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


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YES

Problem:
 a__nats() -> a__adx(a__zeros())
 a__zeros() -> cons(0(),zeros())
 a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
 a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
 a__hd(cons(X,Y)) -> mark(X)
 a__tl(cons(X,Y)) -> mark(Y)
 mark(nats()) -> a__nats()
 mark(adx(X)) -> a__adx(mark(X))
 mark(zeros()) -> a__zeros()
 mark(incr(X)) -> a__incr(mark(X))
 mark(hd(X)) -> a__hd(mark(X))
 mark(tl(X)) -> a__tl(mark(X))
 mark(cons(X1,X2)) -> cons(X1,X2)
 mark(0()) -> 0()
 mark(s(X)) -> s(X)
 a__nats() -> nats()
 a__adx(X) -> adx(X)
 a__zeros() -> zeros()
 a__incr(X) -> incr(X)
 a__hd(X) -> hd(X)
 a__tl(X) -> tl(X)

Proof:
 Matrix Interpretation Processor: dim=1
  
  interpretation:
   [tl](x0) = x0 + 2,
   
   [hd](x0) = x0,
   
   [nats] = 4,
   
   [a__tl](x0) = x0 + 4,
   
   [mark](x0) = 4x0,
   
   [a__hd](x0) = x0,
   
   [adx](x0) = x0,
   
   [incr](x0) = x0,
   
   [s](x0) = x0,
   
   [a__incr](x0) = x0,
   
   [cons](x0, x1) = 6x0 + 4x1,
   
   [zeros] = 0,
   
   [0] = 0,
   
   [a__adx](x0) = x0,
   
   [a__zeros] = 0,
   
   [a__nats] = 5
  orientation:
   a__nats() = 5 >= 0 = a__adx(a__zeros())
   
   a__zeros() = 0 >= 0 = cons(0(),zeros())
   
   a__incr(cons(X,Y)) = 6X + 4Y >= 6X + 4Y = cons(s(X),incr(Y))
   
   a__adx(cons(X,Y)) = 6X + 4Y >= 6X + 4Y = a__incr(cons(X,adx(Y)))
   
   a__hd(cons(X,Y)) = 6X + 4Y >= 4X = mark(X)
   
   a__tl(cons(X,Y)) = 6X + 4Y + 4 >= 4Y = mark(Y)
   
   mark(nats()) = 16 >= 5 = a__nats()
   
   mark(adx(X)) = 4X >= 4X = a__adx(mark(X))
   
   mark(zeros()) = 0 >= 0 = a__zeros()
   
   mark(incr(X)) = 4X >= 4X = a__incr(mark(X))
   
   mark(hd(X)) = 4X >= 4X = a__hd(mark(X))
   
   mark(tl(X)) = 4X + 8 >= 4X + 4 = a__tl(mark(X))
   
   mark(cons(X1,X2)) = 24X1 + 16X2 >= 6X1 + 4X2 = cons(X1,X2)
   
   mark(0()) = 0 >= 0 = 0()
   
   mark(s(X)) = 4X >= X = s(X)
   
   a__nats() = 5 >= 4 = nats()
   
   a__adx(X) = X >= X = adx(X)
   
   a__zeros() = 0 >= 0 = zeros()
   
   a__incr(X) = X >= X = incr(X)
   
   a__hd(X) = X >= X = hd(X)
   
   a__tl(X) = X + 4 >= X + 2 = tl(X)
  problem:
   a__zeros() -> cons(0(),zeros())
   a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
   a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
   a__hd(cons(X,Y)) -> mark(X)
   mark(adx(X)) -> a__adx(mark(X))
   mark(zeros()) -> a__zeros()
   mark(incr(X)) -> a__incr(mark(X))
   mark(hd(X)) -> a__hd(mark(X))
   mark(cons(X1,X2)) -> cons(X1,X2)
   mark(0()) -> 0()
   mark(s(X)) -> s(X)
   a__adx(X) -> adx(X)
   a__zeros() -> zeros()
   a__incr(X) -> incr(X)
   a__hd(X) -> hd(X)
  Matrix Interpretation Processor: dim=1
   
   interpretation:
    [hd](x0) = x0 + 4,
    
    [mark](x0) = x0,
    
    [a__hd](x0) = x0 + 4,
    
    [adx](x0) = 2x0,
    
    [incr](x0) = x0,
    
    [s](x0) = x0,
    
    [a__incr](x0) = x0,
    
    [cons](x0, x1) = 2x0 + 2x1,
    
    [zeros] = 0,
    
    [0] = 0,
    
    [a__adx](x0) = 2x0,
    
    [a__zeros] = 0
   orientation:
    a__zeros() = 0 >= 0 = cons(0(),zeros())
    
    a__incr(cons(X,Y)) = 2X + 2Y >= 2X + 2Y = cons(s(X),incr(Y))
    
    a__adx(cons(X,Y)) = 4X + 4Y >= 2X + 4Y = a__incr(cons(X,adx(Y)))
    
    a__hd(cons(X,Y)) = 2X + 2Y + 4 >= X = mark(X)
    
    mark(adx(X)) = 2X >= 2X = a__adx(mark(X))
    
    mark(zeros()) = 0 >= 0 = a__zeros()
    
    mark(incr(X)) = X >= X = a__incr(mark(X))
    
    mark(hd(X)) = X + 4 >= X + 4 = a__hd(mark(X))
    
    mark(cons(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = cons(X1,X2)
    
    mark(0()) = 0 >= 0 = 0()
    
    mark(s(X)) = X >= X = s(X)
    
    a__adx(X) = 2X >= 2X = adx(X)
    
    a__zeros() = 0 >= 0 = zeros()
    
    a__incr(X) = X >= X = incr(X)
    
    a__hd(X) = X + 4 >= X + 4 = hd(X)
   problem:
    a__zeros() -> cons(0(),zeros())
    a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
    a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
    mark(adx(X)) -> a__adx(mark(X))
    mark(zeros()) -> a__zeros()
    mark(incr(X)) -> a__incr(mark(X))
    mark(hd(X)) -> a__hd(mark(X))
    mark(cons(X1,X2)) -> cons(X1,X2)
    mark(0()) -> 0()
    mark(s(X)) -> s(X)
    a__adx(X) -> adx(X)
    a__zeros() -> zeros()
    a__incr(X) -> incr(X)
    a__hd(X) -> hd(X)
   Matrix Interpretation Processor: dim=1
    
    interpretation:
     [hd](x0) = x0,
     
     [mark](x0) = 6x0 + 2,
     
     [a__hd](x0) = x0,
     
     [adx](x0) = 3x0 + 1,
     
     [incr](x0) = x0,
     
     [s](x0) = x0,
     
     [a__incr](x0) = x0,
     
     [cons](x0, x1) = x0 + 2x1,
     
     [zeros] = 0,
     
     [0] = 0,
     
     [a__adx](x0) = 3x0 + 2,
     
     [a__zeros] = 0
    orientation:
     a__zeros() = 0 >= 0 = cons(0(),zeros())
     
     a__incr(cons(X,Y)) = X + 2Y >= X + 2Y = cons(s(X),incr(Y))
     
     a__adx(cons(X,Y)) = 3X + 6Y + 2 >= X + 6Y + 2 = a__incr(cons(X,adx(Y)))
     
     mark(adx(X)) = 18X + 8 >= 18X + 8 = a__adx(mark(X))
     
     mark(zeros()) = 2 >= 0 = a__zeros()
     
     mark(incr(X)) = 6X + 2 >= 6X + 2 = a__incr(mark(X))
     
     mark(hd(X)) = 6X + 2 >= 6X + 2 = a__hd(mark(X))
     
     mark(cons(X1,X2)) = 6X1 + 12X2 + 2 >= X1 + 2X2 = cons(X1,X2)
     
     mark(0()) = 2 >= 0 = 0()
     
     mark(s(X)) = 6X + 2 >= X = s(X)
     
     a__adx(X) = 3X + 2 >= 3X + 1 = adx(X)
     
     a__zeros() = 0 >= 0 = zeros()
     
     a__incr(X) = X >= X = incr(X)
     
     a__hd(X) = X >= X = hd(X)
    problem:
     a__zeros() -> cons(0(),zeros())
     a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
     a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
     mark(adx(X)) -> a__adx(mark(X))
     mark(incr(X)) -> a__incr(mark(X))
     mark(hd(X)) -> a__hd(mark(X))
     a__zeros() -> zeros()
     a__incr(X) -> incr(X)
     a__hd(X) -> hd(X)
    Matrix Interpretation Processor: dim=3
     
     interpretation:
                 [1 1 1]     [0]
      [hd](x0) = [0 0 0]x0 + [0]
                 [0 0 0]     [1],
      
                   [1 1 1]     [0]
      [mark](x0) = [0 0 0]x0 + [0]
                   [0 0 0]     [1],
      
                    [1 1 1]     [0]
      [a__hd](x0) = [0 0 0]x0 + [0]
                    [0 0 0]     [1],
      
                       [0]
      [adx](x0) = x0 + [1]
                       [0],
      
                   [1 1 0]  
      [incr](x0) = [0 0 0]x0
                   [0 0 1]  ,
      
                [1 0 0]  
      [s](x0) = [0 0 0]x0
                [0 0 0]  ,
      
                      [1 1 0]  
      [a__incr](x0) = [0 0 0]x0
                      [0 0 1]  ,
      
                       [1 0 0]     [1 0 0]     [0]
      [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [0]
                       [0 0 0]     [0 0 0]     [1],
      
                [0]
      [zeros] = [1]
                [0],
      
            [0]
      [0] = [0]
            [0],
      
                     [1 1 1]     [0]
      [a__adx](x0) = [0 1 0]x0 + [0]
                     [0 0 0]     [1],
      
                   [1]
      [a__zeros] = [1]
                   [1]
     orientation:
                   [1]    [0]                    
      a__zeros() = [1] >= [1] = cons(0(),zeros())
                   [1]    [1]                    
      
                           [1 0 0]    [1 1 0]    [0]    [1 0 0]    [1 1 0]    [0]                     
      a__incr(cons(X,Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = cons(s(X),incr(Y))
                           [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                     
      
                          [1 0 0]    [1 1 0]    [1]    [1 0 0]    [1 1 0]    [1]                          
      a__adx(cons(X,Y)) = [0 0 0]X + [0 1 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = a__incr(cons(X,adx(Y)))
                          [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                          
      
                     [1 1 1]    [1]    [1 1 1]    [1]                  
      mark(adx(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__adx(mark(X))
                     [0 0 0]    [1]    [0 0 0]    [1]                  
      
                      [1 1 1]    [0]    [1 1 1]    [0]                   
      mark(incr(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__incr(mark(X))
                      [0 0 0]    [1]    [0 0 0]    [1]                   
      
                    [1 1 1]    [1]    [1 1 1]    [1]                 
      mark(hd(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = a__hd(mark(X))
                    [0 0 0]    [1]    [0 0 0]    [1]                 
      
                   [1]    [0]          
      a__zeros() = [1] >= [1] = zeros()
                   [1]    [0]          
      
                   [1 1 0]     [1 1 0]           
      a__incr(X) = [0 0 0]X >= [0 0 0]X = incr(X)
                   [0 0 1]     [0 0 1]           
      
                 [1 1 1]    [0]    [1 1 1]    [0]        
      a__hd(X) = [0 0 0]X + [0] >= [0 0 0]X + [0] = hd(X)
                 [0 0 0]    [1]    [0 0 0]    [1]        
     problem:
      a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
      a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
      mark(adx(X)) -> a__adx(mark(X))
      mark(incr(X)) -> a__incr(mark(X))
      mark(hd(X)) -> a__hd(mark(X))
      a__incr(X) -> incr(X)
      a__hd(X) -> hd(X)
     Matrix Interpretation Processor: dim=3
      
      interpretation:
                  [1 0 0]  
       [hd](x0) = [0 1 1]x0
                  [0 0 0]  ,
       
                    [1 1 1]     [0]
       [mark](x0) = [0 0 0]x0 + [1]
                    [0 0 0]     [0],
       
                     [1 0 0]  
       [a__hd](x0) = [0 1 1]x0
                     [0 0 0]  ,
       
                   [1 0 0]     [0]
       [adx](x0) = [0 0 0]x0 + [1]
                   [0 1 1]     [0],
       
                    [1 1 0]     [0]
       [incr](x0) = [0 0 0]x0 + [1]
                    [0 0 1]     [0],
       
                 [1 0 0]  
       [s](x0) = [0 0 0]x0
                 [0 0 0]  ,
       
                       [1 1 0]     [0]
       [a__incr](x0) = [0 0 0]x0 + [1]
                       [0 0 1]     [0],
       
                        [1 0 0]     [1 0 0]     [0]
       [cons](x0, x1) = [0 0 0]x0 + [0 1 0]x1 + [0]
                        [0 0 0]     [0 0 0]     [1],
       
                      [1 0 1]     [0]
       [a__adx](x0) = [0 0 0]x0 + [1]
                      [0 0 1]     [0]
      orientation:
                            [1 0 0]    [1 1 0]    [0]    [1 0 0]    [1 1 0]    [0]                     
       a__incr(cons(X,Y)) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = cons(s(X),incr(Y))
                            [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                     
       
                           [1 0 0]    [1 0 0]    [1]    [1 0 0]    [1 0 0]    [1]                          
       a__adx(cons(X,Y)) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = a__incr(cons(X,adx(Y)))
                           [0 0 0]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                          
       
                      [1 1 1]    [1]    [1 1 1]    [0]                  
       mark(adx(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__adx(mark(X))
                      [0 0 0]    [0]    [0 0 0]    [0]                  
       
                       [1 1 1]    [1]    [1 1 1]    [1]                   
       mark(incr(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__incr(mark(X))
                       [0 0 0]    [0]    [0 0 0]    [0]                   
       
                     [1 1 1]    [0]    [1 1 1]    [0]                 
       mark(hd(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = a__hd(mark(X))
                     [0 0 0]    [0]    [0 0 0]    [0]                 
       
                    [1 1 0]    [0]    [1 1 0]    [0]          
       a__incr(X) = [0 0 0]X + [1] >= [0 0 0]X + [1] = incr(X)
                    [0 0 1]    [0]    [0 0 1]    [0]          
       
                  [1 0 0]     [1 0 0]         
       a__hd(X) = [0 1 1]X >= [0 1 1]X = hd(X)
                  [0 0 0]     [0 0 0]         
      problem:
       a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
       a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
       mark(incr(X)) -> a__incr(mark(X))
       mark(hd(X)) -> a__hd(mark(X))
       a__incr(X) -> incr(X)
       a__hd(X) -> hd(X)
      Matrix Interpretation Processor: dim=1
       
       interpretation:
        [hd](x0) = 2x0 + 2,
        
        [mark](x0) = 2x0,
        
        [a__hd](x0) = 2x0 + 4,
        
        [adx](x0) = x0 + 1,
        
        [incr](x0) = 4x0,
        
        [s](x0) = 4x0 + 3,
        
        [a__incr](x0) = 4x0,
        
        [cons](x0, x1) = 3x0 + x1 + 3,
        
        [a__adx](x0) = 4x0 + 4
       orientation:
        a__incr(cons(X,Y)) = 12X + 4Y + 12 >= 12X + 4Y + 12 = cons(s(X),incr(Y))
        
        a__adx(cons(X,Y)) = 12X + 4Y + 16 >= 12X + 4Y + 16 = a__incr(cons(X,adx(Y)))
        
        mark(incr(X)) = 8X >= 8X = a__incr(mark(X))
        
        mark(hd(X)) = 4X + 4 >= 4X + 4 = a__hd(mark(X))
        
        a__incr(X) = 4X >= 4X = incr(X)
        
        a__hd(X) = 2X + 4 >= 2X + 2 = hd(X)
       problem:
        a__incr(cons(X,Y)) -> cons(s(X),incr(Y))
        a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
        mark(incr(X)) -> a__incr(mark(X))
        mark(hd(X)) -> a__hd(mark(X))
        a__incr(X) -> incr(X)
       Matrix Interpretation Processor: dim=3
        
        interpretation:
                    [1 0 0]  
         [hd](x0) = [0 1 0]x0
                    [0 0 0]  ,
         
                      [1 1 0]  
         [mark](x0) = [0 0 1]x0
                      [0 1 0]  ,
         
                       [1 0 0]  
         [a__hd](x0) = [0 0 0]x0
                       [0 0 0]  ,
         
                     [1 1 0]  
         [adx](x0) = [0 0 1]x0
                     [0 0 0]  ,
         
                           [0]
         [incr](x0) = x0 + [1]
                           [1],
         
                   [1 0 0]  
         [s](x0) = [0 0 0]x0
                   [0 0 0]  ,
         
                              [1]
         [a__incr](x0) = x0 + [1]
                              [1],
         
                          [1 0 0]     [1 0 0]  
         [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1
                          [0 0 0]     [0 1 0]  ,
         
                        [1 0 1]     [1]
         [a__adx](x0) = [0 0 0]x0 + [1]
                        [0 1 0]     [1]
        orientation:
                              [1 0 0]    [1 0 0]    [1]    [1 0 0]    [1 0 0]    [0]                     
         a__incr(cons(X,Y)) = [0 0 0]X + [0 0 1]Y + [1] >= [0 0 0]X + [0 0 1]Y + [1] = cons(s(X),incr(Y))
                              [0 0 0]    [0 1 0]    [1]    [0 0 0]    [0 1 0]    [1]                     
         
                             [1 0 0]    [1 1 0]    [1]    [1 0 0]    [1 1 0]    [1]                          
         a__adx(cons(X,Y)) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = a__incr(cons(X,adx(Y)))
                             [0 0 0]    [0 0 1]    [1]    [0 0 0]    [0 0 1]    [1]                          
         
                         [1 1 0]    [1]    [1 1 0]    [1]                   
         mark(incr(X)) = [0 0 1]X + [1] >= [0 0 1]X + [1] = a__incr(mark(X))
                         [0 1 0]    [1]    [0 1 0]    [1]                   
         
                       [1 1 0]     [1 1 0]                  
         mark(hd(X)) = [0 0 0]X >= [0 0 0]X = a__hd(mark(X))
                       [0 1 0]     [0 0 0]                  
         
                          [1]        [0]          
         a__incr(X) = X + [1] >= X + [1] = incr(X)
                          [1]        [1]          
        problem:
         a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
         mark(incr(X)) -> a__incr(mark(X))
         mark(hd(X)) -> a__hd(mark(X))
        Matrix Interpretation Processor: dim=1
         
         interpretation:
          [hd](x0) = x0 + 1,
          
          [mark](x0) = x0 + 4,
          
          [a__hd](x0) = x0,
          
          [adx](x0) = x0 + 3,
          
          [incr](x0) = x0,
          
          [a__incr](x0) = x0,
          
          [cons](x0, x1) = x0 + x1 + 1,
          
          [a__adx](x0) = x0 + 3
         orientation:
          a__adx(cons(X,Y)) = X + Y + 4 >= X + Y + 4 = a__incr(cons(X,adx(Y)))
          
          mark(incr(X)) = X + 4 >= X + 4 = a__incr(mark(X))
          
          mark(hd(X)) = X + 5 >= X + 4 = a__hd(mark(X))
         problem:
          a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
          mark(incr(X)) -> a__incr(mark(X))
         Matrix Interpretation Processor: dim=1
          
          interpretation:
           [mark](x0) = x0 + 6,
           
           [adx](x0) = x0 + 5,
           
           [incr](x0) = x0 + 3,
           
           [a__incr](x0) = x0 + 2,
           
           [cons](x0, x1) = x0 + x1 + 7,
           
           [a__adx](x0) = x0 + 7
          orientation:
           a__adx(cons(X,Y)) = X + Y + 14 >= X + Y + 14 = a__incr(cons(X,adx(Y)))
           
           mark(incr(X)) = X + 9 >= X + 8 = a__incr(mark(X))
          problem:
           a__adx(cons(X,Y)) -> a__incr(cons(X,adx(Y)))
          Matrix Interpretation Processor: dim=1
           
           interpretation:
            [adx](x0) = x0,
            
            [a__incr](x0) = 3x0 + 2,
            
            [cons](x0, x1) = 3x0 + 4x1 + 6,
            
            [a__adx](x0) = 3x0 + 5
           orientation:
            a__adx(cons(X,Y)) = 9X + 12Y + 23 >= 9X + 12Y + 20 = a__incr(cons(X,adx(Y)))
           problem:
            
           Qed