X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Ffactorization.ma;h=92f64e56d9a9c7a556a21f1cacd6d60edaf3b81e;hb=4dc47c9675ffd5fa50296ffaa9b5997501518c98;hp=f17fe03080e1e7b793e72cce687064449f811f40;hpb=a7f664e6b4cda35865cb3619800527204a651ba3;p=helm.git

diff --git a/helm/software/matita/library/nat/factorization.ma b/helm/software/matita/library/nat/factorization.ma
index f17fe0308..92f64e56d 100644
--- a/helm/software/matita/library/nat/factorization.ma
+++ b/helm/software/matita/library/nat/factorization.ma
@@ -74,6 +74,18 @@ cut (\exists i. nth_prime i = smallest_factor n);
     assumption; *) ] 
 qed.
 
+lemma divides_to_prime_divides : \forall n,m.1 < m \to m < n \to m \divides n \to
+ \exists p.p \leq m \land prime p \land p \divides n.
+intros;apply (ex_intro ? ? (nth_prime (max_prime_factor m)));split
+  [split
+     [apply divides_to_le
+        [apply lt_to_le;assumption
+        |apply divides_max_prime_factor_n;assumption]
+     |apply prime_nth_prime;]
+  |apply (transitive_divides ? ? ? ? H2);apply divides_max_prime_factor_n;
+   assumption]
+qed.
+
 theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to 
 max_prime_factor n \le max_prime_factor m.
 intros.unfold max_prime_factor.
@@ -514,7 +526,7 @@ split
       [apply lt_O_nth_prime_n
       |apply (lt_O_n_elim ? H).
        intro.
-       apply (witness ? ? (r*(nth_prime p \sup m))).
+       apply (witness ? ? ((r*(nth_prime p) \sup m))).
        rewrite < assoc_times.
        rewrite < sym_times in \vdash (? ? ? (? % ?)).
        rewrite > exp_n_SO in \vdash (? ? ? (? (? ? %) ?)).
@@ -636,21 +648,19 @@ theorem eq_defactorize_aux_to_eq: \forall f,g:nat_fact.\forall i:nat.
 defactorize_aux f i = defactorize_aux g i \to f = g.
 intro.
 elim f.
-generalize in match H.
-elim g.
+elim g in H ⊢ %.
 apply eq_f.
 apply inj_S. apply (inj_exp_r (nth_prime i)).
 apply lt_SO_nth_prime_n.
 assumption.
 apply False_ind.
-apply (not_eq_nf_last_nf_cons n2 n n1 i H2).
-generalize in match H1.
-elim g.
+apply (not_eq_nf_last_nf_cons n2 n n1 i H1).
+elim g in H1 ⊢ %.
 apply False_ind.
 apply (not_eq_nf_last_nf_cons n1 n2 n i).
 apply sym_eq. assumption.
-simplify in H3.
-generalize in match H3.
+simplify in H2.
+generalize in match H2.
 apply (nat_elim2 (\lambda n,n2.
 ((nth_prime i) \sup n)*(defactorize_aux n1 (S i)) =
 ((nth_prime i) \sup n2)*(defactorize_aux n3 (S i)) \to
@@ -682,7 +692,7 @@ change with
 [ (nf_last m) \Rightarrow m
 | (nf_cons m g) \Rightarrow m ] = m).
 rewrite > Hcut.simplify.reflexivity.
-apply H4.simplify in H5.
+apply H3.simplify in H4.
 apply (inj_times_r1 (nth_prime i)).
 apply lt_O_nth_prime_n.
 rewrite < assoc_times.rewrite < assoc_times.assumption.
@@ -700,22 +710,21 @@ injective nat_fact_all nat defactorize.
 unfold injective.
 change with (\forall f,g.defactorize f = defactorize g \to f=g).
 intro.elim f.
-generalize in match H.elim g.
+elim g in H ⊢ %.
 (* zero - zero *)
 reflexivity.
 (* zero - one *)
 simplify in H1.
 apply False_ind.
-apply (not_eq_O_S O H1).
+apply (not_eq_O_S O H).
 (* zero - proper *)
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n O).
-rewrite > H1 in \vdash (? ? %).
+rewrite > H in \vdash (? ? %).
 change with (O < defactorize_aux n O).
 apply lt_O_defactorize_aux.
-generalize in match H.
-elim g.
+elim g in H ⊢ %.
 (* one - zero *)
 simplify in H1.
 apply False_ind.
@@ -726,28 +735,28 @@ reflexivity.
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n (S O)).
-rewrite > H1 in \vdash (? ? %).
+rewrite > H in \vdash (? ? %).
 change with ((S O) < defactorize_aux n O).
 apply lt_SO_defactorize_aux.
-generalize in match H.elim g.
+elim g in H ⊢ %.
 (* proper - zero *)
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n O).
-rewrite < H1 in \vdash (? ? %).
+rewrite < H in \vdash (? ? %).
 change with (O < defactorize_aux n O).
 apply lt_O_defactorize_aux.
 (* proper - one *)
 simplify in H1.
 apply False_ind.
 apply (not_le_Sn_n (S O)).
-rewrite < H1 in \vdash (? ? %).
+rewrite < H in \vdash (? ? %).
 change with ((S O) < defactorize_aux n O).
 apply lt_SO_defactorize_aux.
 (* proper - proper *)
 apply eq_f.
 apply (injective_defactorize_aux O).
-exact H1.
+exact H.
 qed.
 
 theorem factorize_defactorize: