MAYBE

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=1
  
  interpretation:
   [double](x0) = x0,
   
   [exp](x0) = x0,
   
   [half](x0) = x0,
   
   [b] = 0,
   
   [f](x0, x1, x2) = 4x0 + 2x1 + 2x2,
   
   [s](x0) = x0,
   
   [0] = 2,
   
   [a] = 0,
   
   [tower](x0) = 3x0 + 4
  orientation:
   tower(x) = 3x + 4 >= 2x + 4 = f(a(),x,s(0()))
   
   f(a(),0(),y) = 2y + 4 >= y = y
   
   f(a(),s(x),y) = 2x + 2y >= 2x + 2y = f(b(),y,s(x))
   
   f(b(),y,x) = 2x + 2y >= 2x + 2y = f(a(),half(x),exp(y))
   
   exp(0()) = 2 >= 2 = s(0())
   
   exp(s(x)) = x >= x = double(exp(x))
   
   double(0()) = 2 >= 2 = 0()
   
   double(s(x)) = x >= x = s(s(double(x)))
   
   half(0()) = 2 >= 2 = double(0())
   
   half(s(0())) = 2 >= 2 = half(0())
   
   half(s(s(x))) = x >= x = s(half(x))
  problem:
   tower(x) -> f(a(),x,s(0()))
   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=1
   
   interpretation:
    [double](x0) = x0,
    
    [exp](x0) = x0,
    
    [half](x0) = x0,
    
    [b] = 0,
    
    [f](x0, x1, x2) = 4x0 + x1 + x2,
    
    [s](x0) = x0,
    
    [0] = 0,
    
    [a] = 0,
    
    [tower](x0) = 2x0 + 4
   orientation:
    tower(x) = 2x + 4 >= x = f(a(),x,s(0()))
    
    f(a(),s(x),y) = x + y >= x + y = f(b(),y,s(x))
    
    f(b(),y,x) = x + y >= x + y = f(a(),half(x),exp(y))
    
    exp(0()) = 0 >= 0 = s(0())
    
    exp(s(x)) = x >= x = double(exp(x))
    
    double(0()) = 0 >= 0 = 0()
    
    double(s(x)) = x >= x = s(s(double(x)))
    
    half(0()) = 0 >= 0 = double(0())
    
    half(s(0())) = 0 >= 0 = half(0())
    
    half(s(s(x))) = x >= x = s(half(x))
   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))
       Open
      
      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