X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fnlibrary%2Fnat%2Forder.ma;h=c21cbc64c7bf79fd5a068dcfffac921aa95647f1;hb=cf89508024f0a19023bb1bac343012d54e860d9d;hp=dd8c544ae34727eab6f8cf36a694d842ab60318b;hpb=c6d3537eee27d05490a9555cc7326bc954b356c5;p=helm.git

diff --git a/helm/software/matita/nlibrary/nat/order.ma b/helm/software/matita/nlibrary/nat/order.ma
index dd8c544ae..c21cbc64c 100644
--- a/helm/software/matita/nlibrary/nat/order.ma
+++ b/helm/software/matita/nlibrary/nat/order.ma
@@ -13,12 +13,19 @@
 (**************************************************************************)
 
 include "nat/nat.ma".
+include "sets/sets.ma".
 
 ninductive le (n: nat) : nat → CProp[0] ≝
    le_n: le n n
  | le_S: ∀m. le n m → le n (S m).
 
-ndefinition lt ≝ λn,m. le (S n) m.
+interpretation "natural 'less or equal to'" 'leq x y = (le x y).
+interpretation "natural 'neither less nor equal to'" 'nleq x y = (Not (le x y)).
+
+ndefinition lt ≝ λn,m. S n ≤ m.
+
+interpretation "natural 'less than'" 'lt x y = (lt x y).
+interpretation "natural 'not less than'" 'nless x y = (Not (lt x y)).
 
 nlemma le_O_n: ∀n. le O n.
  #n; nelim n [ napply le_n | #n'; #H; napply le_S; nassumption ]
@@ -28,7 +35,15 @@ nqed.
 naxiom le_S_S: ∀n,m. le n m → le (S n) (S m).
 naxiom le_S_S_to_le: ∀n,m. le (S n) (S m) → le n m.
 naxiom lt_Sn_m: ∀n,m. lt (S n) m → lt n m.
+naxiom lt_to_le: ∀n,m. lt n m → le n m.
+naxiom lt_le_trans: ∀n,m,p. lt n m → le m p → lt n p.
 
 nlemma lt_O_Sn: ∀n. lt O (S n).
  #n; napply le_S_S; napply le_O_n.
 nqed. 
+
+ndefinition Nat_: nat → ext_powerclass NAT.
+ #n; napply mk_ext_powerclass
+  [ napply {m | m < n}
+  | #x; #x'; #H; nrewrite < H; napply mk_iff; #K; nassumption (* refl non va *) ]
+nqed.