proposition: Theorem Qpositive_le'_to_Qpositive_le : forall w w' : Qpositive, Qpositive_le' w w' -> Qpositive_le w w'.
proof: Proof. intros w w'; generalize w; elim w'; clear w w'. intros w'1 Hrec w; case w; clear w. intros w; unfold Qpositive_le', Qpositive_le in |- *; simpl in |- *. generalize (Hrec w); clear Hrec; unfold Qpositive_le, Qpositive_le' in |- *; simpl in |- *. elim (interp_non_zero w); intros p Hex; elim Hex; intros q Heq; clear Hex; rewrite Heq. elim (interp_non_zero w'1); intros p' Hex; elim Hex; intros q' Heq1; clear Hex; rewrite Heq1. intros H H1; (lapply H; [ intros H0; try exact H0 | idtac ]). apply (fun p n m : nat => Nat.add_le_mono_l n m p) with (S q * S q'). replace (S q * S q' + S p * S q') with ((S p + S q) * S q'). replace (S q * S q' + S p' * S q) with ((S p' + S q') * S q). auto. ring. ring. unfold Qpositive_le in |- *; simpl in |- *; auto. unfold Qpositive_le in |- *; simpl in |- *. auto. intros w'1 Hrec w; case w; clear w; unfold Qpositive_le in |- *; simpl in |- *; auto. intros w1; unfold Qpositive_le' in |- *; simpl in |- *. elim (interp_non_zero w1); intros p Hex; elim Hex; intros q Heq; clear Hex; rewrite Heq. elim (interp_non_zero w'1); intros p' Hex; elim Hex; intros q' Heq1; clear Hex; rewrite Heq1. rewrite Nat.mul_add_distr_r. repeat rewrite <- (Nat.mul_comm (S p' + S q')). repeat rewrite Nat.mul_add_distr_r. repeat rewrite <- (Nat.add_comm (S p' * S q)) || rewrite Nat.add_assoc; repeat rewrite <- Nat.add_assoc. pattern (S p' * S q) at 2 in |- *; rewrite plus_n_O. intros H; pose proof H as H1; apply Nat.add_le_mono_l in H1; simpl in |- *; inversion H1. intros w; unfold Qpositive_le', Qpositive_le in |- *; simpl in |- *. generalize (Hrec w); clear Hrec; unfold Qpositive_le, Qpositive_le' in |- *; simpl in |- *. elim (interp_non_zero w); intros p Hex; elim Hex; intros q Heq; clear Hex; rewrite Heq. elim (interp_non_zero w'1); intros p' Hex; elim Hex; intros q' Heq1; clear Hex; rewrite Heq1. intros H H1; (lapply H; [ intros H0; try exact H0 | idtac ]). apply (fun p n m : nat => Nat.add_le_mono_l n m p) with (S p * S p'). replace (S p * S p' + S p * S q') with (S p * (S p' + S q')). replace (S p * S p' + S p' * S q) with (S p' * (S p + S q)). auto. ring. ring. unfold Qpositive_le' in |- *; simpl in |- *. elim (interp_non_zero w'1); intros p' Hex; elim Hex; intros q' Heq1; clear Hex; rewrite Heq1. rewrite <- plus_n_O; rewrite <- (Nat.mul_comm 1); simpl in |- *; intros H. pose proof (le_S_n _ _ H) as H1; apply Nat.add_le_mono_l in H1; inversion H1. intros w; case w; clear w; unfold Qpositive_le', Qpositive_le in |- *; simpl in |- *. intros w; elim (interp_non_zero w). intros p Hex; elim Hex; intros q Heq; rewrite Heq. rewrite Nat.mul_1_r. rewrite (Nat.add_comm (S p)). intros H; apply Nat.add_le_mono_l in H; inversion H. auto. auto. Qed.