X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fcompare.ma;h=3a8e4a87ca50a7416e2df5f72920ea151977f41c;hb=b9af9f1c0de6a1735b492f5c793a87a8fce218cc;hp=d148dfd310b2b8be46fdb8a5e4b854e71d6ac34f;hpb=f263e4ec717d5ec2e7f9c057855f8223f81baae8;p=helm.git

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index d148dfd31..3a8e4a87c 100644
--- a/helm/matita/library/nat/compare.ma
+++ b/helm/matita/library/nat/compare.ma
@@ -16,6 +16,7 @@ set "baseuri" "cic:/matita/nat/compare".
 
 include "nat/orders.ma".
 include "datatypes/bool.ma".
+include "datatypes/compare.ma".
 
 let rec leb n m \def 
 match n with 
@@ -27,13 +28,13 @@ match n with
 	
 theorem leb_to_Prop: \forall n,m:nat. 
 match (leb n m) with
-[ true  \Rightarrow (le n m) 
-| false \Rightarrow (Not (le n m))].
+[ true  \Rightarrow n \leq m 
+| false \Rightarrow \lnot (n \leq m)].
 intros.
 apply nat_elim2
 (\lambda n,m:nat.match (leb n m) with
-[ true  \Rightarrow (le n m) 
-| false \Rightarrow (Not (le n m))]).
+[ true  \Rightarrow n \leq m 
+| false \Rightarrow \lnot (n \leq m)]).
 simplify.exact le_O_n.
 simplify.exact not_le_Sn_O.
 intros 2.simplify.elim (leb n1 m1).
@@ -41,19 +42,99 @@ simplify.apply le_S_S.apply H.
 simplify.intros.apply H.apply le_S_S_to_le.assumption.
 qed.
 
-theorem le_elim: \forall n,m:nat. \forall P:bool \to Prop. 
-((le n m) \to (P true)) \to ((Not (le n m)) \to (P false)) \to
+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
 P (leb n m).
 intros.
 cut 
 match (leb n m) with
-[ true  \Rightarrow (le n m) 
-| false \Rightarrow (Not (le n m))] \to (P (leb n m)).
+[ true  \Rightarrow n \leq m
+| false \Rightarrow \lnot (n \leq m)] \to (P (leb n m)).
 apply Hcut.apply leb_to_Prop.
 elim leb n m.
 apply (H H2).
 apply (H1 H2).
 qed.
 
+let rec nat_compare n m: compare \def
+match n with
+[ O \Rightarrow 
+    match m with 
+      [ O \Rightarrow EQ
+      | (S q) \Rightarrow LT ]
+| (S p) \Rightarrow 
+    match m with 
+      [ O \Rightarrow GT
+      | (S q) \Rightarrow nat_compare p q]].
 
+theorem nat_compare_n_n: \forall n:nat. nat_compare n n = EQ.
+intro.elim n.
+simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem nat_compare_S_S: \forall n,m:nat. 
+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
+  | EQ \Rightarrow n=m
+  | GT \Rightarrow m < n ].
+intros.
+apply nat_elim2 (\lambda n,m.match (nat_compare n m) with
+  [ LT \Rightarrow n < m
+  | EQ \Rightarrow n=m
+  | GT \Rightarrow m < n ]).
+intro.elim n1.simplify.reflexivity.
+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.
+qed.
+
+theorem nat_compare_n_m_m_n: \forall n,m:nat. 
+nat_compare n m = compare_invert (nat_compare m n).
+intros. 
+apply nat_elim2 (\lambda n,m. nat_compare n m = compare_invert (nat_compare m n)).
+intros.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intro.elim n1.simplify.reflexivity.
+simplify.reflexivity.
+intros.simplify.elim H.reflexivity.
+qed.
+     
+theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
+(n < m \to P LT) \to (n=m \to P EQ) \to (m < n \to P GT) \to 
+(P (nat_compare n m)).
+intros.
+cut match (nat_compare n m) with
+[ LT \Rightarrow n < m
+| EQ \Rightarrow n=m
+| GT \Rightarrow m < n] \to
+(P (nat_compare n m)).
+apply Hcut.apply nat_compare_to_Prop.
+elim (nat_compare n m).
+apply (H H3).
+apply (H2 H3).
+apply (H1 H3).
+qed.