/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:
 tower(x) -> f(a(),x,s(0()))
 f(a(),0(),y) -> y
 f(a(),s(x),y) -> f(b(),y,s(x))
 f(b(),y,x) -> f(a(),half(x),exp(y))
 exp(0()) -> s(0())
 exp(s(x)) -> double(exp(x))
 double(0()) -> 0()
 double(s(x)) -> s(s(double(x)))
 half(0()) -> double(0())
 half(s(0())) -> half(0())
 half(s(s(x))) -> s(half(x))

Proof:
 Matrix Interpretation Processor: dim=3
  
  interpretation:
                  [1 0 0]  
   [double](x0) = [0 0 0]x0
                  [0 0 0]  ,
   
               [1 0 0]  
   [exp](x0) = [0 0 0]x0
               [0 0 0]  ,
   
                [1 1 0]  
   [half](x0) = [0 0 0]x0
                [1 0 0]  ,
   
         [0]
   [b] = [0]
         [1],
   
                     [1 0 0]     [1 0 0]     [1 1 0]  
   [f](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 1 0]x2
                     [0 0 1]     [0 0 0]     [0 1 1]  ,
   
             [1 1 0]  
   [s](x0) = [0 0 0]x0
             [0 0 0]  ,
   
         [0]
   [0] = [0]
         [0],
   
         [0]
   [a] = [0]
         [1],
   
                 [1 1 1]     [1]
   [tower](x0) = [1 1 0]x0 + [0]
                 [0 0 0]     [1]
  orientation:
              [1 1 1]    [1]    [1 0 0]    [0]                  
   tower(x) = [1 1 0]x + [0] >= [0 0 0]x + [0] = f(a(),x,s(0()))
              [0 0 0]    [1]    [0 0 0]    [1]                  
   
                  [1 1 0]    [0]         
   f(a(),0(),y) = [0 1 0]y + [0] >= y = y
                  [0 1 1]    [1]         
   
                   [1 1 0]    [1 1 0]    [0]    [1 1 0]    [1 0 0]    [0]                
   f(a(),s(x),y) = [0 0 0]x + [0 1 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = f(b(),y,s(x))
                   [0 0 0]    [0 1 1]    [1]    [0 0 0]    [0 0 0]    [1]                
   
                [1 1 0]    [1 0 0]    [0]    [1 1 0]    [1 0 0]    [0]                        
   f(b(),y,x) = [0 1 0]x + [0 0 0]y + [0] >= [0 0 0]x + [0 0 0]y + [0] = f(a(),half(x),exp(y))
                [0 1 1]    [0 0 0]    [1]    [0 0 0]    [0 0 0]    [1]                        
   
              [0]    [0]         
   exp(0()) = [0] >= [0] = s(0())
              [0]    [0]         
   
               [1 1 0]     [1 0 0]                  
   exp(s(x)) = [0 0 0]x >= [0 0 0]x = double(exp(x))
               [0 0 0]     [0 0 0]                  
   
                 [0]    [0]      
   double(0()) = [0] >= [0] = 0()
                 [0]    [0]      
   
                  [1 1 0]     [1 0 0]                   
   double(s(x)) = [0 0 0]x >= [0 0 0]x = s(s(double(x)))
                  [0 0 0]     [0 0 0]                   
   
               [0]    [0]              
   half(0()) = [0] >= [0] = double(0())
               [0]    [0]              
   
                  [0]    [0]            
   half(s(0())) = [0] >= [0] = half(0())
                  [0]    [0]            
   
                   [1 1 0]     [1 1 0]              
   half(s(s(x))) = [0 0 0]x >= [0 0 0]x = s(half(x))
                   [1 1 0]     [0 0 0]              
  problem:
   f(a(),0(),y) -> y
   f(a(),s(x),y) -> f(b(),y,s(x))
   f(b(),y,x) -> f(a(),half(x),exp(y))
   exp(0()) -> s(0())
   exp(s(x)) -> double(exp(x))
   double(0()) -> 0()
   double(s(x)) -> s(s(double(x)))
   half(0()) -> double(0())
   half(s(0())) -> half(0())
   half(s(s(x))) -> s(half(x))
  Matrix Interpretation Processor: dim=3
   
   interpretation:
                   [1 0 0]     [0]
    [double](x0) = [0 0 0]x0 + [1]
                   [0 0 0]     [0],
    
                [1 0 0]  
    [exp](x0) = [0 1 0]x0
                [0 0 0]  ,
    
                   
    [half](x0) = x0
                   ,
    
          [0]
    [b] = [0]
          [0],
    
                      [1 0 0]     [1 1 0]     [1 1 1]  
    [f](x0, x1, x2) = [0 0 0]x0 + [0 1 0]x1 + [0 1 0]x2
                      [0 0 0]     [0 0 1]     [0 0 1]  ,
    
              [1 0 1]     [0]
    [s](x0) = [0 0 0]x0 + [1]
              [0 0 0]     [0],
    
          [0]
    [0] = [1]
          [0],
    
          [0]
    [a] = [0]
          [0]
   orientation:
                   [1 1 1]    [1]         
    f(a(),0(),y) = [0 1 0]y + [1] >= y = y
                   [0 0 1]    [0]         
    
                    [1 0 1]    [1 1 1]    [1]    [1 0 1]    [1 1 0]    [1]                
    f(a(),s(x),y) = [0 0 0]x + [0 1 0]y + [1] >= [0 0 0]x + [0 1 0]y + [1] = f(b(),y,s(x))
                    [0 0 0]    [0 0 1]    [0]    [0 0 0]    [0 0 1]    [0]                
    
                 [1 1 1]    [1 1 0]     [1 1 0]    [1 1 0]                         
    f(b(),y,x) = [0 1 0]x + [0 1 0]y >= [0 1 0]x + [0 1 0]y = f(a(),half(x),exp(y))
                 [0 0 1]    [0 0 1]     [0 0 1]    [0 0 0]                         
    
               [0]    [0]         
    exp(0()) = [1] >= [1] = s(0())
               [0]    [0]         
    
                [1 0 1]    [0]    [1 0 0]    [0]                 
    exp(s(x)) = [0 0 0]x + [1] >= [0 0 0]x + [1] = double(exp(x))
                [0 0 0]    [0]    [0 0 0]    [0]                 
    
                  [0]    [0]      
    double(0()) = [1] >= [1] = 0()
                  [0]    [0]      
    
                   [1 0 1]    [0]    [1 0 0]    [0]                  
    double(s(x)) = [0 0 0]x + [1] >= [0 0 0]x + [1] = s(s(double(x)))
                   [0 0 0]    [0]    [0 0 0]    [0]                  
    
                [0]    [0]              
    half(0()) = [1] >= [1] = double(0())
                [0]    [0]              
    
                   [0]    [0]            
    half(s(0())) = [1] >= [1] = half(0())
                   [0]    [0]            
    
                    [1 0 1]    [0]    [1 0 1]    [0]             
    half(s(s(x))) = [0 0 0]x + [1] >= [0 0 0]x + [1] = s(half(x))
                    [0 0 0]    [0]    [0 0 0]    [0]             
   problem:
    f(a(),s(x),y) -> f(b(),y,s(x))
    f(b(),y,x) -> f(a(),half(x),exp(y))
    exp(0()) -> s(0())
    exp(s(x)) -> double(exp(x))
    double(0()) -> 0()
    double(s(x)) -> s(s(double(x)))
    half(0()) -> double(0())
    half(s(0())) -> half(0())
    half(s(s(x))) -> s(half(x))
   DP Processor:
    DPs:
     f#(a(),s(x),y) -> f#(b(),y,s(x))
     f#(b(),y,x) -> exp#(y)
     f#(b(),y,x) -> half#(x)
     f#(b(),y,x) -> f#(a(),half(x),exp(y))
     exp#(s(x)) -> exp#(x)
     exp#(s(x)) -> double#(exp(x))
     double#(s(x)) -> double#(x)
     half#(0()) -> double#(0())
     half#(s(0())) -> half#(0())
     half#(s(s(x))) -> half#(x)
    TRS:
     f(a(),s(x),y) -> f(b(),y,s(x))
     f(b(),y,x) -> f(a(),half(x),exp(y))
     exp(0()) -> s(0())
     exp(s(x)) -> double(exp(x))
     double(0()) -> 0()
     double(s(x)) -> s(s(double(x)))
     half(0()) -> double(0())
     half(s(0())) -> half(0())
     half(s(s(x))) -> s(half(x))
    TDG Processor:
     DPs:
      f#(a(),s(x),y) -> f#(b(),y,s(x))
      f#(b(),y,x) -> exp#(y)
      f#(b(),y,x) -> half#(x)
      f#(b(),y,x) -> f#(a(),half(x),exp(y))
      exp#(s(x)) -> exp#(x)
      exp#(s(x)) -> double#(exp(x))
      double#(s(x)) -> double#(x)
      half#(0()) -> double#(0())
      half#(s(0())) -> half#(0())
      half#(s(s(x))) -> half#(x)
     TRS:
      f(a(),s(x),y) -> f(b(),y,s(x))
      f(b(),y,x) -> f(a(),half(x),exp(y))
      exp(0()) -> s(0())
      exp(s(x)) -> double(exp(x))
      double(0()) -> 0()
      double(s(x)) -> s(s(double(x)))
      half(0()) -> double(0())
      half(s(0())) -> half(0())
      half(s(s(x))) -> s(half(x))
     graph:
      double#(s(x)) -> double#(x) -> double#(s(x)) -> double#(x)
      half#(s(s(x))) -> half#(x) -> half#(s(s(x))) -> half#(x)
      half#(s(s(x))) -> half#(x) -> half#(s(0())) -> half#(0())
      half#(s(s(x))) -> half#(x) -> half#(0()) -> double#(0())
      half#(s(0())) -> half#(0()) -> half#(s(s(x))) -> half#(x)
      half#(s(0())) -> half#(0()) -> half#(s(0())) -> half#(0())
      half#(s(0())) -> half#(0()) -> half#(0()) -> double#(0())
      half#(0()) -> double#(0()) -> double#(s(x)) -> double#(x)
      exp#(s(x)) -> double#(exp(x)) -> double#(s(x)) -> double#(x)
      exp#(s(x)) -> exp#(x) -> exp#(s(x)) -> double#(exp(x))
      exp#(s(x)) -> exp#(x) -> exp#(s(x)) -> exp#(x)
      f#(b(),y,x) -> half#(x) -> half#(s(s(x))) -> half#(x)
      f#(b(),y,x) -> half#(x) -> half#(s(0())) -> half#(0())
      f#(b(),y,x) -> half#(x) -> half#(0()) -> double#(0())
      f#(b(),y,x) -> exp#(y) -> exp#(s(x)) -> double#(exp(x))
      f#(b(),y,x) -> exp#(y) -> exp#(s(x)) -> exp#(x)
      f#(b(),y,x) -> f#(a(),half(x),exp(y)) ->
      f#(b(),y,x) -> f#(a(),half(x),exp(y))
      f#(b(),y,x) -> f#(a(),half(x),exp(y)) ->
      f#(b(),y,x) -> half#(x)
      f#(b(),y,x) -> f#(a(),half(x),exp(y)) ->
      f#(b(),y,x) -> exp#(y)
      f#(b(),y,x) -> f#(a(),half(x),exp(y)) ->
      f#(a(),s(x),y) -> f#(b(),y,s(x))
      f#(a(),s(x),y) -> f#(b(),y,s(x)) ->
      f#(b(),y,x) -> f#(a(),half(x),exp(y))
      f#(a(),s(x),y) -> f#(b(),y,s(x)) -> f#(b(),y,x) -> half#(x)
      f#(a(),s(x),y) -> f#(b(),y,s(x)) -> f#(b(),y,x) -> exp#(y)
      f#(a(),s(x),y) -> f#(b(),y,s(x)) -> f#(a(),s(x),y) -> f#(b(),y,s(x))
     SCC Processor:
      #sccs: 4
      #rules: 6
      #arcs: 24/100
      DPs:
       f#(b(),y,x) -> f#(a(),half(x),exp(y))
       f#(a(),s(x),y) -> f#(b(),y,s(x))
      TRS:
       f(a(),s(x),y) -> f(b(),y,s(x))
       f(b(),y,x) -> f(a(),half(x),exp(y))
       exp(0()) -> s(0())
       exp(s(x)) -> double(exp(x))
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       half(0()) -> double(0())
       half(s(0())) -> half(0())
       half(s(s(x))) -> s(half(x))
      EDG Processor:
       DPs:
        f#(b(),y,x) -> f#(a(),half(x),exp(y))
        f#(a(),s(x),y) -> f#(b(),y,s(x))
       TRS:
        f(a(),s(x),y) -> f(b(),y,s(x))
        f(b(),y,x) -> f(a(),half(x),exp(y))
        exp(0()) -> s(0())
        exp(s(x)) -> double(exp(x))
        double(0()) -> 0()
        double(s(x)) -> s(s(double(x)))
        half(0()) -> double(0())
        half(s(0())) -> half(0())
        half(s(s(x))) -> s(half(x))
       graph:
        f#(b(),y,x) -> f#(a(),half(x),exp(y)) ->
        f#(a(),s(x),y) -> f#(b(),y,s(x))
        f#(a(),s(x),y) -> f#(b(),y,s(x)) -> f#(b(),y,x) -> f#(a(),half(x),exp(y))
       Extended Uncurrying Processor:
        application symbol: f#
        symbol table:
         double ==> double0/1
         exp ==> exp0/1
         half ==> half0/1
         b ==> b0/0 b_f#_1#/2
         f ==> f0/3
         s ==> s0/1
         0 ==> 00/0
         a ==> a0/0 a_f#_1#/2
         
        uncurry-rules:
         
        eta-rules:
         
        problem:
         DPs:
          b_f#_1#(y,x) -> a_f#_1#(half0(x),exp0(y))
          a_f#_1#(s0(x),y) -> b_f#_1#(y,s0(x))
         TRS:
          f0(a0(),s0(x),y) -> f0(b0(),y,s0(x))
          f0(b0(),y,x) -> f0(a0(),half0(x),exp0(y))
          exp0(00()) -> s0(00())
          exp0(s0(x)) -> double0(exp0(x))
          double0(00()) -> 00()
          double0(s0(x)) -> s0(s0(double0(x)))
          half0(00()) -> double0(00())
          half0(s0(00())) -> half0(00())
          half0(s0(s0(x))) -> s0(half0(x))
        Usable Rule Processor:
         DPs:
          b_f#_1#(y,x) -> a_f#_1#(half0(x),exp0(y))
          a_f#_1#(s0(x),y) -> b_f#_1#(y,s0(x))
         TRS:
          exp0(00()) -> s0(00())
          exp0(s0(x)) -> double0(exp0(x))
          double0(00()) -> 00()
          double0(s0(x)) -> s0(s0(double0(x)))
          half0(00()) -> double0(00())
          half0(s0(00())) -> half0(00())
          half0(s0(s0(x))) -> s0(half0(x))
         Matrix Interpretation Processor: dim=1
          
          usable rules:
           double0(00()) -> 00()
           double0(s0(x)) -> s0(s0(double0(x)))
           half0(00()) -> double0(00())
           half0(s0(00())) -> half0(00())
           half0(s0(s0(x))) -> s0(half0(x))
          interpretation:
           [double0](x0) = 2x0,
           
           [exp0](x0) = 0,
           
           [half0](x0) = 1/2x0,
           
           [b_f#_1#](x0, x1) = 1/2x1 + 1/2,
           
           [s0](x0) = x0 + 1,
           
           [00] = 0,
           
           [a_f#_1#](x0, x1) = x0
          orientation:
           b_f#_1#(y,x) = 1/2x + 1/2 >= 1/2x = a_f#_1#(half0(x),exp0(y))
           
           a_f#_1#(s0(x),y) = x + 1 >= 1/2x + 1 = b_f#_1#(y,s0(x))
           
           exp0(00()) = 0 >= 1 = s0(00())
           
           exp0(s0(x)) = 0 >= 0 = double0(exp0(x))
           
           double0(00()) = 0 >= 0 = 00()
           
           double0(s0(x)) = 2x + 2 >= 2x + 2 = s0(s0(double0(x)))
           
           half0(00()) = 0 >= 0 = double0(00())
           
           half0(s0(00())) = 1/2 >= 0 = half0(00())
           
           half0(s0(s0(x))) = 1/2x + 1 >= 1/2x + 1 = s0(half0(x))
          problem:
           DPs:
            a_f#_1#(s0(x),y) -> b_f#_1#(y,s0(x))
           TRS:
            exp0(00()) -> s0(00())
            exp0(s0(x)) -> double0(exp0(x))
            double0(00()) -> 00()
            double0(s0(x)) -> s0(s0(double0(x)))
            half0(00()) -> double0(00())
            half0(s0(00())) -> half0(00())
            half0(s0(s0(x))) -> s0(half0(x))
          Restore Modifier:
           DPs:
            a_f#_1#(s0(x),y) -> b_f#_1#(y,s0(x))
           TRS:
            f0(a0(),s0(x),y) -> f0(b0(),y,s0(x))
            f0(b0(),y,x) -> f0(a0(),half0(x),exp0(y))
            exp0(00()) -> s0(00())
            exp0(s0(x)) -> double0(exp0(x))
            double0(00()) -> 00()
            double0(s0(x)) -> s0(s0(double0(x)))
            half0(00()) -> double0(00())
            half0(s0(00())) -> half0(00())
            half0(s0(s0(x))) -> s0(half0(x))
           SCC Processor:
            #sccs: 0
            #rules: 0
            #arcs: 2/1
            
      
      DPs:
       exp#(s(x)) -> exp#(x)
      TRS:
       f(a(),s(x),y) -> f(b(),y,s(x))
       f(b(),y,x) -> f(a(),half(x),exp(y))
       exp(0()) -> s(0())
       exp(s(x)) -> double(exp(x))
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       half(0()) -> double(0())
       half(s(0())) -> half(0())
       half(s(s(x))) -> s(half(x))
      Subterm Criterion Processor:
       simple projection:
        pi(exp#) = 0
       problem:
        DPs:
         
        TRS:
         f(a(),s(x),y) -> f(b(),y,s(x))
         f(b(),y,x) -> f(a(),half(x),exp(y))
         exp(0()) -> s(0())
         exp(s(x)) -> double(exp(x))
         double(0()) -> 0()
         double(s(x)) -> s(s(double(x)))
         half(0()) -> double(0())
         half(s(0())) -> half(0())
         half(s(s(x))) -> s(half(x))
       Qed
      
      DPs:
       half#(s(s(x))) -> half#(x)
       half#(s(0())) -> half#(0())
      TRS:
       f(a(),s(x),y) -> f(b(),y,s(x))
       f(b(),y,x) -> f(a(),half(x),exp(y))
       exp(0()) -> s(0())
       exp(s(x)) -> double(exp(x))
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       half(0()) -> double(0())
       half(s(0())) -> half(0())
       half(s(s(x))) -> s(half(x))
      Subterm Criterion Processor:
       simple projection:
        pi(half#) = 0
       problem:
        DPs:
         
        TRS:
         f(a(),s(x),y) -> f(b(),y,s(x))
         f(b(),y,x) -> f(a(),half(x),exp(y))
         exp(0()) -> s(0())
         exp(s(x)) -> double(exp(x))
         double(0()) -> 0()
         double(s(x)) -> s(s(double(x)))
         half(0()) -> double(0())
         half(s(0())) -> half(0())
         half(s(s(x))) -> s(half(x))
       Qed
      
      DPs:
       double#(s(x)) -> double#(x)
      TRS:
       f(a(),s(x),y) -> f(b(),y,s(x))
       f(b(),y,x) -> f(a(),half(x),exp(y))
       exp(0()) -> s(0())
       exp(s(x)) -> double(exp(x))
       double(0()) -> 0()
       double(s(x)) -> s(s(double(x)))
       half(0()) -> double(0())
       half(s(0())) -> half(0())
       half(s(s(x))) -> s(half(x))
      Subterm Criterion Processor:
       simple projection:
        pi(double#) = 0
       problem:
        DPs:
         
        TRS:
         f(a(),s(x),y) -> f(b(),y,s(x))
         f(b(),y,x) -> f(a(),half(x),exp(y))
         exp(0()) -> s(0())
         exp(s(x)) -> double(exp(x))
         double(0()) -> 0()
         double(s(x)) -> s(s(double(x)))
         half(0()) -> double(0())
         half(s(0())) -> half(0())
         half(s(s(x))) -> s(half(x))
       Qed