X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fcompare.ma;h=55e526c8fad0f80eade4775141c99326975efee8;hb=8b55faddb06e3c4b0a13839210bb49170939b33e;hp=fec46ae2fe1f65f8106020bc4a038553beb1667b;hpb=b5ab291cf2ad1431128dd0b1ab629169a3cdeb50;p=helm.git

diff --git a/helm/matita/library/nat/compare.ma b/helm/matita/library/nat/compare.ma
index fec46ae2f..55e526c8f 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 +14,74 @@
 
 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 n \neq m].
+intros.
+apply nat_elim2
+(\lambda n,m:nat.match (eqb n m) with
+[ true  \Rightarrow n = m 
+| false \Rightarrow n \neq 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 (n \neq m \to (P false)) \to (P (eqb n m)). 
+intros.
+cut 
+match (eqb n m) with
+[ true  \Rightarrow n = m
+| false \Rightarrow n \neq m] \to (P (eqb n m)).
+apply Hcut.apply eqb_to_Prop.
+elim eqb n m.
+apply (H H2).
+apply (H1 H2).
+qed.
+
+theorem eqb_n_n: \forall n. eqb n n = true.
+intro.elim n.simplify.reflexivity.
+simplify.assumption.
+qed.
+
+theorem eq_to_eqb_true: \forall n,m:nat.
+n = m \to eqb n m = true.
+intros.apply eqb_elim n m.
+intros. reflexivity.
+intros.apply False_ind.apply H1 H.
+qed.
+
+theorem not_eq_to_eqb_false: \forall n,m:nat.
+\lnot (n = m) \to eqb n m = false.
+intros.apply eqb_elim n m.
+intros. apply False_ind.apply H H1.
+intros.reflexivity.
+qed.
 
 let rec leb n m \def 
 match n with 
@@ -28,13 +93,13 @@ 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 n \nleq 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 n \nleq m]).
 simplify.exact le_O_n.
 simplify.exact not_le_Sn_O.
 intros 2.simplify.elim (leb n1 m1).
@@ -43,13 +108,13 @@ simplify.intros.apply H.apply le_S_S_to_le.assumption.
 qed.
 
 theorem leb_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
+(n \leq m \to (P true)) \to (n \nleq 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 n \nleq m] \to (P (leb n m)).
 apply Hcut.apply leb_to_Prop.
 elim leb n m.
 apply (H H2).
@@ -67,40 +132,55 @@ match n with
       [ O \Rightarrow GT
       | (S q) \Rightarrow nat_compare p q]].
 
-theorem nat_compare_n_n: \forall n:nat.(eq compare (nat_compare n n) EQ).
+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. 
-eq compare (nat_compare n m) (nat_compare (S n) (S m)).
+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 (lt n m)
-  | EQ \Rightarrow (eq nat n m)
-  | GT \Rightarrow (lt m n) ].
+  [ 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 (lt n m)
-  | EQ \Rightarrow (eq nat n m)
-  | GT \Rightarrow (lt m n) ]).
+  [ 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.
+simplify. apply le_S_S.apply H.
 qed.
 
 theorem nat_compare_n_m_m_n: \forall n,m:nat. 
-eq compare (nat_compare n m) (compare_invert (nat_compare m n)).
+nat_compare n m = compare_invert (nat_compare m n).
 intros. 
-apply nat_elim2 (\lambda n,m.eq compare (nat_compare n m) (compare_invert (nat_compare m n))).
+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.
@@ -109,17 +189,17 @@ intros.simplify.elim H.reflexivity.
 qed.
      
 theorem nat_compare_elim : \forall n,m:nat. \forall P:compare \to Prop.
-((lt n m) \to (P LT)) \to ((eq nat n m) \to (P EQ)) \to ((lt m n) \to (P GT)) \to 
+(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 (lt n m)
-| EQ \Rightarrow (eq nat n m)
-| GT \Rightarrow (lt m n)] \to
+[ 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).
+apply (H2 H3).
 qed.