/export/starexec/sandbox2/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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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=1
  
  interpretation:
   [pred](x0) = x0,
   
   [s](x0) = x0,
   
   [0] = 4,
   
   [plus](x0, x1) = x0 + x1,
   
   [sum](x0) = x0,
   
   [cons](x0, x1) = x0 + x1,
   
   [app](x0, x1) = 4x0 + 2x1,
   
   [nil] = 1
  orientation:
   app(nil(),k) = 2k + 4 >= k = k
   
   app(l,nil()) = 4l + 2 >= l = l
   
   app(cons(x,l),k) = 2k + 4l + 4x >= 2k + 4l + x = cons(x,app(l,k))
   
   sum(cons(x,nil())) = x + 1 >= x + 1 = 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)))) = 2k + 4l + 2x + 2y >= 2k + 4l + 2x + 2y = sum(app(l,sum(cons(x,cons(y,k)))))
   
   plus(0(),y) = y + 4 >= y = y
   
   plus(s(x),y) = x + y >= x + y = s(plus(x,y))
   
   sum(plus(cons(0(),x),cons(y,l))) = l + x + y + 4 >= l + x + y = pred(sum(cons(s(x),cons(y,l))))
   
   pred(cons(s(x),nil())) = x + 1 >= x + 1 = 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))
   pred(cons(s(x),nil())) -> cons(x,nil())
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [pred](x0) = 4x0,
    
    [s](x0) = x0,
    
    [plus](x0, x1) = x0 + x1,
    
    [sum](x0) = x0,
    
    [cons](x0, x1) = x0 + x1,
    
    [app](x0, x1) = 4x0 + 4x1,
    
    [nil] = 4
   orientation:
    app(cons(x,l),k) = 4k + 4l + 4x >= 4k + 4l + x = 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 >= l + x + y = sum(cons(plus(x,y),l))
    
    sum(app(l,cons(x,cons(y,k)))) = 4k + 4l + 4x + 4y >= 4k + 4l + 4x + 4y = sum(app(l,sum(cons(x,cons(y,k)))))
    
    plus(s(x),y) = x + y >= x + y = s(plus(x,y))
    
    pred(cons(s(x),nil())) = 4x + 16 >= x + 4 = 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))
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [s](x0) = x0 + 2,
     
     [plus](x0, x1) = x0 + x1,
     
     [sum](x0) = x0,
     
     [cons](x0, x1) = 4x0 + x1 + 2,
     
     [app](x0, x1) = 4x0 + 4x1 + 2,
     
     [nil] = 2
    orientation:
     app(cons(x,l),k) = 4k + 4l + 16x + 10 >= 4k + 4l + 4x + 4 = cons(x,app(l,k))
     
     sum(cons(x,nil())) = 4x + 4 >= 4x + 4 = cons(x,nil())
     
     sum(cons(x,cons(y,l))) = l + 4x + 4y + 4 >= l + 4x + 4y + 2 = sum(cons(plus(x,y),l))
     
     sum(app(l,cons(x,cons(y,k)))) = 4k + 4l + 16x + 16y + 18 >= 4k + 4l + 16x + 16y + 18 = sum(app(l,sum(cons(x,cons(y,k)))))
     
     plus(s(x),y) = x + y + 2 >= x + y + 2 = s(plus(x,y))
    problem:
     sum(cons(x,nil())) -> cons(x,nil())
     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))
    Matrix Interpretation Processor: dim=1
     
     interpretation:
      [s](x0) = x0 + 3,
      
      [plus](x0, x1) = 3x0 + x1 + 2,
      
      [sum](x0) = x0,
      
      [cons](x0, x1) = x0 + x1,
      
      [app](x0, x1) = 4x0 + x1,
      
      [nil] = 4
     orientation:
      sum(cons(x,nil())) = x + 4 >= x + 4 = cons(x,nil())
      
      sum(app(l,cons(x,cons(y,k)))) = k + 4l + x + y >= k + 4l + x + y = sum(app(l,sum(cons(x,cons(y,k)))))
      
      plus(s(x),y) = 3x + y + 11 >= 3x + y + 5 = s(plus(x,y))
     problem:
      sum(cons(x,nil())) -> cons(x,nil())
      sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
     Matrix Interpretation Processor: dim=3
      
      interpretation:
                   [1 1 0]     [0]
       [sum](x0) = [0 0 0]x0 + [1]
                   [1 0 0]     [1],
       
                        [1 1 0]     [1 1 0]  
       [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1
                        [0 0 0]     [0 0 0]  ,
       
                       [1 0 0]     [1 0 0]     [0]
       [app](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                       [0 0 0]     [0 1 0]     [0],
       
               [0]
       [nil] = [0]
               [1]
      orientation:
                            [1 1 0]    [1]    [1 1 0]    [0]                
       sum(cons(x,nil())) = [0 0 0]x + [1] >= [0 0 0]x + [1] = cons(x,nil())
                            [1 1 0]    [1]    [0 0 0]    [0]                
       
                                       [1 1 1]    [1 0 0]    [1 1 0]    [1 1 0]    [1]    [1 1 1]    [1 0 0]    [1 1 0]    [1 1 0]    [1]                                     
       sum(app(l,cons(x,cons(y,k)))) = [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [1] = sum(app(l,sum(cons(x,cons(y,k)))))
                                       [1 1 1]    [1 0 0]    [1 1 0]    [1 1 0]    [1]    [1 1 1]    [1 0 0]    [1 1 0]    [1 1 0]    [1]                                     
      problem:
       sum(app(l,cons(x,cons(y,k)))) -> sum(app(l,sum(cons(x,cons(y,k)))))
      Matrix Interpretation Processor: dim=3
       
       interpretation:
                    [1 0 0]  
        [sum](x0) = [0 0 0]x0
                    [0 0 0]  ,
        
                         [1 0 0]     [1 0 0]     [0]
        [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1]
                         [0 0 0]     [0 0 0]     [0],
        
                        [1 0 0]     [1 1 0]     [0]
        [app](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0]
                        [0 0 0]     [0 0 0]     [1]
       orientation:
                                        [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]                                      
        sum(app(l,cons(x,cons(y,k)))) = [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y + [0] >= [0 0 0]k + [0 0 0]l + [0 0 0]x + [0 0 0]y = sum(app(l,sum(cons(x,cons(y,k)))))
                                        [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0]    [0]    [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0]                                      
       problem:
        
       Qed