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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "ordered_set.ma".
17 include "nat/compare.ma".
18 include "cprop_connectives.ma".
20 definition nat_excess : nat → nat → CProp ≝ λn,m. m<n.
24 (∀ n:nat. R O n) → (∀n:nat. R (S n) O) → (∀n,m:nat. R n m → R (S n) (S m)) →
26 intros 5;elim n; [apply H]
27 cases m;[ apply H1| apply H2; apply H3 ]
30 lemma nat_discriminable: ∀x,y:nat.x < y ∨ x = y ∨ y < x.
31 intros (x y); apply (nat_elim2 ???? x y);
32 [1: intro;left;cases n; [right;reflexivity] left; apply lt_O_S;
33 |2: intro;right;apply lt_O_S;
35 [1: cases H1; [left; left; apply le_S_S; assumption]
36 left;right;rewrite > H2; reflexivity;
37 |2: right;apply le_S_S; assumption]]
40 lemma nat_excess_cotransitive: cotransitive ? nat_excess.
41 intros 3 (x y z); unfold nat_excess; simplify; intros;
42 cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1;
43 [1: right; apply (trans_lt ??? H H2);
44 |2: right; rewrite < H2; assumption;]
47 lemma nat_ordered_set : ordered_set.
48 apply (mk_ordered_set ? nat_excess);
49 [1: intro x; intro; apply (not_le_Sn_n ? H);
50 |2: apply nat_excess_cotransitive]