more work on supremum

[helm.git] / helm / software / matita / contribs / dama / dama / nat_ordered_set.ma

diff --git a/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma b/helm/software/matita/contribs/dama/dama/nat_ordered_set.ma
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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "ordered_set.ma".
+
+include "nat/compare.ma".
+include "cprop_connectives.ma".
+
+definition nat_excess : nat → nat → CProp ≝ λn,m. m<n.
+
+lemma nat_elim2: 
+  ∀R:nat → nat → CProp.
+  (∀ n:nat. R O n) → (∀n:nat. R (S n) O) → (∀n,m:nat. R n m → R (S n) (S m)) →
+    ∀n,m:nat. R n m.
+intros 5;elim n; [apply H]
+cases m;[ apply H1| apply H2; apply H3 ]
+qed.
+
+lemma nat_discriminable: ∀x,y:nat.x < y ∨ x = y ∨ y < x.
+intros (x y); apply (nat_elim2 ???? x y); 
+[1: intro;left;cases n; [right;reflexivity] left; apply lt_O_S;
+|2: intro;right;apply lt_O_S;
+|3: intros; cases H; 
+    [1: cases H1; [left; left; apply le_S_S; assumption]
+        left;right;rewrite > H2; reflexivity;
+    |2: right;apply le_S_S; assumption]]
+qed.
+        
+lemma nat_excess_cotransitive: cotransitive ? nat_excess.
+intros 3 (x y z); unfold nat_excess; simplify; intros;
+cases (nat_discriminable x z); [2: left; assumption] cases H1; clear H1;
+[1: right; apply (trans_lt ??? H H2);
+|2: right; rewrite < H2; assumption;]
+qed.
+  
+lemma nat_ordered_set : ordered_set.
+apply (mk_ordered_set ? nat_excess);
+[1: intro x; intro; apply (not_le_Sn_n ? H);
+|2: apply nat_excess_cotransitive]
+qed.