YES

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:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                [1 0 0]  
   [pred](x0) = [0 0 0]x0
                [0 0 1]  ,
   
             [1 0 0]  
   [s](x0) = [0 0 0]x0
             [0 0 0]  ,
   
         [0]
   [0] = [0]
         [0],
   
                    [1 0 0]       
   [plus](x0, x1) = [0 0 0]x0 + x1
                    [0 0 0]       ,
   
               [1 0 0]     [0]
   [sum](x0) = [0 1 0]x0 + [0]
               [0 0 0]     [1],
   
                    [1 0 0]     [1 0 0]     [0]
   [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                    [0 0 0]     [0 1 0]     [1],
   
                   [1 0 0]     [1 1 1]  
   [app](x0, x1) = [1 1 0]x0 + [1 1 1]x1
                   [1 1 1]     [1 1 1]  ,
   
           [1]
   [nil] = [0]
           [0]
  orientation:
                  [1 1 1]    [1]         
   app(nil(),k) = [1 1 1]k + [1] >= k = k
                  [1 1 1]    [1]         
   
                  [1 0 0]    [1]         
   app(l,nil()) = [1 1 0]l + [1] >= l = l
                  [1 1 1]    [1]         
   
                      [1 1 1]    [1 0 0]    [1 0 0]    [0]    [1 1 1]    [1 0 0]    [1 0 0]    [0]                   
   app(cons(x,l),k) = [1 1 1]k + [1 0 0]l + [1 0 0]x + [0] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0] = cons(x,app(l,k))
                      [1 1 1]    [1 1 0]    [1 0 0]    [1]    [1 1 1]    [1 1 0]    [0 0 0]    [1]                   
   
                        [1 0 0]    [1]    [1 0 0]    [1]                
   sum(cons(x,nil())) = [0 0 0]x + [0] >= [0 0 0]x + [0] = cons(x,nil())
                        [0 0 0]    [1]    [0 0 0]    [1]                
   
                            [1 0 0]    [1 0 0]    [1 0 0]    [0]    [1 0 0]    [1 0 0]    [1 0 0]    [0]                         
   sum(cons(x,cons(y,l))) = [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] = sum(cons(plus(x,y),l))
                            [0 0 0]    [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [0 0 0]    [1]                         
   
                                   [1 0 0]    [1 0 0]    [1 0 0]    [1 0 0]    [1]    [1 0 0]    [1 0 0]    [1 0 0]    [1 0 0]    [1]                                     
   sum(app(l,cons(x,cons(y,k)))) = [1 0 0]k + [1 1 0]l + [1 0 0]x + [1 0 0]y + [1] >= [1 0 0]k + [1 1 0]l + [1 0 0]x + [1 0 0]y + [1] = sum(app(l,sum(cons(x,cons(y,k)))))
                                   [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0]    [1]                                     
   
                           
   plus(0(),y) = y >= y = y
                           
   
                  [1 0 0]         [1 0 0]    [1 0 0]                
   plus(s(x),y) = [0 0 0]x + y >= [0 0 0]x + [0 0 0]y = s(plus(x,y))
                  [0 0 0]         [0 0 0]    [0 0 0]                
   
                                      [1 0 0]    [1 0 0]    [1 0 0]    [0]    [1 0 0]    [1 0 0]    [1 0 0]    [0]                                  
   sum(plus(cons(0(),x),cons(y,l))) = [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] = pred(sum(cons(s(x),cons(y,l))))
                                      [0 0 0]    [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [0 0 0]    [1]                                  
   
                            [1 0 0]    [1]    [1 0 0]    [1]                
   pred(cons(s(x),nil())) = [0 0 0]x + [0] >= [0 0 0]x + [0] = cons(x,nil())
                            [0 0 0]    [1]    [0 0 0]    [1]                
  problem:
   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())
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [pred](x0) = x0 + 4,
    
    [s](x0) = x0 + 2,
    
    [0] = 6,
    
    [plus](x0, x1) = x0 + x1,
    
    [sum](x0) = x0,
    
    [cons](x0, x1) = x0 + x1,
    
    [app](x0, x1) = x0 + x1 + 2,
    
    [nil] = 0
   orientation:
    app(cons(x,l),k) = k + l + x + 2 >= k + l + x + 2 = cons(x,app(l,k))
    
    sum(cons(x,nil())) = x >= x = cons(x,nil())
    
    sum(cons(x,cons(y,l))) = l + x + y >= l + x + y = sum(cons(plus(x,y),l))
    
    sum(app(l,cons(x,cons(y,k)))) = k + l + x + y + 2 >= k + l + x + y + 2 = sum(app(l,sum(cons(x,cons(y,k)))))
    
    plus(0(),y) = y + 6 >= y = y
    
    plus(s(x),y) = x + y + 2 >= x + y + 2 = s(plus(x,y))
    
    sum(plus(cons(0(),x),cons(y,l))) = l + x + y + 6 >= l + x + y + 6 = pred(sum(cons(s(x),cons(y,l))))
    
    pred(cons(s(x),nil())) = x + 6 >= x = cons(x,nil())
   problem:
    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))))
   DP Processor:
    DPs:
     app#(cons(x,l),k) -> app#(l,k)
     sum#(cons(x,cons(y,l))) -> plus#(x,y)
     sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
     sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
     sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
     sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
     plus#(s(x),y) -> plus#(x,y)
     sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
    TRS:
     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))))
    TDG Processor:
     DPs:
      app#(cons(x,l),k) -> app#(l,k)
      sum#(cons(x,cons(y,l))) -> plus#(x,y)
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      plus#(s(x),y) -> plus#(x,y)
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
     TRS:
      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))))
     graph:
      plus#(s(x),y) -> plus#(x,y) ->
      plus#(s(x),y) -> plus#(x,y)
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
      sum#(cons(x,cons(y,l))) -> plus#(x,y)
      sum#(cons(x,cons(y,l))) -> plus#(x,y) ->
      plus#(s(x),y) -> plus#(x,y)
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
      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)))))
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
      sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
      sum#(cons(x,cons(y,l))) -> plus#(x,y)
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k))) ->
      sum#(cons(x,cons(y,l))) -> plus#(x,y)
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k))))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
      sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,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))))) ->
      sum#(cons(x,cons(y,l))) -> plus#(x,y)
      sum#(app(l,cons(x,cons(y,k)))) -> app#(l,sum(cons(x,cons(y,k)))) ->
      app#(cons(x,l),k) -> app#(l,k)
      app#(cons(x,l),k) -> app#(l,k) -> app#(cons(x,l),k) -> app#(l,k)
     SCC Processor:
      #sccs: 3
      #rules: 6
      #arcs: 28/64
      DPs:
       sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
       sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
       sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
       sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
      TRS:
       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))))
      EDG Processor:
       DPs:
        sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
        sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
        sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
        sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
       TRS:
        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))))
       graph:
        sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
        sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
        sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l)) ->
        sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
        sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,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))))) ->
        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))))) ->
        sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
        sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k))))) ->
        sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
       CDG Processor:
        DPs:
         sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l)))
         sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
         sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,k)))
         sum#(app(l,cons(x,cons(y,k)))) -> sum#(app(l,sum(cons(x,cons(y,k)))))
        TRS:
         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))))
        graph:
         sum#(plus(cons(0(),x),cons(y,l))) -> sum#(cons(s(x),cons(y,l))) ->
         sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
         sum#(app(l,cons(x,cons(y,k)))) -> sum#(cons(x,cons(y,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))))) ->
         sum#(cons(x,cons(y,l))) -> sum#(cons(plus(x,y),l))
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 3/16
         
      
      DPs:
       app#(cons(x,l),k) -> app#(l,k)
      TRS:
       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))))
      Subterm Criterion Processor:
       simple projection:
        pi(app#) = 0
       problem:
        DPs:
         
        TRS:
         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))))
       Qed
      
      DPs:
       plus#(s(x),y) -> plus#(x,y)
      TRS:
       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))))
      Subterm Criterion Processor:
       simple projection:
        pi(plus#) = 0
       problem:
        DPs:
         
        TRS:
         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))))
       Qed