X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fcompare.ma;h=e20d824df5ae105129599fda2b93beac3a23fcfb;hb=30b7e65d641fe7243c4f36ed448f56360a1c5e1c;hp=60a9d4194c2dc5b3dbf47936260f9763d4bf577d;hpb=3eff4cc36820df9faddb3cb16390717851db499c;p=helm.git

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index 60a9d4194..e20d824df 100644
--- 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||                                                             *)
@@ -14,9 +14,55 @@
 
 set "baseuri" "cic:/matita/nat/compare".
 
-include "nat/orders.ma".
 include "datatypes/bool.ma".
 include "datatypes/compare.ma".
+include "nat/orders.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 
@@ -43,7 +89,7 @@ simplify.intros.apply H.apply le_S_S_to_le.assumption.
 qed.
 
 theorem leb_elim: \forall n,m:nat. \forall P:bool \to Prop. 
-(n \leq m \to (P true)) \to (\not (n \leq m) \to (P false)) \to
+(n \leq m \to (P true)) \to (\lnot (n \leq m) \to (P false)) \to
 P (leb n m).
 intros.
 cut 
@@ -78,6 +124,21 @@ nat_compare n m = nat_compare (S n) (S m).
 intros.simplify.reflexivity.
 qed.
 
+theorem S_pred: \forall n:nat.lt O n \to eq nat n (S (pred n)).
+intro.elim n.apply False_ind.exact not_le_Sn_O O H.
+apply eq_f.apply pred_Sn.
+qed.
+
+theorem nat_compare_pred_pred: 
+\forall n,m:nat.lt O n \to lt O m \to 
+eq compare (nat_compare n m) (nat_compare (pred n) (pred m)).
+intros.
+apply lt_O_n_elim n H.
+apply lt_O_n_elim m H1.
+intros.
+simplify.reflexivity.
+qed.
+
 theorem nat_compare_to_Prop: \forall n,m:nat. 
 match (nat_compare n m) with
   [ LT \Rightarrow n < m
@@ -93,8 +154,8 @@ simplify.apply le_S_S.apply le_O_n.
 intro.simplify.apply le_S_S. apply le_O_n.
 intros 2.simplify.elim (nat_compare n1 m1).
 simplify. apply le_S_S.apply H.
-simplify. apply le_S_S.apply H.
 simplify. apply eq_f. apply H.
+simplify. apply le_S_S.apply H.
 qed.
 
 theorem nat_compare_n_m_m_n: \forall n,m:nat. 
@@ -120,6 +181,6 @@ cut match (nat_compare n m) with
 apply Hcut.apply nat_compare_to_Prop.
 elim (nat_compare n m).
 apply (H H3).
-apply (H2 H3).
 apply (H1 H3).
+apply (H2 H3).
 qed.