The library grows...

[helm.git] / helm / matita / library / nat / compare.ma

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index 3a8e4a87ca50a7416e2df5f72920ea151977f41c..194b38d84f255e3924a3273919f01bb96d320ca5 100644 (file)
--- a/helm/matita/library/nat/compare.ma
+++ b/helm/matita/library/nat/compare.ma
@@ -1,5 +1,5 @@
 (**************************************************************************)
-(*       ___                                                               *)
+(*       ___                                                             *)
 (*      ||M||                                                             *)
 (*      ||A||       A project by Andrea Asperti                           *)
 (*      ||T||                                                             *)
@@ -18,6 +18,52 @@ include "nat/orders.ma".
 include "datatypes/bool.ma".
 include "datatypes/compare.ma".
 
+let rec eqb n m \def 
+match n with 
+  [ O \Rightarrow 
+     match m with 
+     [ O \Rightarrow true
+          | (S q) \Rightarrow false] 
+  | (S p) \Rightarrow
+          match m with 
+     [ O \Rightarrow false
+          | (S q) \Rightarrow eqb p q]].
+          
+theorem eqb_to_Prop: \forall n,m:nat. 
+match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow \lnot (n = m)].
+intros.
+apply nat_elim2
+(\lambda n,m:nat.match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow \lnot (n = m)]).
+intro.elim n1.
+simplify.reflexivity.
+simplify.apply not_eq_O_S.
+intro.
+simplify.
+intro. apply not_eq_O_S n1 ?.apply sym_eq.assumption.
+intros.simplify.
+generalize in match H.
+elim (eqb n1 m1).
+simplify.apply eq_f.apply H1.
+simplify.intro.apply H1.apply inj_S.assumption.
+qed.
+
+theorem eqb_elim : \forall n,m:nat.\forall P:bool \to Prop.
+(n=m \to (P true)) \to (\lnot n=m \to (P false)) \to (P (eqb n m)). 
+intros.
+cut 
+match (eqb n m) with
+[ true  \Rightarrow n = m
+| false \Rightarrow \lnot (n = m)] \to (P (eqb n m)).
+apply Hcut.apply eqb_to_Prop.
+elim eqb n m.
+apply (H H2).
+apply (H1 H2).
+qed.
+
 let rec leb n m \def 
 match n with 
     [ O \Rightarrow true