Definition t := string.Definition eq       := @Logic.eq       string.Definition eq_refl  := @Logic.eq_refl  string.Definition eq_sym   := @Logic.eq_sym   string.Definition eq_trans := @Logic.eq_trans string.Definition eq_dec   := string_dec.Inductive SOrdering := SLT | SEQ | SGT.Inductive SOrdering := SLT | SEQ | SGT.Inductive SOrdering := SLT | SEQ | SGT.Inductive SOrdering := SLT | SEQ | SGT.Fixpoint strcmp (s1 s2 : string) : SOrdering :=
  match s1, s2 with
    | EmptyString,    EmptyString    => SEQ
    | EmptyString,    String _ _     => SLT
    | String _   _,   EmptyString    => SGT
    | String ch1 s1', String ch2 s2' =>
        match AsciiOT.compare ch1 ch2 with
          | LT _ => SLT
          | EQ _ => strcmp s1' s2'
          | GT _ => SGT
        end
  end.Definition lt (s1 s2 : string) := strcmp s1 s2 = SLT.Local Ltac do_ascii_lt_trans :=
  match goal with
    | [  _ : AsciiOT.lt ?c1 ?c2
      ,  _ : AsciiOT.lt ?c2 ?c3
      |- _ ]
      => assert (AsciiOT.lt c1 c3) by (eapply AsciiOT.lt_trans; eauto)
  end.Local Ltac not_ascii_lt_refl :=
  match goal with
    | [ _ : AsciiOT.lt ?c ?c |- _ ]
      => assert (c <> c) by (apply AsciiOT.lt_not_eq; assumption);
         congruence
  end.

Theorem lt_not_eq : forall s1 s2 : string, lt s1 s2 -> ~ eq s1 s2.