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)
  Usable Rule 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:
    h(0()) -> a()
    i(0()) -> b()
    i(s(y)) -> i(y)
    g(h(x)) -> g(x)
    g(s(x)) -> s(x)
    g(0()) -> s(0())
   EDG 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:
     h(0()) -> a()
     i(0()) -> b()
     i(s(y)) -> i(y)
     g(h(x)) -> g(x)
     g(s(x)) -> s(x)
     g(0()) -> s(0())
    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()) -> i#(y)
     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()) -> g#(h(x))
     f#(s(x),y,b()) -> f#(g(h(x)),y,i(y)) -> f#(s(x),y,b()) -> f#(g(h(x)),y,i(y))
    Restore Modifier:
     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)
     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