MAYBE

Problem:
 f(s(x),y,b()) -> f(g(h(x)),y,i(y))
 g(h(x)) -> g(x)
 g(s(x)) -> s(x)
 g(0()) -> s(0())
 h(0()) -> a()
 i(0()) -> b()
 i(s(y)) -> i(y)

Proof:
 DP Processor:
  DPs:
   f#(s(x),y,b()) -> i#(y)
   f#(s(x),y,b()) -> h#(x)
   f#(s(x),y,b()) -> g#(h(x))
   f#(s(x),y,b()) -> f#(g(h(x)),y,i(y))
   g#(h(x)) -> g#(x)
   i#(s(y)) -> i#(y)
  TRS:
   f(s(x),y,b()) -> f(g(h(x)),y,i(y))
   g(h(x)) -> g(x)
   g(s(x)) -> s(x)
   g(0()) -> s(0())
   h(0()) -> a()
   i(0()) -> b()
   i(s(y)) -> i(y)
  TDG Processor:
   DPs:
    f#(s(x),y,b()) -> i#(y)
    f#(s(x),y,b()) -> h#(x)
    f#(s(x),y,b()) -> g#(h(x))
    f#(s(x),y,b()) -> f#(g(h(x)),y,i(y))
    g#(h(x)) -> g#(x)
    i#(s(y)) -> i#(y)
   TRS:
    f(s(x),y,b()) -> f(g(h(x)),y,i(y))
    g(h(x)) -> g(x)
    g(s(x)) -> s(x)
    g(0()) -> s(0())
    h(0()) -> a()
    i(0()) -> b()
    i(s(y)) -> i(y)
   graph:
    g#(h(x)) -> g#(x) -> g#(h(x)) -> g#(x)
    i#(s(y)) -> i#(y) -> i#(s(y)) -> i#(y)
    f#(s(x),y,b()) -> g#(h(x)) -> g#(h(x)) -> g#(x)
    f#(s(x),y,b()) -> i#(y) -> i#(s(y)) -> i#(y)
    f#(s(x),y,b()) -> f#(g(h(x)),y,i(y)) ->
    f#(s(x),y,b()) -> f#(g(h(x)),y,i(y))
    f#(s(x),y,b()) -> f#(g(h(x)),y,i(y)) ->
    f#(s(x),y,b()) -> g#(h(x))
    f#(s(x),y,b()) -> f#(g(h(x)),y,i(y)) ->
    f#(s(x),y,b()) -> h#(x)
    f#(s(x),y,b()) -> f#(g(h(x)),y,i(y)) -> f#(s(x),y,b()) -> i#(y)
   SCC Processor:
    #sccs: 3
    #rules: 3
    #arcs: 8/36
    DPs:
     f#(s(x),y,b()) -> f#(g(h(x)),y,i(y))
    TRS:
     f(s(x),y,b()) -> f(g(h(x)),y,i(y))
     g(h(x)) -> g(x)
     g(s(x)) -> s(x)
     g(0()) -> s(0())
     h(0()) -> a()
     i(0()) -> b()
     i(s(y)) -> i(y)
    Open
    
    DPs:
     i#(s(y)) -> i#(y)
    TRS:
     f(s(x),y,b()) -> f(g(h(x)),y,i(y))
     g(h(x)) -> g(x)
     g(s(x)) -> s(x)
     g(0()) -> s(0())
     h(0()) -> a()
     i(0()) -> b()
     i(s(y)) -> i(y)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [i#](x0) = x0 + 1,
      
      [a] = 1,
      
      [0] = 0,
      
      [i](x0) = 1,
      
      [g](x0) = x0 + 1,
      
      [h](x0) = x0 + 1,
      
      [f](x0, x1, x2) = 0,
      
      [b] = 0,
      
      [s](x0) = x0 + 1
     orientation:
      i#(s(y)) = y + 2 >= y + 1 = i#(y)
      
      f(s(x),y,b()) = 0 >= 0 = f(g(h(x)),y,i(y))
      
      g(h(x)) = x + 2 >= x + 1 = g(x)
      
      g(s(x)) = x + 2 >= x + 1 = s(x)
      
      g(0()) = 1 >= 1 = s(0())
      
      h(0()) = 1 >= 1 = a()
      
      i(0()) = 1 >= 0 = b()
      
      i(s(y)) = 1 >= 1 = i(y)
     problem:
      DPs:
       
      TRS:
       f(s(x),y,b()) -> f(g(h(x)),y,i(y))
       g(h(x)) -> g(x)
       g(s(x)) -> s(x)
       g(0()) -> s(0())
       h(0()) -> a()
       i(0()) -> b()
       i(s(y)) -> i(y)
     Qed
    
    DPs:
     g#(h(x)) -> g#(x)
    TRS:
     f(s(x),y,b()) -> f(g(h(x)),y,i(y))
     g(h(x)) -> g(x)
     g(s(x)) -> s(x)
     g(0()) -> s(0())
     h(0()) -> a()
     i(0()) -> b()
     i(s(y)) -> i(y)
    Matrix Interpretation Processor:
     dimension: 1
     interpretation:
      [g#](x0) = x0 + 1,
      
      [a] = 1,
      
      [0] = 0,
      
      [i](x0) = 0,
      
      [g](x0) = 0,
      
      [h](x0) = x0 + 1,
      
      [f](x0, x1, x2) = 0,
      
      [b] = 0,
      
      [s](x0) = 0
     orientation:
      g#(h(x)) = x + 2 >= x + 1 = g#(x)
      
      f(s(x),y,b()) = 0 >= 0 = f(g(h(x)),y,i(y))
      
      g(h(x)) = 0 >= 0 = g(x)
      
      g(s(x)) = 0 >= 0 = s(x)
      
      g(0()) = 0 >= 0 = s(0())
      
      h(0()) = 1 >= 1 = a()
      
      i(0()) = 0 >= 0 = b()
      
      i(s(y)) = 0 >= 0 = i(y)
     problem:
      DPs:
       
      TRS:
       f(s(x),y,b()) -> f(g(h(x)),y,i(y))
       g(h(x)) -> g(x)
       g(s(x)) -> s(x)
       g(0()) -> s(0())
       h(0()) -> a()
       i(0()) -> b()
       i(s(y)) -> i(y)
     Qed