intros w w'; generalize w; elim w'; unfold Qpositive_le in |- *;
simpl in |- *; clear w w'.
(* Focus 1 *)
intros w' Hrec w; case w.
(* 1.3 *)
3: intros w''; case w''; simpl in |- *; auto.
(* 1.1 *)
intros w0 w''; case w''; simpl in |- *; auto.
(* 1.2 *)
intros w0 w''; case w''; simpl in |- *; auto.
intros w''0 H H1; discriminate H1.
(* Focus 2 *)
intros w' Hrec w; case w.
(* 2.3 *)
3: intros w'' H; discriminate H.
(* 2.1 *)
intros w0 w'' H; discriminate H.
(* 2.2 *)
intros w0 w''; case w''; simpl in |- *; auto.
(* Focus 3 *)
intros w; case w; clear w; simpl in |- *.
(* 3.3 *)
3: auto.
(* 3.1 *)
intros w w'' H; discriminate H.
(* 3.2 *)
intros w w''; case w''; simpl in |- *; auto.
intros w''0 H H1; discriminate H1.
Qed.

Theorem Qpositive_le_antisym :
forall w w' : Qpositive, Qpositive_le w w' -> Qpositive_le w' w -> w = w'.