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=1
  
  interpretation:
   [f](x0, x1, x2) = 4x0 + 2x1 + 2x2 + 4,
   
   [double](x0) = x0,
   
   [a] = 0,
   
   [s](x0) = x0,
   
   [b] = 0,
   
   [tower](x0) = 2x0 + 5,
   
   [exp](x0) = x0,
   
   [half](x0) = x0,
   
   [0] = 0
  orientation:
   tower(x) = 2x + 5 >= 2x + 4 = f(a(),x,s(0()))
   
   f(a(),0(),y) = 2y + 4 >= y = y
   
   f(a(),s(x),y) = 2x + 2y + 4 >= 2x + 2y + 4 = f(b(),y,s(x))
   
   f(b(),y,x) = 2x + 2y + 4 >= 2x + 2y + 4 = 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))
      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:
          [a_f#_1#](x0, x1) = 2x0 + 1,
          
          [00] = 0,
          
          [b_f#_1#](x0, x1) = x1 + 3,
          
          [s0](x0) = x0 + 2,
          
          [half0](x0) = 1/2x0,
          
          [exp0](x0) = 0,
          
          [double0](x0) = 5/2x0
         orientation:
          b_f#_1#(y,x) = x + 3 >= x + 1 = a_f#_1#(half0(x),exp0(y))
          
          a_f#_1#(s0(x),y) = 2x + 5 >= x + 5 = b_f#_1#(y,s0(x))
          
          exp0(00()) = 0 >= 2 = s0(00())
          
          exp0(s0(x)) = 0 >= 0 = double0(exp0(x))
          
          double0(00()) = 0 >= 0 = 00()
          
          double0(s0(x)) = 5/2x + 5 >= 5/2x + 4 = s0(s0(double0(x)))
          
          half0(00()) = 0 >= 0 = double0(00())
          
          half0(s0(00())) = 1 >= 0 = half0(00())
          
          half0(s0(s0(x))) = 1/2x + 2 >= 1/2x + 2 = 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