MAYBE

Problem:
 a__h(X) -> a__g(mark(X),X)
 a__g(a(),X) -> a__f(b(),X)
 a__f(X,X) -> a__h(a__a())
 a__a() -> b()
 mark(h(X)) -> a__h(mark(X))
 mark(g(X1,X2)) -> a__g(mark(X1),X2)
 mark(a()) -> a__a()
 mark(f(X1,X2)) -> a__f(mark(X1),X2)
 mark(b()) -> b()
 a__h(X) -> h(X)
 a__g(X1,X2) -> g(X1,X2)
 a__a() -> a()
 a__f(X1,X2) -> f(X1,X2)

Proof:
 DP Processor:
  DPs:
   a__h#(X) -> mark#(X)
   a__h#(X) -> a__g#(mark(X),X)
   a__g#(a(),X) -> a__f#(b(),X)
   a__f#(X,X) -> a__a#()
   a__f#(X,X) -> a__h#(a__a())
   mark#(h(X)) -> mark#(X)
   mark#(h(X)) -> a__h#(mark(X))
   mark#(g(X1,X2)) -> mark#(X1)
   mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
   mark#(a()) -> a__a#()
   mark#(f(X1,X2)) -> mark#(X1)
   mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
  TRS:
   a__h(X) -> a__g(mark(X),X)
   a__g(a(),X) -> a__f(b(),X)
   a__f(X,X) -> a__h(a__a())
   a__a() -> b()
   mark(h(X)) -> a__h(mark(X))
   mark(g(X1,X2)) -> a__g(mark(X1),X2)
   mark(a()) -> a__a()
   mark(f(X1,X2)) -> a__f(mark(X1),X2)
   mark(b()) -> b()
   a__h(X) -> h(X)
   a__g(X1,X2) -> g(X1,X2)
   a__a() -> a()
   a__f(X1,X2) -> f(X1,X2)
  TDG Processor:
   DPs:
    a__h#(X) -> mark#(X)
    a__h#(X) -> a__g#(mark(X),X)
    a__g#(a(),X) -> a__f#(b(),X)
    a__f#(X,X) -> a__a#()
    a__f#(X,X) -> a__h#(a__a())
    mark#(h(X)) -> mark#(X)
    mark#(h(X)) -> a__h#(mark(X))
    mark#(g(X1,X2)) -> mark#(X1)
    mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
    mark#(a()) -> a__a#()
    mark#(f(X1,X2)) -> mark#(X1)
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
   TRS:
    a__h(X) -> a__g(mark(X),X)
    a__g(a(),X) -> a__f(b(),X)
    a__f(X,X) -> a__h(a__a())
    a__a() -> b()
    mark(h(X)) -> a__h(mark(X))
    mark(g(X1,X2)) -> a__g(mark(X1),X2)
    mark(a()) -> a__a()
    mark(f(X1,X2)) -> a__f(mark(X1),X2)
    mark(b()) -> b()
    a__h(X) -> h(X)
    a__g(X1,X2) -> g(X1,X2)
    a__a() -> a()
    a__f(X1,X2) -> f(X1,X2)
   graph:
    a__f#(X,X) -> a__h#(a__a()) -> a__h#(X) -> a__g#(mark(X),X)
    a__f#(X,X) -> a__h#(a__a()) -> a__h#(X) -> mark#(X)
    a__g#(a(),X) -> a__f#(b(),X) -> a__f#(X,X) -> a__h#(a__a())
    a__g#(a(),X) -> a__f#(b(),X) -> a__f#(X,X) -> a__a#()
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2) ->
    a__f#(X,X) -> a__h#(a__a())
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2) -> a__f#(X,X) -> a__a#()
    mark#(f(X1,X2)) -> mark#(X1) ->
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    mark#(f(X1,X2)) -> mark#(X1) -> mark#(f(X1,X2)) -> mark#(X1)
    mark#(f(X1,X2)) -> mark#(X1) -> mark#(a()) -> a__a#()
    mark#(f(X1,X2)) -> mark#(X1) ->
    mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
    mark#(f(X1,X2)) -> mark#(X1) -> mark#(g(X1,X2)) -> mark#(X1)
    mark#(f(X1,X2)) -> mark#(X1) -> mark#(h(X)) -> a__h#(mark(X))
    mark#(f(X1,X2)) -> mark#(X1) -> mark#(h(X)) -> mark#(X)
    mark#(g(X1,X2)) -> a__g#(mark(X1),X2) ->
    a__g#(a(),X) -> a__f#(b(),X)
    mark#(g(X1,X2)) -> mark#(X1) ->
    mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    mark#(g(X1,X2)) -> mark#(X1) -> mark#(f(X1,X2)) -> mark#(X1)
    mark#(g(X1,X2)) -> mark#(X1) -> mark#(a()) -> a__a#()
    mark#(g(X1,X2)) -> mark#(X1) ->
    mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
    mark#(g(X1,X2)) -> mark#(X1) -> mark#(g(X1,X2)) -> mark#(X1)
    mark#(g(X1,X2)) -> mark#(X1) -> mark#(h(X)) -> a__h#(mark(X))
    mark#(g(X1,X2)) -> mark#(X1) -> mark#(h(X)) -> mark#(X)
    mark#(h(X)) -> mark#(X) -> mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    mark#(h(X)) -> mark#(X) -> mark#(f(X1,X2)) -> mark#(X1)
    mark#(h(X)) -> mark#(X) -> mark#(a()) -> a__a#()
    mark#(h(X)) -> mark#(X) -> mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
    mark#(h(X)) -> mark#(X) -> mark#(g(X1,X2)) -> mark#(X1)
    mark#(h(X)) -> mark#(X) -> mark#(h(X)) -> a__h#(mark(X))
    mark#(h(X)) -> mark#(X) -> mark#(h(X)) -> mark#(X)
    mark#(h(X)) -> a__h#(mark(X)) -> a__h#(X) -> a__g#(mark(X),X)
    mark#(h(X)) -> a__h#(mark(X)) -> a__h#(X) -> mark#(X)
    a__h#(X) -> a__g#(mark(X),X) -> a__g#(a(),X) -> a__f#(b(),X)
    a__h#(X) -> mark#(X) -> mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    a__h#(X) -> mark#(X) -> mark#(f(X1,X2)) -> mark#(X1)
    a__h#(X) -> mark#(X) -> mark#(a()) -> a__a#()
    a__h#(X) -> mark#(X) -> mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
    a__h#(X) -> mark#(X) -> mark#(g(X1,X2)) -> mark#(X1)
    a__h#(X) -> mark#(X) -> mark#(h(X)) -> a__h#(mark(X))
    a__h#(X) -> mark#(X) -> mark#(h(X)) -> mark#(X)
   SCC Processor:
    #sccs: 1
    #rules: 10
    #arcs: 38/144
    DPs:
     a__f#(X,X) -> a__h#(a__a())
     a__h#(X) -> mark#(X)
     mark#(h(X)) -> mark#(X)
     mark#(h(X)) -> a__h#(mark(X))
     a__h#(X) -> a__g#(mark(X),X)
     a__g#(a(),X) -> a__f#(b(),X)
     mark#(g(X1,X2)) -> mark#(X1)
     mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
     mark#(f(X1,X2)) -> mark#(X1)
     mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
    TRS:
     a__h(X) -> a__g(mark(X),X)
     a__g(a(),X) -> a__f(b(),X)
     a__f(X,X) -> a__h(a__a())
     a__a() -> b()
     mark(h(X)) -> a__h(mark(X))
     mark(g(X1,X2)) -> a__g(mark(X1),X2)
     mark(a()) -> a__a()
     mark(f(X1,X2)) -> a__f(mark(X1),X2)
     mark(b()) -> b()
     a__h(X) -> h(X)
     a__g(X1,X2) -> g(X1,X2)
     a__a() -> a()
     a__f(X1,X2) -> f(X1,X2)
    Arctic Interpretation Processor:
     dimension: 1
     interpretation:
      [a__f#](x0, x1) = x0 + 0,
      
      [a__g#](x0, x1) = 0,
      
      [mark#](x0) = x0 + 0,
      
      [a__h#](x0) = x0 + 0,
      
      [f](x0, x1) = 3x0 + 4,
      
      [g](x0, x1) = x0,
      
      [h](x0) = 3x0 + 4,
      
      [a__a] = 0,
      
      [a__f](x0, x1) = 3x0 + 4,
      
      [b] = 0,
      
      [a] = 0,
      
      [a__g](x0, x1) = x0 + 4,
      
      [mark](x0) = 3x0 + 4,
      
      [a__h](x0) = 3x0 + 4
     orientation:
      a__f#(X,X) = X + 0 >= 0 = a__h#(a__a())
      
      a__h#(X) = X + 0 >= X + 0 = mark#(X)
      
      mark#(h(X)) = 3X + 4 >= X + 0 = mark#(X)
      
      mark#(h(X)) = 3X + 4 >= 3X + 4 = a__h#(mark(X))
      
      a__h#(X) = X + 0 >= 0 = a__g#(mark(X),X)
      
      a__g#(a(),X) = 0 >= 0 = a__f#(b(),X)
      
      mark#(g(X1,X2)) = X1 + 0 >= X1 + 0 = mark#(X1)
      
      mark#(g(X1,X2)) = X1 + 0 >= 0 = a__g#(mark(X1),X2)
      
      mark#(f(X1,X2)) = 3X1 + 4 >= X1 + 0 = mark#(X1)
      
      mark#(f(X1,X2)) = 3X1 + 4 >= 3X1 + 4 = a__f#(mark(X1),X2)
      
      a__h(X) = 3X + 4 >= 3X + 4 = a__g(mark(X),X)
      
      a__g(a(),X) = 4 >= 4 = a__f(b(),X)
      
      a__f(X,X) = 3X + 4 >= 4 = a__h(a__a())
      
      a__a() = 0 >= 0 = b()
      
      mark(h(X)) = 6X + 7 >= 6X + 7 = a__h(mark(X))
      
      mark(g(X1,X2)) = 3X1 + 4 >= 3X1 + 4 = a__g(mark(X1),X2)
      
      mark(a()) = 4 >= 0 = a__a()
      
      mark(f(X1,X2)) = 6X1 + 7 >= 6X1 + 7 = a__f(mark(X1),X2)
      
      mark(b()) = 4 >= 0 = b()
      
      a__h(X) = 3X + 4 >= 3X + 4 = h(X)
      
      a__g(X1,X2) = X1 + 4 >= X1 = g(X1,X2)
      
      a__a() = 0 >= 0 = a()
      
      a__f(X1,X2) = 3X1 + 4 >= 3X1 + 4 = f(X1,X2)
     problem:
      DPs:
       a__f#(X,X) -> a__h#(a__a())
       a__h#(X) -> mark#(X)
       mark#(h(X)) -> a__h#(mark(X))
       a__h#(X) -> a__g#(mark(X),X)
       a__g#(a(),X) -> a__f#(b(),X)
       mark#(g(X1,X2)) -> mark#(X1)
       mark#(g(X1,X2)) -> a__g#(mark(X1),X2)
       mark#(f(X1,X2)) -> a__f#(mark(X1),X2)
      TRS:
       a__h(X) -> a__g(mark(X),X)
       a__g(a(),X) -> a__f(b(),X)
       a__f(X,X) -> a__h(a__a())
       a__a() -> b()
       mark(h(X)) -> a__h(mark(X))
       mark(g(X1,X2)) -> a__g(mark(X1),X2)
       mark(a()) -> a__a()
       mark(f(X1,X2)) -> a__f(mark(X1),X2)
       mark(b()) -> b()
       a__h(X) -> h(X)
       a__g(X1,X2) -> g(X1,X2)
       a__a() -> a()
       a__f(X1,X2) -> f(X1,X2)
     Arctic Interpretation Processor:
      dimension: 1
      interpretation:
       [a__f#](x0, x1) = 1x0 + 1,
       
       [a__g#](x0, x1) = x1 + 1,
       
       [mark#](x0) = 1x0,
       
       [a__h#](x0) = 1x0 + 1,
       
       [f](x0, x1) = 2x0 + 1,
       
       [g](x0, x1) = 1x0 + 1x1 + 1,
       
       [h](x0) = 2x0 + 1,
       
       [a__a] = 0,
       
       [a__f](x0, x1) = 2x0 + 2,
       
       [b] = 0,
       
       [a] = 0,
       
       [a__g](x0, x1) = 1x0 + 2x1 + 2,
       
       [mark](x0) = 1x0 + 0,
       
       [a__h](x0) = 2x0 + 2
      orientation:
       a__f#(X,X) = 1X + 1 >= 1 = a__h#(a__a())
       
       a__h#(X) = 1X + 1 >= 1X = mark#(X)
       
       mark#(h(X)) = 3X + 2 >= 2X + 1 = a__h#(mark(X))
       
       a__h#(X) = 1X + 1 >= X + 1 = a__g#(mark(X),X)
       
       a__g#(a(),X) = X + 1 >= 1 = a__f#(b(),X)
       
       mark#(g(X1,X2)) = 2X1 + 2X2 + 2 >= 1X1 = mark#(X1)
       
       mark#(g(X1,X2)) = 2X1 + 2X2 + 2 >= X2 + 1 = a__g#(mark(X1),X2)
       
       mark#(f(X1,X2)) = 3X1 + 2 >= 2X1 + 1 = a__f#(mark(X1),X2)
       
       a__h(X) = 2X + 2 >= 2X + 2 = a__g(mark(X),X)
       
       a__g(a(),X) = 2X + 2 >= 2 = a__f(b(),X)
       
       a__f(X,X) = 2X + 2 >= 2 = a__h(a__a())
       
       a__a() = 0 >= 0 = b()
       
       mark(h(X)) = 3X + 2 >= 3X + 2 = a__h(mark(X))
       
       mark(g(X1,X2)) = 2X1 + 2X2 + 2 >= 2X1 + 2X2 + 2 = a__g(mark(X1),X2)
       
       mark(a()) = 1 >= 0 = a__a()
       
       mark(f(X1,X2)) = 3X1 + 2 >= 3X1 + 2 = a__f(mark(X1),X2)
       
       mark(b()) = 1 >= 0 = b()
       
       a__h(X) = 2X + 2 >= 2X + 1 = h(X)
       
       a__g(X1,X2) = 1X1 + 2X2 + 2 >= 1X1 + 1X2 + 1 = g(X1,X2)
       
       a__a() = 0 >= 0 = a()
       
       a__f(X1,X2) = 2X1 + 2 >= 2X1 + 1 = f(X1,X2)
      problem:
       DPs:
        a__f#(X,X) -> a__h#(a__a())
        a__h#(X) -> mark#(X)
        a__h#(X) -> a__g#(mark(X),X)
        a__g#(a(),X) -> a__f#(b(),X)
       TRS:
        a__h(X) -> a__g(mark(X),X)
        a__g(a(),X) -> a__f(b(),X)
        a__f(X,X) -> a__h(a__a())
        a__a() -> b()
        mark(h(X)) -> a__h(mark(X))
        mark(g(X1,X2)) -> a__g(mark(X1),X2)
        mark(a()) -> a__a()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
        mark(b()) -> b()
        a__h(X) -> h(X)
        a__g(X1,X2) -> g(X1,X2)
        a__a() -> a()
        a__f(X1,X2) -> f(X1,X2)
      SCC Processor:
       #sccs: 1
       #rules: 3
       #arcs: 32/16
       DPs:
        a__f#(X,X) -> a__h#(a__a())
        a__h#(X) -> a__g#(mark(X),X)
        a__g#(a(),X) -> a__f#(b(),X)
       TRS:
        a__h(X) -> a__g(mark(X),X)
        a__g(a(),X) -> a__f(b(),X)
        a__f(X,X) -> a__h(a__a())
        a__a() -> b()
        mark(h(X)) -> a__h(mark(X))
        mark(g(X1,X2)) -> a__g(mark(X1),X2)
        mark(a()) -> a__a()
        mark(f(X1,X2)) -> a__f(mark(X1),X2)
        mark(b()) -> b()
        a__h(X) -> h(X)
        a__g(X1,X2) -> g(X1,X2)
        a__a() -> a()
        a__f(X1,X2) -> f(X1,X2)
       Open