cons(x,l),k) = [0 1]k + l + [0 0]x + [13] >= [0 1]k + l + [0 0]x + [13] = cons(x,app(l,k))
        
                                 [1 0]    [1 0]    [1 0]    [0]    [1 0]    [1 0]    [1 0]    [0]                         
        sum(cons(x,cons(y,l))) = [0 0]l + [0 0]x + [0 0]y + [9] >= [0 0]l + [0 0]x + [0 0]y + [9] = sum(cons(plus(x,y),l))
        
                             [1 0]    [4]    [1 0]    [4]                
        sum(cons(x,nil())) = [0 0]x + [9] >= [0 0]x + [9] = cons(x,nil())
        
                                           [1 0]    [1 0]    [1 0]    [10]    [1 0]    [1 0]    [1 0]    [10]                                  
        sum(plus(cons(0(),x),cons(y,l))) = [0 0]l + [0 0]x + [0 0]y + [9 ] >= [0 0]l + [0 0]x + [0 0]y + [9 ] = pred(sum(cons(s(x),cons(y,l))))
        
                       [1 1]    [19]         
        app(nil(),k) = [0 1]k + [6 ] >= k = k
        
                           [20]         
        app(l,nil()) = l + [6 ] >= l = l
        
                          [10]         
        plus(0(),y) = y + [0 ] >= y = y
        
                                 [1 0]    [14]    [1 0]    [4]                
        pred(cons(s(x),nil())) = [0 0]x + [9 ] >= [0 0]x + [9] = cons(x,nil())
       problem:
        strict:
         plus(s(x),y) -> s(plus(x,y))
        weak:
         sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
         app(cons(x,l),k) -> cons(x,app(l,k))
         sum(cons(x,cons(y,l))) -> sum(cons(plus(x,y),l))
         sum(cons(x,nil())) -> cons(x,nil())
         sum(plus(cons(0(),x),cons(y,l))) -> pred(sum(cons(s(x),cons(y,l))))
         app(nil(),k) -> k
         app(l,nil()) -> l
         plus(0(),y) -> y
         pred(cons(s(x),nil())) -> cons(x,nil())
       Open