X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Ford.ma;h=24874c08af2fd10d2e3286f01e388940f3390d51;hb=97c2d258a5c524eb5c4b85208899d80751a2c82f;hp=7b246be0ee48d9b6aa633012c40ba1c89e42f34d;hpb=27bd932eeab546b36d2750bd6f4d047ebf2964f6;p=helm.git

diff --git a/helm/matita/library/nat/ord.ma b/helm/matita/library/nat/ord.ma
index 7b246be0e..24874c08a 100644
--- a/helm/matita/library/nat/ord.ma
+++ b/helm/matita/library/nat/ord.ma
@@ -22,13 +22,13 @@ include "nat/primes.ma".
 (* this definition of log is based on pairs, with a remainder *)
 
 let rec p_ord_aux p n m \def
-  match (mod n m) with
+  match n \mod m with
   [ O \Rightarrow 
     match p with
       [ O \Rightarrow pair nat nat O n
       | (S p) \Rightarrow 
-       match (p_ord_aux p (div n m) m) with
-       [ (pair q r) \Rightarrow pair nat nat (S q) r]]
+       match (p_ord_aux p (n / m) m) with
+       [ (pair q r) \Rightarrow pair nat nat (S q) r] ]
   | (S a) \Rightarrow pair nat nat O n].
   
 (* p_ord n m = <q,r> if m divides n q times, with remainder r *)
@@ -41,36 +41,36 @@ intro.
 elim p.
 change with 
 match (
-match (mod n m) with
+match n \mod m with
   [ O \Rightarrow pair nat nat O n
   | (S a) \Rightarrow pair nat nat O n] )
 with
   [ (pair q r) \Rightarrow n = m \sup q * r ].
-apply nat_case (mod n m).
+apply (nat_case (n \mod m)).
 simplify.apply plus_n_O.
 intros.
 simplify.apply plus_n_O. 
 change with 
 match (
-match (mod n1 m) with
+match n1 \mod m with
   [ O \Rightarrow 
-     match (p_ord_aux n (div n1 m) m) with
+     match (p_ord_aux n (n1 / m) m) with
        [ (pair q r) \Rightarrow pair nat nat (S q) r]
   | (S a) \Rightarrow pair nat nat O n1] )
 with
   [ (pair q r) \Rightarrow n1 = m \sup q * r].
-apply nat_case1 (mod n1 m).intro.
+apply (nat_case1 (n1 \mod m)).intro.
 change with 
 match (
- match (p_ord_aux n (div n1 m) m) with
+ match (p_ord_aux n (n1 / m) m) with
    [ (pair q r) \Rightarrow pair nat nat (S q) r])
 with
   [ (pair q r) \Rightarrow n1 = m \sup q * r].
-generalize in match (H (div n1 m) m).
-elim p_ord_aux n (div n1 m) m.
+generalize in match (H (n1 / m) m).
+elim (p_ord_aux n (n1 / m) m).
 simplify.
 rewrite > assoc_times.
-rewrite < H3.rewrite > plus_n_O (m*(div n1 m)).
+rewrite < H3.rewrite > (plus_n_O (m*(n1 / m))).
 rewrite < H2.
 rewrite > sym_times.
 rewrite < div_mod.reflexivity.
@@ -89,105 +89,103 @@ apply p_ord_aux_to_Prop.
 assumption.
 qed.
 (* questo va spostato in primes1.ma *)
-theorem p_ord_exp: \forall n,m,i. O < m \to mod n m \neq O \to
+theorem p_ord_exp: \forall n,m,i. O < m \to n \mod m \neq O \to
 \forall p. i \le p \to p_ord_aux p (m \sup i * n) m = pair nat nat i n.
 intros 5.
 elim i.
 simplify.
 rewrite < plus_n_O.
-apply nat_case p.
+apply (nat_case p).
 change with
- match (mod n m) with
+ (match n \mod m with
   [ O \Rightarrow pair nat nat O n
   | (S a) \Rightarrow pair nat nat O n]
-  = pair nat nat O n.
-elim (mod n m).simplify.reflexivity.simplify.reflexivity.
+  = pair nat nat O n).
+elim (n \mod m).simplify.reflexivity.simplify.reflexivity.
 intro.
 change with
- match (mod n m) with
+ (match n \mod m with
   [ O \Rightarrow 
-       match (p_ord_aux m1 (div n m) m) with
+       match (p_ord_aux m1 (n / m) m) with
        [ (pair q r) \Rightarrow pair nat nat (S q) r]
   | (S a) \Rightarrow pair nat nat O n]
-  = pair nat nat O n.
-cut O < mod n m \lor O = mod n m.
-elim Hcut.apply lt_O_n_elim (mod n m) H3.
+  = pair nat nat O n).
+cut (O < n \mod m \lor O = n \mod m).
+elim Hcut.apply (lt_O_n_elim (n \mod m) H3).
 intros. simplify.reflexivity.
 apply False_ind.
 apply H1.apply sym_eq.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 generalize in match H3.
-apply nat_case p.intro.apply False_ind.apply not_le_Sn_O n1 H4.
+apply (nat_case p).intro.apply False_ind.apply (not_le_Sn_O n1 H4).
 intros.
 change with
- match (mod (m \sup (S n1) *n) m) with
+ (match ((m \sup (S n1) *n) \mod m) with
   [ O \Rightarrow 
-       match (p_ord_aux m1 (div (m \sup (S n1) *n) m) m) with
+       match (p_ord_aux m1 ((m \sup (S n1) *n) / m) m) with
        [ (pair q r) \Rightarrow pair nat nat (S q) r]
   | (S a) \Rightarrow pair nat nat O (m \sup (S n1) *n)]
-  = pair nat nat (S n1) n.
-cut (mod (m \sup (S n1)*n) m) = O.
+  = pair nat nat (S n1) n).
+cut (((m \sup (S n1)*n) \mod m) = O).
 rewrite > Hcut.
 change with
-match (p_ord_aux m1 (div (m \sup (S n1)*n) m) m) with
+(match (p_ord_aux m1 ((m \sup (S n1)*n) / m) m) with
        [ (pair q r) \Rightarrow pair nat nat (S q) r] 
-  = pair nat nat (S n1) n. 
-cut div (m \sup (S n1) *n) m = m \sup n1 *n.
+  = pair nat nat (S n1) n). 
+cut ((m \sup (S n1) *n) / m = m \sup n1 *n).
 rewrite > Hcut1.
-rewrite > H2 m1. simplify.reflexivity.
+rewrite > (H2 m1). simplify.reflexivity.
 apply le_S_S_to_le.assumption.
 (* div_exp *)
-change with div (m* m \sup n1 *n) m = m \sup n1 * n.
+change with ((m* m \sup n1 *n) / m = m \sup n1 * n).
 rewrite > assoc_times.
-apply lt_O_n_elim m H.
+apply (lt_O_n_elim m H).
 intro.apply div_times.
 (* mod_exp = O *)
 apply divides_to_mod_O.
 assumption.
 simplify.rewrite > assoc_times.
-apply witness ? ? (m \sup n1 *n).reflexivity.
+apply (witness ? ? (m \sup n1 *n)).reflexivity.
 qed.
 
 theorem p_ord_aux_to_Prop1: \forall p,n,m. (S O) < m \to O < n \to n \le p \to
   match p_ord_aux p n m with
-  [ (pair q r) \Rightarrow mod r m \neq O].
-intro.elim p.absurd O < n.assumption.
+  [ (pair q r) \Rightarrow r \mod m \neq O].
+intro.elim p.absurd (O < n).assumption.
 apply le_to_not_lt.assumption.
 change with
 match 
-  (match (mod n1 m) with
+  (match n1 \mod m with
     [ O \Rightarrow 
-        match (p_ord_aux n(div n1 m) m) with
+        match (p_ord_aux n(n1 / m) m) with
         [ (pair q r) \Rightarrow pair nat nat (S q) r]
     | (S a) \Rightarrow pair nat nat O n1])
 with
-  [ (pair q r) \Rightarrow mod r m \neq O].
-apply nat_case1 (mod n1 m).intro.
-generalize in match (H (div n1 m) m).
-elim (p_ord_aux n (div n1 m) m).
+  [ (pair q r) \Rightarrow r \mod m \neq O].
+apply (nat_case1 (n1 \mod m)).intro.
+generalize in match (H (n1 / m) m).
+elim (p_ord_aux n (n1 / m) m).
 apply H5.assumption.
 apply eq_mod_O_to_lt_O_div.
-apply trans_lt ? (S O).simplify.apply le_n.
+apply (trans_lt ? (S O)).unfold lt.apply le_n.
 assumption.assumption.assumption.
 apply le_S_S_to_le.
-apply trans_le ? n1.change with div n1 m < n1.
+apply (trans_le ? n1).change with (n1 / m < n1).
 apply lt_div_n_m_n.assumption.assumption.assumption.
 intros.
-change with mod n1 m \neq O.
+change with (n1 \mod m \neq O).
 rewrite > H4.
-(* Andrea: META NOT FOUND !!!  
-rewrite > sym_eq. *)
-simplify.intro.
-apply not_eq_O_S m1.
+unfold Not.intro.
+apply (not_eq_O_S m1).
 rewrite > H5.reflexivity.
 qed.
 
 theorem p_ord_aux_to_not_mod_O: \forall p,n,m,q,r. (S O) < m \to O < n \to n \le p \to
- pair nat nat q r = p_ord_aux p n m \to mod r m \neq O.
+ pair nat nat q r = p_ord_aux p n m \to r \mod m \neq O.
 intros.
 change with 
   match (pair nat nat q r) with
-  [ (pair q r) \Rightarrow mod r m \neq O].
+  [ (pair q r) \Rightarrow r \mod m \neq O].
 rewrite > H3. 
 apply p_ord_aux_to_Prop1.
 assumption.assumption.assumption.