X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fgcd.ma;h=90de2013a5e9f9ba02f681e2cfe7d4761179897b;hb=8b55faddb06e3c4b0a13839210bb49170939b33e;hp=8475fc06aa026aacff1fd29c46c264ff25575947;hpb=30b7e65d641fe7243c4f36ed448f56360a1c5e1c;p=helm.git

diff --git a/helm/matita/library/nat/gcd.ma b/helm/matita/library/nat/gcd.ma
index 8475fc06a..90de2013a 100644
--- a/helm/matita/library/nat/gcd.ma
+++ b/helm/matita/library/nat/gcd.ma
@@ -22,7 +22,7 @@ match divides_b n m with
 | false \Rightarrow 
   match p with
   [O \Rightarrow n
-  |(S q) \Rightarrow gcd_aux q n (mod m n)]].
+  |(S q) \Rightarrow gcd_aux q n (m \mod n)]].
   
 definition gcd : nat \to nat \to nat \def
 \lambda n,m:nat.
@@ -37,28 +37,24 @@ definition gcd : nat \to nat \to nat \def
     | (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
 
 theorem divides_mod: \forall p,m,n:nat. O < n \to p \divides m \to p \divides n \to
-p \divides mod m n.
+p \divides (m \mod n).
 intros.elim H1.elim H2.
-apply witness ? ? (n2 - n1*(div m n)).
+apply witness ? ? (n2 - n1*(m / n)).
 rewrite > distr_times_minus.
 rewrite < H3.
 rewrite < assoc_times.
 rewrite < H4.
 apply sym_eq.
 apply plus_to_minus.
-rewrite > div_mod m n in \vdash (? ? %).
 rewrite > sym_times.
-apply eq_plus_to_le ? ? (mod m n).
-reflexivity.
+apply div_mod.
 assumption.
-rewrite > sym_times.
-apply div_mod.assumption.
 qed.
 
 theorem divides_mod_to_divides: \forall p,m,n:nat. O < n \to
-p \divides mod m n \to p \divides n \to p \divides m. 
+p \divides (m \mod n) \to p \divides n \to p \divides m. 
 intros.elim H1.elim H2.
-apply witness p m ((n1*(div m n))+n2).
+apply witness p m ((n1*(m / n))+n2).
 rewrite > distr_times_plus.
 rewrite < H3.
 rewrite < assoc_times.
@@ -74,25 +70,25 @@ cut (n1 \divides m) \lor (n1 \ndivides m).
 change with 
 (match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]) \divides m \land
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]) \divides m \land
 (match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]) \divides n1.
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]) \divides n1.
 elim Hcut.rewrite > divides_to_divides_b_true.
 simplify.
 split.assumption.apply witness n1 n1 (S O).apply times_n_SO.
 assumption.assumption.
 rewrite > not_divides_to_divides_b_false.
 change with 
-gcd_aux n n1 (mod m n1) \divides m \land
-gcd_aux n n1 (mod m n1) \divides n1.
-cut gcd_aux n n1 (mod m n1) \divides n1 \land
-gcd_aux n n1 (mod m n1) \divides mod m n1.
+gcd_aux n n1 (m \mod n1) \divides m \land
+gcd_aux n n1 (m \mod n1) \divides n1.
+cut gcd_aux n n1 (m \mod n1) \divides n1 \land
+gcd_aux n n1 (m \mod n1) \divides mod m n1.
 elim Hcut1.
 split.apply divides_mod_to_divides ? ? n1.
 assumption.assumption.assumption.assumption.
 apply H.
-cut O \lt mod m n1 \lor O = mod m n1.
+cut O \lt m \mod n1 \lor O = mod m n1.
 elim Hcut1.assumption.
 apply False_ind.apply H4.apply mod_O_to_divides.
 assumption.apply sym_eq.assumption.
@@ -101,7 +97,7 @@ apply lt_to_le.
 apply lt_mod_m_m.assumption.
 apply le_S_S_to_le.
 apply trans_le ? n1.
-change with mod m n1 < n1.
+change with m \mod n1 < n1.
 apply lt_mod_m_m.assumption.assumption.
 assumption.assumption.
 apply decidable_divides n1 m.assumption.
@@ -119,7 +115,7 @@ match leb n m with
   | false \Rightarrow 
     match m with 
     [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]] \divides m
+    | (S p) \Rightarrow gcd_aux (S p) n (S p) ] ] \divides m
 \land
 match leb n m with
   [ true \Rightarrow 
@@ -129,7 +125,7 @@ match leb n m with
   | false \Rightarrow 
     match m with 
     [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]] \divides n. 
+    | (S p) \Rightarrow gcd_aux (S p) n (S p) ] ] \divides n. 
 apply leb_elim n m.
 apply nat_case1 n.
 simplify.intros.split.
@@ -180,16 +176,16 @@ change with
 d \divides
 (match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]).
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]).
 cut n1 \divides m \lor n1 \ndivides m.
 elim Hcut.
 rewrite > divides_to_divides_b_true.
 simplify.assumption.
 assumption.assumption.
 rewrite > not_divides_to_divides_b_false.
-change with d \divides gcd_aux n n1 (mod m n1).
+change with d \divides gcd_aux n n1 (m \mod n1).
 apply H.
-cut O \lt mod m n1 \lor O = mod m n1.
+cut O \lt m \mod n1 \lor O = m \mod n1.
 elim Hcut1.assumption.
 absurd n1 \divides m.apply mod_O_to_divides.
 assumption.apply sym_eq.assumption.assumption.
@@ -198,7 +194,7 @@ apply lt_to_le.
 apply lt_mod_m_m.assumption.
 apply le_S_S_to_le.
 apply trans_le ? n1.
-change with mod m n1 < n1.
+change with m \mod n1 < n1.
 apply lt_mod_m_m.assumption.assumption.
 assumption.
 apply divides_mod.assumption.assumption.assumption.
@@ -247,11 +243,11 @@ change with
 \exists a,b.
 a*n1 - b*m = match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]
+| false \Rightarrow gcd_aux n n1 (m \mod n1)]
 \lor 
 b*m - a*n1 = match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)].
+| false \Rightarrow gcd_aux n n1 (m \mod n1)].
 elim Hcut1.
 rewrite > divides_to_divides_b_true.
 simplify.
@@ -263,18 +259,18 @@ assumption.assumption.
 rewrite > not_divides_to_divides_b_false.
 change with
 \exists a,b.
-a*n1 - b*m = gcd_aux n n1 (mod m n1)
+a*n1 - b*m = gcd_aux n n1 (m \mod n1)
 \lor 
-b*m - a*n1 = gcd_aux n n1 (mod m n1).
+b*m - a*n1 = gcd_aux n n1 (m \mod n1).
 cut 
 \exists a,b.
-a*(mod m n1) - b*n1= gcd_aux n n1 (mod m n1)
+a*(m \mod n1) - b*n1= gcd_aux n n1 (m \mod n1)
 \lor
-b*n1 - a*(mod m n1) = gcd_aux n n1 (mod m n1).
+b*n1 - a*(m \mod n1) = gcd_aux n n1 (m \mod n1).
 elim Hcut2.elim H5.elim H6.
 (* first case *)
 rewrite < H7.
-apply ex_intro ? ? (a1+a*(div m n1)).
+apply ex_intro ? ? (a1+a*(m / n1)).
 apply ex_intro ? ? a.
 right.
 rewrite < sym_plus.
@@ -294,9 +290,10 @@ apply le_n.
 assumption.
 (* second case *)
 rewrite < H7.
-apply ex_intro ? ? (a1+a*(div m n1)).
+apply ex_intro ? ? (a1+a*(m / n1)).
 apply ex_intro ? ? a.
 left.
+(* clear Hcut2.clear H5.clear H6.clear H. *)
 rewrite > sym_times.
 rewrite > distr_times_plus.
 rewrite > sym_times.
@@ -310,8 +307,8 @@ rewrite < plus_minus.
 rewrite < minus_n_n.reflexivity.
 apply le_n.
 assumption.
-apply H n1 (mod m n1).
-cut O \lt mod m n1 \lor O = mod m n1.
+apply H n1 (m \mod n1).
+cut O \lt m \mod n1 \lor O = m \mod n1.
 elim Hcut2.assumption. 
 absurd n1 \divides m.apply mod_O_to_divides.
 assumption.
@@ -321,7 +318,7 @@ apply lt_to_le.
 apply lt_mod_m_m.assumption.
 apply le_S_S_to_le.
 apply trans_le ? n1.
-change with mod m n1 < n1.
+change with m \mod n1 < n1.
 apply lt_mod_m_m.
 assumption.assumption.assumption.assumption.
 apply decidable_divides n1 m.assumption.
@@ -331,28 +328,7 @@ qed.
 theorem eq_minus_gcd:
  \forall m,n.\exists a,b.a*n - b*m = (gcd n m) \lor b*m - a*n = (gcd n m).
 intros.
-change with
-\exists a,b.
-a*n - b*m = 
-match leb n m with
-  [ true \Rightarrow 
-    match n with 
-    [ O \Rightarrow m
-    | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
-  | false \Rightarrow 
-    match m with 
-    [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]
-\lor b*m - a*n = 
-match leb n m with
-  [ true \Rightarrow 
-    match n with 
-    [ O \Rightarrow m
-    | (S p) \Rightarrow gcd_aux (S p) m (S p) ]
-  | false \Rightarrow 
-    match m with 
-    [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
+unfold gcd.
 apply leb_elim n m.
 apply nat_case1 n.
 simplify.intros.