MAYBE
TRS:
 { app(app(app(if(), true()), x), y) -> x,
  app(app(app(if(), false()), x), y) -> y,
     app(app(app(until(), p), f), x) -> app(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x)))}
 DP:
  Strict:
   {app#(app(app(until(), p), f), x) -> app#(p, x),
    app#(app(app(until(), p), f), x) -> app#(f, x),
    app#(app(app(until(), p), f), x) -> app#(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x))),
    app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)),
    app#(app(app(until(), p), f), x) -> app#(app(if(), app(p, x)), x),
    app#(app(app(until(), p), f), x) -> app#(if(), app(p, x))}
  Weak:
  { app(app(app(if(), true()), x), y) -> x,
   app(app(app(if(), false()), x), y) -> y,
      app(app(app(until(), p), f), x) -> app(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x)))}
  EDG:
   {(app#(app(app(until(), p), f), x) -> app#(f, x), app#(app(app(until(), p), f), x) -> app#(if(), app(p, x)))
    (app#(app(app(until(), p), f), x) -> app#(f, x), app#(app(app(until(), p), f), x) -> app#(app(if(), app(p, x)), x))
    (app#(app(app(until(), p), f), x) -> app#(f, x), app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)))
    (app#(app(app(until(), p), f), x) -> app#(f, x), app#(app(app(until(), p), f), x) -> app#(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x))))
    (app#(app(app(until(), p), f), x) -> app#(f, x), app#(app(app(until(), p), f), x) -> app#(f, x))
    (app#(app(app(until(), p), f), x) -> app#(f, x), app#(app(app(until(), p), f), x) -> app#(p, x))
    (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(p, x))
    (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(f, x))
    (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x))))
    (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)))
    (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(app(if(), app(p, x)), x))
    (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(if(), app(p, x)))
    (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(p, x))
    (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(f, x))
    (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x))))
    (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)))
    (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(app(if(), app(p, x)), x))
    (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(if(), app(p, x)))}
   SCCS:
    Scc:
     {app#(app(app(until(), p), f), x) -> app#(p, x),
      app#(app(app(until(), p), f), x) -> app#(f, x),
      app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
    SCC:
     Strict:
      {app#(app(app(until(), p), f), x) -> app#(p, x),
       app#(app(app(until(), p), f), x) -> app#(f, x),
       app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
     Weak:
     { app(app(app(if(), true()), x), y) -> x,
      app(app(app(if(), false()), x), y) -> y,
         app(app(app(until(), p), f), x) -> app(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x)))}
     SPSC:
      Simple Projection:
       pi(app#) = 0
      Strict:
       {app#(app(app(until(), p), f), x) -> app#(p, x),
        app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
      EDG:
       {(app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)))
        (app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(p, x))
        (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(p, x))
        (app#(app(app(until(), p), f), x) -> app#(p, x), app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)))}
       SCCS:
        Scc:
         {app#(app(app(until(), p), f), x) -> app#(p, x),
          app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
        SCC:
         Strict:
          {app#(app(app(until(), p), f), x) -> app#(p, x),
           app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
         Weak:
         { app(app(app(if(), true()), x), y) -> x,
          app(app(app(if(), false()), x), y) -> y,
             app(app(app(until(), p), f), x) -> app(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x)))}
         SPSC:
          Simple Projection:
           pi(app#) = 0
          Strict:
           {app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
          EDG:
           {(app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)), app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x)))}
           SCCS:
            Scc:
             {app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
            SCC:
             Strict:
              {app#(app(app(until(), p), f), x) -> app#(app(app(until(), p), f), app(f, x))}
             Weak:
             { app(app(app(if(), true()), x), y) -> x,
              app(app(app(if(), false()), x), y) -> y,
                 app(app(app(until(), p), f), x) -> app(app(app(if(), app(p, x)), x), app(app(app(until(), p), f), app(f, x)))}
             Fail