theory First

  type t
  constant c : t
  predicate a
  predicate p t
  predicate q t
  function f (t) : t

  goal P1 : (forall x:t. p(x)) -> (exists x:t. p(x))
  goal P5 : (forall x:t. p(x) -> p(f(x))) -> forall x:t.p(x) -> p(f(f(x)))

end
