qed.
lemma nat_ordered_set : ordered_set.
-letin hos ≝ (mk_half_ordered_set nat nat_excess ? nat_excess_cotransitive);[
- intro x; intro H; apply (not_le_Sn_n ? H);]
+letin hos ≝ (mk_half_ordered_set nat (λT:Type.λf:T→T→CProp.f) ? nat_excess ? nat_excess_cotransitive);
+[ intros; left; intros; reflexivity;
+| intro x; intro H; apply (not_le_Sn_n ? H);]
constructor 1;
[ apply hos; | apply (dual_hos hos); | reflexivity]
qed.
intros 3; apply (le_to_not_lt a b);assumption;
qed.
-(*
-lemma nat_lt_to_os_lt:
- ∀a,b:nat_ordered_set.a < b → lt nat_ordered_set a b.
-intros 3; split;
-[1: apply nat_le_to_os_le; apply lt_to_le;assumption;
-|2: right; apply H;]
-qed.
-
-lemma os_lt_to_nat_lt:
- ∀a,b:nat_ordered_set. lt nat_ordered_set a b → a < b.
-intros; cases H; clear H; cases H2;
-[2: apply H;| cases (H1 H)]
-qed.
-*)
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