Ltac id_comp x y H1 H2 H3 :=
destruct (CompSpec2Type (Ident.compare_spec x y)) as [H1|H2|H3].

Lemma id2pos_inj x y : id2pos x = id2pos y -> x=y.
introv H; id_comp x y H1 H2 H3; [auto|elimtype False; gen H2|gen H3];
done (tapp Ilt_morphism; rewrite H; tapp Pos.lt_irrefl).