X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Fgcd.ma;h=99c97b09add8d506bc1566c5250da6fd3b76969f;hb=28ac70d3f475442cda4ef30e0e9c0e6d012b2527;hp=be1d79b1d265c3fb50f059015b8a5fda4058f1f1;hpb=ab44166935d77276c04fcce50aa8281292776e29;p=helm.git

diff --git a/helm/matita/library/nat/gcd.ma b/helm/matita/library/nat/gcd.ma
index be1d79b1d..99c97b09a 100644
--- a/helm/matita/library/nat/gcd.ma
+++ b/helm/matita/library/nat/gcd.ma
@@ -36,8 +36,8 @@ definition gcd : nat \to nat \to nat \def
     [ O \Rightarrow n
     | (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
 
-theorem divides_mod: \forall p,m,n:nat. O < n \to divides p m \to divides p n \to
-divides p (mod m n).
+theorem divides_mod: \forall p,m,n:nat. O < n \to p \divides m \to p \divides n \to
+p \divides mod m n.
 intros.elim H1.elim H2.
 apply witness ? ? (n2 - n1*(div m n)).
 rewrite > distr_times_minus.
@@ -56,7 +56,7 @@ apply div_mod.assumption.
 qed.
 
 theorem divides_mod_to_divides: \forall p,m,n:nat. O < n \to
-divides p (mod m n) \to divides p n \to divides p m. 
+p \divides mod m n \to p \divides n \to p \divides m. 
 intros.elim H1.elim H2.
 apply witness p m ((n1*(div m n))+n2).
 rewrite > distr_times_plus.
@@ -67,29 +67,27 @@ apply div_mod.assumption.
 qed.
 
 theorem divides_gcd_aux_mn: \forall p,m,n. O < n \to n \le m \to n \le p \to
-divides (gcd_aux p m n) m \land divides (gcd_aux p m n) n. 
+gcd_aux p m n \divides m \land gcd_aux p m n \divides n. 
 intro.elim p.
 absurd O < n.assumption.apply le_to_not_lt.assumption.
-cut (divides n1 m) \lor \not (divides n1 m).
+cut (n1 \divides m) \lor (n1 \ndivides m).
 change with 
-(divides 
 (match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]) m) \land
-(divides 
+| false \Rightarrow gcd_aux n n1 (mod m n1)]) \divides m \land
 (match divides_b n1 m with
 [ true \Rightarrow n1
-| false \Rightarrow gcd_aux n n1 (mod m n1)]) n1).
+| false \Rightarrow gcd_aux n n1 (mod m 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 
-(divides (gcd_aux n n1 (mod m n1)) m) \land
-(divides (gcd_aux n n1 (mod m n1)) n1).
-cut (divides (gcd_aux n n1 (mod m n1)) n1) \land
-(divides (gcd_aux n n1 (mod m n1)) (mod m n1)).
+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.
 elim Hcut1.
 split.apply divides_mod_to_divides ? ? n1.
 assumption.assumption.assumption.assumption.
@@ -110,11 +108,10 @@ apply decidable_divides n1 m.assumption.
 qed.
 
 theorem divides_gcd_nm: \forall n,m.
-divides (gcd n m) m \land divides (gcd n m) n.
+gcd n m \divides m \land gcd n m \divides n.
 intros.
 change with
-divides 
-(match leb n m with
+match leb n m with
   [ true \Rightarrow 
     match n with 
     [ O \Rightarrow m
@@ -122,10 +119,9 @@ divides
   | false \Rightarrow 
     match m with 
     [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]) m
+    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]] \divides m
 \land
- divides 
-(match leb n m with
+match leb n m with
   [ true \Rightarrow 
     match n with 
     [ O \Rightarrow m
@@ -133,16 +129,16 @@ divides
   | false \Rightarrow 
     match m with 
     [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]) 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.
 apply witness m m (S O).apply times_n_SO.
 apply witness m O O.apply times_n_O.
 intros.change with
-divides (gcd_aux (S m1) m (S m1)) m
+gcd_aux (S m1) m (S m1) \divides m
 \land 
-divides (gcd_aux (S m1) m (S m1)) (S m1).
+gcd_aux (S m1) m (S m1) \divides (S m1).
 apply divides_gcd_aux_mn.
 simplify.apply le_S_S.apply le_O_n.
 assumption.apply le_n.
@@ -152,12 +148,12 @@ simplify.intros.split.
 apply witness n O O.apply times_n_O.
 apply witness n n (S O).apply times_n_SO.
 intros.change with
-divides (gcd_aux (S m1) n (S m1)) (S m1)
+gcd_aux (S m1) n (S m1) \divides (S m1)
 \land 
-divides (gcd_aux (S m1) n (S m1)) n.
-cut divides (gcd_aux (S m1) n (S m1)) n
+gcd_aux (S m1) n (S m1) \divides n.
+cut gcd_aux (S m1) n (S m1) \divides n
 \land 
-divides (gcd_aux (S m1) n (S m1)) (S m1).
+gcd_aux (S m1) n (S m1) \divides S m1.
 elim Hcut.split.assumption.assumption.
 apply divides_gcd_aux_mn.
 simplify.apply le_S_S.apply le_O_n.
@@ -166,38 +162,36 @@ rewrite > H1.apply trans_le ? (S n).
 apply le_n_Sn.assumption.apply le_n.
 qed.
 
-theorem divides_gcd_n: \forall n,m.
-divides (gcd n m) n.
+theorem divides_gcd_n: \forall n,m. gcd n m \divides n.
 intros. 
 exact proj2  ? ? (divides_gcd_nm n m).
 qed.
 
-theorem divides_gcd_m: \forall n,m.
-divides (gcd n m) m.
+theorem divides_gcd_m: \forall n,m. gcd n m \divides m.
 intros. 
 exact proj1 ? ? (divides_gcd_nm n m).
 qed.
 
 theorem divides_gcd_aux: \forall p,m,n,d. O < n \to n \le m \to n \le p \to
-divides d m \to divides d n \to divides d (gcd_aux p m n). 
+d \divides m \to d \divides n \to d \divides gcd_aux p m n. 
 intro.elim p.
 absurd O < n.assumption.apply le_to_not_lt.assumption.
 change with
-divides d 
+d \divides
 (match divides_b n1 m with
 [ true \Rightarrow n1
 | false \Rightarrow gcd_aux n n1 (mod m n1)]).
-cut divides n1 m \lor \not (divides n1 m).
+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 divides d (gcd_aux n n1 (mod m n1)).
+change with d \divides gcd_aux n n1 (mod m n1).
 apply H.
 cut O \lt mod m n1 \lor O = mod m n1.
 elim Hcut1.assumption.
-absurd divides n1 m.apply mod_O_to_divides.
+absurd n1 \divides m.apply mod_O_to_divides.
 assumption.apply sym_eq.assumption.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 apply lt_to_le.
@@ -213,10 +207,10 @@ apply decidable_divides n1 m.assumption.
 qed.
 
 theorem divides_d_gcd: \forall m,n,d. 
-divides d m \to divides d n \to divides d (gcd n m). 
+d \divides m \to d \divides n \to d \divides gcd n m. 
 intros.
 change with
-divides d (
+d \divides
 match leb n m with
   [ true \Rightarrow 
     match n with 
@@ -225,17 +219,17 @@ match leb n m with
   | false \Rightarrow 
     match m with 
     [ O \Rightarrow n
-    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]]).
+    | (S p) \Rightarrow gcd_aux (S p) n (S p) ]].
 apply leb_elim n m.
 apply nat_case1 n.simplify.intros.assumption.
 intros.
-change with divides d (gcd_aux (S m1) m (S m1)).
+change with d \divides gcd_aux (S m1) m (S m1).
 apply divides_gcd_aux.
 simplify.apply le_S_S.apply le_O_n.assumption.apply le_n.assumption.
 rewrite < H2.assumption.
 apply nat_case1 m.simplify.intros.assumption.
 intros.
-change with divides d (gcd_aux (S m1) n (S m1)).
+change with d \divides gcd_aux (S m1) n (S m1).
 apply divides_gcd_aux.
 simplify.apply le_S_S.apply le_O_n.
 apply lt_to_le.apply not_le_to_lt.assumption.apply le_n.assumption.
@@ -248,7 +242,7 @@ intro.
 elim p.
 absurd O < n.assumption.apply le_to_not_lt.assumption.
 cut O < m.
-cut (divides n1 m) \lor \not (divides n1 m).
+cut n1 \divides m \lor  n1 \ndivides m.
 change with
 \exists a,b.
 a*n1 - b*m = match divides_b n1 m with
@@ -303,6 +297,7 @@ rewrite < H7.
 apply ex_intro ? ? (a1+a*(div 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.
@@ -319,7 +314,7 @@ assumption.
 apply H n1 (mod m n1).
 cut O \lt mod m n1 \lor O = mod m n1.
 elim Hcut2.assumption. 
-absurd divides n1 m.apply mod_O_to_divides.
+absurd n1 \divides m.apply mod_O_to_divides.
 assumption.
 symmetry.assumption.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
@@ -337,28 +332,7 @@ 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.
@@ -413,7 +387,7 @@ qed.
 
 theorem gcd_O_to_eq_O:\forall m,n:nat. (gcd m n) = O \to
 m = O \land n = O.
-intros.cut divides O n \land divides O m.
+intros.cut O \divides n \land O \divides m.
 elim Hcut.elim H2.split.
 assumption.elim H1.assumption.
 rewrite < H.
@@ -462,7 +436,7 @@ apply gcd_O_to_eq_O.apply sym_eq.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 qed.
 
-theorem prime_to_gcd_SO: \forall n,m:nat. prime n \to \not (divides n m) \to
+theorem prime_to_gcd_SO: \forall n,m:nat. prime n \to n \ndivides m \to
 gcd n m = (S O).
 intros.simplify in H.change with gcd n m = (S O). 
 elim H.
@@ -482,10 +456,10 @@ apply gcd_O_to_eq_O.apply sym_eq.assumption.
 apply le_to_or_lt_eq.apply le_O_n.
 qed.
 
-theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to divides n (p*q) \to
-divides n p \lor divides n q.
+theorem divides_times_to_divides: \forall n,p,q:nat.prime n \to n \divides p*q \to
+n \divides p \lor n \divides q.
 intros.
-cut divides n p \lor \not (divides n p).
+cut n \divides p \lor n \ndivides p.
 elim Hcut.
 left.assumption.
 right.