X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Fnat%2Ffactorization.ma;h=6a309430302ee18b576e2def76303d3333a6fdfd;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=1b946be522c143b6be9e1082dab079c9b82a86dd;hpb=30b7e65d641fe7243c4f36ed448f56360a1c5e1c;p=helm.git

diff --git a/helm/matita/library/nat/factorization.ma b/helm/matita/library/nat/factorization.ma
index 1b946be52..6a3094303 100644
--- a/helm/matita/library/nat/factorization.ma
+++ b/helm/matita/library/nat/factorization.ma
@@ -14,36 +14,36 @@
 
 set "baseuri" "cic:/matita/nat/factorization".
 
-include "nat/log.ma".
+include "nat/ord.ma".
 include "nat/gcd.ma".
 include "nat/nth_prime.ma".
 
 (* the following factorization algorithm looks for the largest prime
    factor. *)
 definition max_prime_factor \def \lambda n:nat.
-(max n (\lambda p:nat.eqb (mod n (nth_prime p)) O)).
+(max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)).
 
 (* max_prime_factor is indeed a factor *)
 theorem divides_max_prime_factor_n: \forall n:nat. (S O) < n \to
 nth_prime (max_prime_factor n) \divides n.
 intros.apply divides_b_true_to_divides.
 apply lt_O_nth_prime_n.
-apply f_max_true  (\lambda p:nat.eqb (mod n (nth_prime p)) O) n.
-cut \exists i. nth_prime i = smallest_factor n.
+apply (f_max_true  (\lambda p:nat.eqb (n \mod (nth_prime p)) O) n).
+cut (\exists i. nth_prime i = smallest_factor n).
 elim Hcut.
-apply ex_intro nat ? a.
+apply (ex_intro nat ? a).
 split.
-apply trans_le a (nth_prime a).
+apply (trans_le a (nth_prime a)).
 apply le_n_fn.
 exact lt_nth_prime_n_nth_prime_Sn.
 rewrite > H1. apply le_smallest_factor_n.
 rewrite > H1.
-change with divides_b (smallest_factor n) n = true.
+change with (divides_b (smallest_factor n) n = true).
 apply divides_to_divides_b_true.
-apply trans_lt ? (S O).simplify. apply le_n.
+apply (trans_lt ? (S O)).unfold lt. apply le_n.
 apply lt_SO_smallest_factor.assumption.
 apply divides_smallest_factor_n.
-apply trans_lt ? (S O). simplify. apply le_n. assumption.
+apply (trans_lt ? (S O)). unfold lt. apply le_n. assumption.
 apply prime_to_nth_prime.
 apply prime_smallest_factor_n.assumption.
 qed.
@@ -51,17 +51,17 @@ 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.change with
-(max n (\lambda p:nat.eqb (mod n (nth_prime p)) O)) \le
-(max m (\lambda p:nat.eqb (mod m (nth_prime p)) O)).
+((max n (\lambda p:nat.eqb (n \mod (nth_prime p)) O)) \le
+(max m (\lambda p:nat.eqb (m \mod (nth_prime p)) O))).
 apply f_m_to_le_max.
-apply trans_le ? n.
+apply (trans_le ? n).
 apply le_max_n.apply divides_to_le.assumption.assumption.
-change with divides_b (nth_prime (max_prime_factor n)) m = true.
+change with (divides_b (nth_prime (max_prime_factor n)) m = true).
 apply divides_to_divides_b_true.
-cut prime (nth_prime (max_prime_factor n)).
+cut (prime (nth_prime (max_prime_factor n))).
 apply lt_O_nth_prime_n.apply prime_nth_prime.
-cut nth_prime (max_prime_factor n) \divides n.
-apply transitive_divides ? n.
+cut (nth_prime (max_prime_factor n) \divides n).
+apply (transitive_divides ? n).
 apply divides_max_prime_factor_n.
 assumption.assumption.
 apply divides_b_true_to_divides.
@@ -72,53 +72,53 @@ apply divides_max_prime_factor_n.
 assumption.
 qed.
 
-theorem plog_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
+theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
 p = max_prime_factor n \to 
-(pair nat nat q r) = plog n (nth_prime p) \to
+(pair nat nat q r) = p_ord n (nth_prime p) \to
 (S O) < r \to max_prime_factor r < p.
 intros.
 rewrite > H1.
-cut max_prime_factor r \lt max_prime_factor n \lor
-    max_prime_factor r = max_prime_factor n.
+cut (max_prime_factor r \lt max_prime_factor n \lor
+    max_prime_factor r = max_prime_factor n).
 elim Hcut.assumption.
-absurd nth_prime (max_prime_factor n) \divides r.
+absurd (nth_prime (max_prime_factor n) \divides r).
 rewrite < H4.
 apply divides_max_prime_factor_n.
 assumption.
-change with nth_prime (max_prime_factor n) \divides r \to False.
+change with (nth_prime (max_prime_factor n) \divides r \to False).
 intro.
-cut \lnot (mod r (nth_prime (max_prime_factor n))) = O.
+cut (r \mod (nth_prime (max_prime_factor n)) \neq O).
 apply Hcut1.apply divides_to_mod_O.
 apply lt_O_nth_prime_n.assumption.
-apply plog_aux_to_not_mod_O n n ? q r.
+apply (p_ord_aux_to_not_mod_O n n ? q r).
 apply lt_SO_nth_prime_n.assumption.
 apply le_n.
 rewrite < H1.assumption.
-apply le_to_or_lt_eq (max_prime_factor r)  (max_prime_factor n).
+apply (le_to_or_lt_eq (max_prime_factor r)  (max_prime_factor n)).
 apply divides_to_max_prime_factor.
 assumption.assumption.
-apply witness r n ((nth_prime p) \sup q).
+apply (witness r n ((nth_prime p) \sup q)).
 rewrite < sym_times.
-apply plog_aux_to_exp n n ? q r.
+apply (p_ord_aux_to_exp n n ? q r).
 apply lt_O_nth_prime_n.assumption.
 qed.
 
-theorem plog_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
+theorem p_ord_to_lt_max_prime_factor1: \forall n,p,q,r. O < n \to
 max_prime_factor n \le p \to 
-(pair nat nat q r) = plog n (nth_prime p) \to
+(pair nat nat q r) = p_ord n (nth_prime p) \to
 (S O) < r \to max_prime_factor r < p.
 intros.
-cut max_prime_factor n < p \lor max_prime_factor n = p.
-elim Hcut.apply le_to_lt_to_lt ? (max_prime_factor n).
+cut (max_prime_factor n < p \lor max_prime_factor n = p).
+elim Hcut.apply (le_to_lt_to_lt ? (max_prime_factor n)).
 apply divides_to_max_prime_factor.assumption.assumption.
-apply witness r n ((nth_prime p) \sup q).
+apply (witness r n ((nth_prime p) \sup q)).
 rewrite > sym_times.
-apply plog_aux_to_exp n n.
+apply (p_ord_aux_to_exp n n).
 apply lt_O_nth_prime_n.
 assumption.assumption.
-apply plog_to_lt_max_prime_factor n ? q.
+apply (p_ord_to_lt_max_prime_factor n ? q).
 assumption.apply sym_eq.assumption.assumption.assumption.
-apply le_to_or_lt_eq ? p H1.
+apply (le_to_or_lt_eq ? p H1).
 qed.
 
 (* datatypes and functions *)
@@ -136,7 +136,7 @@ let rec factorize_aux p n acc \def
   match p with 
   [ O \Rightarrow acc
   | (S p1) \Rightarrow 
-    match plog n (nth_prime p1) with
+    match p_ord n (nth_prime p1) with
     [ (pair q r) \Rightarrow 
       factorize_aux p1 r (nf_cons q acc)]].
   
@@ -147,8 +147,8 @@ definition factorize : nat \to nat_fact_all \def \lambda n:nat.
       match n1 with
       [ O \Rightarrow nfa_one
     | (S n2) \Rightarrow 
-      let p \def (max (S(S n2)) (\lambda p:nat.eqb (mod (S(S n2)) (nth_prime p)) O)) in
-      match plog (S(S n2)) (nth_prime p) with
+      let p \def (max (S(S n2)) (\lambda p:nat.eqb ((S(S n2)) \mod (nth_prime p)) O)) in
+      match p_ord (S(S n2)) (nth_prime p) with
       [ (pair q r) \Rightarrow 
            nfa_proper (factorize_aux p r (nf_last (pred q)))]]].
            
@@ -165,6 +165,47 @@ match f with
 | nfa_one \Rightarrow (S O)
 | (nfa_proper g) \Rightarrow defactorize_aux g O]. 
 
+theorem lt_O_defactorize_aux: \forall f:nat_fact.\forall i:nat.
+O < defactorize_aux f i.
+intro.elim f.simplify.unfold lt. 
+rewrite > times_n_SO.
+apply le_times.
+change with (O < nth_prime i).
+apply lt_O_nth_prime_n.
+change with (O < exp (nth_prime i) n).
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+simplify.unfold lt.
+rewrite > times_n_SO.
+apply le_times.
+change with (O < exp (nth_prime i) n).
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+change with (O < defactorize_aux n1 (S i)).
+apply H.
+qed.
+
+theorem lt_SO_defactorize_aux: \forall f:nat_fact.\forall i:nat.
+S O < defactorize_aux f i.
+intro.elim f.simplify.unfold lt.
+rewrite > times_n_SO.
+apply le_times.
+change with (S O < nth_prime i).
+apply lt_SO_nth_prime_n.
+change with (O < exp (nth_prime i) n).
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+simplify.unfold lt.
+rewrite > times_n_SO.
+rewrite > sym_times.
+apply le_times.
+change with (O < exp (nth_prime i) n).
+apply lt_O_exp.
+apply lt_O_nth_prime_n.
+change with (S O < defactorize_aux n1 (S i)).
+apply H.
+qed.
+
 theorem defactorize_aux_factorize_aux : 
 \forall p,n:nat.\forall acc:nat_fact.O < n \to
 ((n=(S O) \land p=O) \lor max_prime_factor n < p) \to
@@ -172,150 +213,150 @@ defactorize_aux (factorize_aux p n acc) O = n*(defactorize_aux acc p).
 intro.elim p.simplify.
 elim H1.elim H2.rewrite > H3.
 rewrite > sym_times. apply times_n_SO.
-apply False_ind.apply not_le_Sn_O (max_prime_factor n) H2.
+apply False_ind.apply (not_le_Sn_O (max_prime_factor n) H2).
 simplify.
 (* generalizing the goal: I guess there exists a better way *)
-cut \forall q,r.(pair nat nat q r) = (plog_aux n1 n1 (nth_prime n)) \to
-defactorize_aux match (plog_aux n1 n1 (nth_prime n)) with
+cut (\forall q,r.(pair nat nat q r) = (p_ord_aux n1 n1 (nth_prime n)) \to
+defactorize_aux match (p_ord_aux n1 n1 (nth_prime n)) with
 [(pair q r)  \Rightarrow (factorize_aux n r (nf_cons q acc))] O =
-n1*defactorize_aux acc (S n).
-apply Hcut (fst ? ? (plog_aux n1 n1 (nth_prime n)))
-(snd ? ? (plog_aux n1 n1 (nth_prime n))).
+n1*defactorize_aux acc (S n)).
+apply (Hcut (fst ? ? (p_ord_aux n1 n1 (nth_prime n)))
+(snd ? ? (p_ord_aux n1 n1 (nth_prime n)))).
 apply sym_eq.apply eq_pair_fst_snd.
 intros.
 rewrite < H3.
 simplify.
-cut n1 = r * (nth_prime n) \sup q.
+cut (n1 = r * (nth_prime n) \sup q).
 rewrite > H.
 simplify.rewrite < assoc_times.
 rewrite < Hcut.reflexivity.
-cut O < r \lor O = r.
-elim Hcut1.assumption.absurd n1 = O.
+cut (O < r \lor O = r).
+elim Hcut1.assumption.absurd (n1 = O).
 rewrite > Hcut.rewrite < H4.reflexivity.
-simplify. intro.apply not_le_Sn_O O.
+unfold Not. intro.apply (not_le_Sn_O O).
 rewrite < H5 in \vdash (? ? %).assumption.
 apply le_to_or_lt_eq.apply le_O_n.
-cut (S O) < r \lor \lnot (S O) < r.
+cut ((S O) < r \lor (S O) \nlt r).
 elim Hcut1.
 right.
-apply plog_to_lt_max_prime_factor1 n1 ? q r.
+apply (p_ord_to_lt_max_prime_factor1 n1 ? q r).
 assumption.elim H2.
 elim H5.
 apply False_ind.
-apply not_eq_O_S n.apply sym_eq.assumption.
+apply (not_eq_O_S n).apply sym_eq.assumption.
 apply le_S_S_to_le.
 exact H5.
 assumption.assumption.
-cut r=(S O).
-apply nat_case n.
+cut (r=(S O)).
+apply (nat_case n).
 left.split.assumption.reflexivity.
 intro.right.rewrite > Hcut2.
-simplify.apply le_S_S.apply le_O_n.
-cut r \lt (S O) \or r=(S O).
-elim Hcut2.absurd O=r.
+simplify.unfold lt.apply le_S_S.apply le_O_n.
+cut (r \lt (S O) \or r=(S O)).
+elim Hcut2.absurd (O=r).
 apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
-simplify.intro.
-cut O=n1.
-apply not_le_Sn_O O.
+unfold Not.intro.
+cut (O=n1).
+apply (not_le_Sn_O O).
 rewrite > Hcut3 in \vdash (? ? %).
 assumption.rewrite > Hcut. 
 rewrite < H6.reflexivity.
 assumption.
-apply le_to_or_lt_eq r (S O).
+apply (le_to_or_lt_eq r (S O)).
 apply not_lt_to_le.assumption.
-apply decidable_lt (S O) r.
+apply (decidable_lt (S O) r).
 rewrite > sym_times.
-apply plog_aux_to_exp n1 n1.
+apply (p_ord_aux_to_exp n1 n1).
 apply lt_O_nth_prime_n.assumption.
 qed.
 
 theorem defactorize_factorize: \forall n:nat.defactorize (factorize n) = n.
 intro.
-apply nat_case n.reflexivity.
-intro.apply nat_case m.reflexivity.
+apply (nat_case n).reflexivity.
+intro.apply (nat_case m).reflexivity.
 intro.change with  
-let p \def (max (S(S m1)) (\lambda p:nat.eqb (mod (S(S m1)) (nth_prime p)) O)) in
-defactorize (match plog (S(S m1)) (nth_prime p) with
+(let p \def (max (S(S m1)) (\lambda p:nat.eqb ((S(S m1)) \mod (nth_prime p)) O)) in
+defactorize (match p_ord (S(S m1)) (nth_prime p) with
 [ (pair q r) \Rightarrow 
-   nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
+   nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1))).
 intro.
 (* generalizing the goal; find a better way *)
-cut \forall q,r.(pair nat nat q r) = (plog (S(S m1)) (nth_prime p)) \to
-defactorize (match plog (S(S m1)) (nth_prime p) with
+cut (\forall q,r.(pair nat nat q r) = (p_ord (S(S m1)) (nth_prime p)) \to
+defactorize (match p_ord (S(S m1)) (nth_prime p) with
 [ (pair q r) \Rightarrow 
-   nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1)).
-apply Hcut (fst ? ? (plog (S(S m1)) (nth_prime p)))
-(snd ? ? (plog (S(S m1)) (nth_prime p))).
+   nfa_proper (factorize_aux p r (nf_last (pred q)))])=(S(S m1))).
+apply (Hcut (fst ? ? (p_ord (S(S m1)) (nth_prime p)))
+(snd ? ? (p_ord (S(S m1)) (nth_prime p)))).
 apply sym_eq.apply eq_pair_fst_snd.
 intros.
 rewrite < H.
 change with 
-defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1)).
-cut (S(S m1)) = (nth_prime p) \sup q *r.
-cut O <r.
+(defactorize_aux (factorize_aux p r (nf_last (pred q))) O = (S(S m1))).
+cut ((S(S m1)) = (nth_prime p) \sup q *r).
+cut (O<r).
 rewrite > defactorize_aux_factorize_aux.
-change with r*(nth_prime p) \sup (S (pred q)) = (S(S m1)).
-cut (S (pred q)) = q.
+change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
+cut ((S (pred q)) = q).
 rewrite > Hcut2.
 rewrite > sym_times.
 apply sym_eq.
-apply plog_aux_to_exp (S(S m1)).
+apply (p_ord_aux_to_exp (S(S m1))).
 apply lt_O_nth_prime_n.
 assumption.
 (* O < q *)
 apply sym_eq. apply S_pred.
-cut O < q \lor O = q.
+cut (O < q \lor O = q).
 elim Hcut2.assumption.
-absurd nth_prime p \divides S (S m1).
-apply divides_max_prime_factor_n (S (S m1)).
-simplify.apply le_S_S.apply le_S_S. apply le_O_n.
-cut (S(S m1)) = r.
+absurd (nth_prime p \divides S (S m1)).
+apply (divides_max_prime_factor_n (S (S m1))).
+unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
+cut ((S(S m1)) = r).
 rewrite > Hcut3 in \vdash (? (? ? %)).
-change with nth_prime p \divides r \to False.
+change with (nth_prime p \divides r \to False).
 intro.
-apply plog_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r.
+apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
 apply lt_SO_nth_prime_n.
-simplify.apply le_S_S.apply le_O_n.apply le_n.
+unfold lt.apply le_S_S.apply le_O_n.apply le_n.
 assumption.
 apply divides_to_mod_O.apply lt_O_nth_prime_n.assumption.
 rewrite > times_n_SO in \vdash (? ? ? %).
 rewrite < sym_times.
-rewrite > exp_n_O (nth_prime p).
+rewrite > (exp_n_O (nth_prime p)).
 rewrite > H1 in \vdash (? ? ? (? (? ? %) ?)).
 assumption.
 apply le_to_or_lt_eq.apply le_O_n.assumption.
 (* e adesso l'ultimo goal. TASSI: che ora non e' piu' l'ultimo :P *)
-cut (S O) < r \lor \lnot (S O) < r.
+cut ((S O) < r \lor S O \nlt r).
 elim Hcut2.
 right. 
-apply plog_to_lt_max_prime_factor1 (S(S m1)) ? q r.
-simplify.apply le_S_S. apply le_O_n.
+apply (p_ord_to_lt_max_prime_factor1 (S(S m1)) ? q r).
+unfold lt.apply le_S_S. apply le_O_n.
 apply le_n.
 assumption.assumption.
-cut r=(S O).
-apply nat_case p.
+cut (r=(S O)).
+apply (nat_case p).
 left.split.assumption.reflexivity.
 intro.right.rewrite > Hcut3.
-simplify.apply le_S_S.apply le_O_n.
-cut r \lt (S O) \or r=(S O).
-elim Hcut3.absurd O=r.
+simplify.unfold lt.apply le_S_S.apply le_O_n.
+cut (r \lt (S O) \or r=(S O)).
+elim Hcut3.absurd (O=r).
 apply le_n_O_to_eq.apply le_S_S_to_le.exact H2.
-simplify.intro.
-apply not_le_Sn_O O.
+unfold Not.intro.
+apply (not_le_Sn_O O).
 rewrite > H3 in \vdash (? ? %).assumption.assumption.
-apply le_to_or_lt_eq r (S O).
+apply (le_to_or_lt_eq r (S O)).
 apply not_lt_to_le.assumption.
-apply decidable_lt (S O) r.
+apply (decidable_lt (S O) r).
 (* O < r *)
-cut O < r \lor O = r.
+cut (O < r \lor O = r).
 elim Hcut1.assumption. 
 apply False_ind.
-apply not_eq_O_S (S m1).
+apply (not_eq_O_S (S m1)).
 rewrite > Hcut.rewrite < H1.rewrite < times_n_O.reflexivity.
 apply le_to_or_lt_eq.apply le_O_n.
 (* prova del cut *)
 goal 20.
-apply plog_aux_to_exp (S(S m1)).
+apply (p_ord_aux_to_exp (S(S m1))).
 apply lt_O_nth_prime_n.
 assumption.
 (* fine prova cut *)
@@ -334,17 +375,17 @@ match f with
 theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat. 
 nth_prime ((max_p f)+i) \divides defactorize_aux f i.
 intro.
-elim f.simplify.apply witness ? ? ((nth_prime i) \sup n).
+elim f.simplify.apply (witness ? ? ((nth_prime i) \sup n)).
 reflexivity.
 change with 
-nth_prime (S(max_p n1)+i) \divides
-(nth_prime i) \sup n *(defactorize_aux n1 (S i)).
-elim H (S i).
+(nth_prime (S(max_p n1)+i) \divides
+(nth_prime i) \sup n *(defactorize_aux n1 (S i))).
+elim (H (S i)).
 rewrite > H1.
 rewrite < sym_times.
 rewrite > assoc_times.
 rewrite < plus_n_Sm.
-apply witness ? ? (n2* (nth_prime i) \sup n).
+apply (witness ? ? (n2* (nth_prime i) \sup n)).
 reflexivity.
 qed.
 
@@ -352,9 +393,9 @@ theorem divides_exp_to_divides:
 \forall p,n,m:nat. prime p \to 
 p \divides n \sup m \to p \divides n.
 intros 3.elim m.simplify in H1.
-apply transitive_divides p (S O).assumption.
+apply (transitive_divides p (S O)).assumption.
 apply divides_SO_n.
-cut p \divides n \lor p \divides n \sup n1.
+cut (p \divides n \lor p \divides n \sup n1).
 elim Hcut.assumption.
 apply H.assumption.assumption.
 apply divides_times_to_divides.assumption.
@@ -365,48 +406,83 @@ theorem divides_exp_to_eq:
 \forall p,q,m:nat. prime p \to prime q \to
 p \divides q \sup m \to p = q.
 intros.
-simplify in H1.
+unfold prime in H1.
 elim H1.apply H4.
-apply divides_exp_to_divides p q m.
+apply (divides_exp_to_divides p q m).
 assumption.assumption.
-simplify in H.elim H.assumption.
+unfold prime in H.elim H.assumption.
 qed.
 
 theorem  not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
 i < j \to nth_prime i \ndivides defactorize_aux f j.
 intro.elim f.
 change with
-nth_prime i \divides (nth_prime j) \sup (S n) \to False.
-intro.absurd (nth_prime i) = (nth_prime j).
-apply divides_exp_to_eq ? ? (S n).
+(nth_prime i \divides (nth_prime j) \sup (S n) \to False).
+intro.absurd ((nth_prime i) = (nth_prime j)).
+apply (divides_exp_to_eq ? ? (S n)).
 apply prime_nth_prime.apply prime_nth_prime.
 assumption.
-change with (nth_prime i) = (nth_prime j) \to False.
-intro.cut i = j.
-apply not_le_Sn_n i.rewrite > Hcut in \vdash (? ? %).assumption.
-apply injective_nth_prime ? ? H2.
+change with ((nth_prime i) = (nth_prime j) \to False).
+intro.cut (i = j).
+apply (not_le_Sn_n i).rewrite > Hcut in \vdash (? ? %).assumption.
+apply (injective_nth_prime ? ? H2).
 change with 
-nth_prime i \divides (nth_prime j) \sup n *(defactorize_aux n1 (S j)) \to False.
+(nth_prime i \divides (nth_prime j) \sup n *(defactorize_aux n1 (S j)) \to False).
 intro.
-cut nth_prime i \divides (nth_prime j) \sup n
-\lor nth_prime i \divides defactorize_aux n1 (S j).
+cut (nth_prime i \divides (nth_prime j) \sup n
+\lor nth_prime i \divides defactorize_aux n1 (S j)).
 elim Hcut.
-absurd (nth_prime i) = (nth_prime j).
-apply divides_exp_to_eq ? ? n.
+absurd ((nth_prime i) = (nth_prime j)).
+apply (divides_exp_to_eq ? ? n).
 apply prime_nth_prime.apply prime_nth_prime.
 assumption.
-change with (nth_prime i) = (nth_prime j) \to False.
+change with ((nth_prime i) = (nth_prime j) \to False).
 intro.
-cut i = j.
-apply not_le_Sn_n i.rewrite > Hcut1 in \vdash (? ? %).assumption.
-apply injective_nth_prime ? ? H4.
-apply H i (S j).
-apply trans_lt ? j.assumption.simplify.apply le_n.
+cut (i = j).
+apply (not_le_Sn_n i).rewrite > Hcut1 in \vdash (? ? %).assumption.
+apply (injective_nth_prime ? ? H4).
+apply (H i (S j)).
+apply (trans_lt ? j).assumption.unfold lt.apply le_n.
 assumption.
 apply divides_times_to_divides.
 apply prime_nth_prime.assumption.
 qed.
 
+lemma not_eq_nf_last_nf_cons: \forall g:nat_fact.\forall n,m,i:nat.
+\lnot (defactorize_aux (nf_last n) i= defactorize_aux (nf_cons m g) i).
+intros.
+change with 
+(exp (nth_prime i) (S n) = defactorize_aux (nf_cons m g) i \to False).
+intro.
+cut (S(max_p g)+i= i).
+apply (not_le_Sn_n i).
+rewrite < Hcut in \vdash (? ? %).
+simplify.apply le_S_S.
+apply le_plus_n.
+apply injective_nth_prime.
+(* uffa, perche' semplifica ? *)
+change with (nth_prime (S(max_p g)+i)= nth_prime i).
+apply (divides_exp_to_eq ? ? (S n)).
+apply prime_nth_prime.apply prime_nth_prime.
+rewrite > H.
+change with (divides (nth_prime ((max_p (nf_cons m g))+i)) 
+(defactorize_aux (nf_cons m g) i)).
+apply divides_max_p_defactorize.
+qed.
+
+lemma not_eq_nf_cons_O_nf_cons: \forall f,g:nat_fact.\forall n,i:nat.
+\lnot (defactorize_aux (nf_cons O f) i= defactorize_aux (nf_cons (S n) g) i).
+intros.
+simplify.unfold Not.rewrite < plus_n_O.
+intro.
+apply (not_divides_defactorize_aux f i (S i) ?).
+unfold lt.apply le_n.
+rewrite > H.
+rewrite > assoc_times.
+apply (witness ? ? ((exp (nth_prime i) n)*(defactorize_aux g (S i)))).
+reflexivity.
+qed.
+
 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.
@@ -414,116 +490,124 @@ elim f.
 generalize in match H.
 elim g.
 apply eq_f.
-apply inj_S. apply inj_exp_r (nth_prime i).
+apply inj_S. apply (inj_exp_r (nth_prime i)).
 apply lt_SO_nth_prime_n.
 assumption.
-absurd defactorize_aux (nf_last n) i =
-defactorize_aux (nf_cons n1 n2) i.
-rewrite > H2.reflexivity.
-absurd nth_prime (S(max_p n2)+i) \divides defactorize_aux (nf_cons n1 n2) i.
-apply divides_max_p_defactorize.
-rewrite < H2.
-change with 
-(nth_prime (S(max_p n2)+i) \divides (nth_prime i) \sup (S n)) \to False.
-intro.
-absurd nth_prime (S (max_p n2) + i) = nth_prime i.
-apply divides_exp_to_eq ? ? (S n).
-apply prime_nth_prime.apply prime_nth_prime.assumption.
-change with nth_prime (S (max_p n2) + i) = nth_prime i \to False.
-intro.apply not_le_Sn_n i.
-cut S(max_p n2)+i= i.
-rewrite < Hcut in \vdash (? ? %).
-simplify.apply le_S_S.
-apply le_plus_n.
-apply injective_nth_prime ? ? H4.
+apply False_ind.
+apply (not_eq_nf_last_nf_cons n2 n n1 i H2).
 generalize in match H1.
 elim g.
-absurd defactorize_aux (nf_last n2) i =
-defactorize_aux (nf_cons n n1) i.
+apply False_ind.
+apply (not_eq_nf_last_nf_cons n1 n2 n i).
 apply sym_eq. assumption.
-absurd nth_prime (S(max_p n1)+i) \divides defactorize_aux (nf_cons n n1) i.
-apply divides_max_p_defactorize.
-rewrite > H2.
-change with 
-(nth_prime (S(max_p n1)+i) \divides (nth_prime i) \sup (S n2)) \to False.
-intro.
-absurd nth_prime (S (max_p n1) + i) = nth_prime i.
-apply divides_exp_to_eq ? ? (S n2).
-apply prime_nth_prime.apply prime_nth_prime.assumption.
-change with nth_prime (S (max_p n1) + i) = nth_prime i \to False.
-intro.apply not_le_Sn_n i.
-cut S(max_p n1)+i= i.
-rewrite < Hcut in \vdash (? ? %).
-simplify.apply le_S_S.
-apply le_plus_n.
-apply injective_nth_prime ? ? H4.
 simplify in H3.
 generalize in match H3.
-apply nat_elim2 (\lambda n,n2.
+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
-nf_cons n n1 = nf_cons n2 n3).
+nf_cons n n1 = nf_cons n2 n3)).
 intro.
 elim n4. apply eq_f.
-apply H n3 (S i).
+apply (H n3 (S i)).
 simplify in H4.
 rewrite > plus_n_O.
-rewrite > plus_n_O (defactorize_aux n3 (S i)).assumption.
+rewrite > (plus_n_O (defactorize_aux n3 (S i))).assumption.
 apply False_ind.
-apply not_divides_defactorize_aux n1 i (S i).
-simplify. apply le_n.
-simplify in H5.
-rewrite > plus_n_O (defactorize_aux n1 (S i)).
-rewrite > H5.
-rewrite > assoc_times.
-apply witness ? ? (((nth_prime i) \sup n5)*(defactorize_aux n3 (S i))).
-reflexivity.
+apply (not_eq_nf_cons_O_nf_cons n1 n3 n5 i).assumption.
 intros.
 apply False_ind.
-apply not_divides_defactorize_aux n3 i (S i).
-simplify. apply le_n.
-simplify in H4.
-rewrite > plus_n_O (defactorize_aux n3 (S i)).
-rewrite < H4.
-rewrite > assoc_times.
-apply witness ? ? (((nth_prime i) \sup n4)*(defactorize_aux n1 (S i))).
-reflexivity.
+apply (not_eq_nf_cons_O_nf_cons n3 n1 n4 i).
+apply sym_eq.assumption.
 intros.
-cut nf_cons n4 n1 = nf_cons m n3.
-cut n4=m.
-cut n1=n3.
+cut (nf_cons n4 n1 = nf_cons m n3).
+cut (n4=m).
+cut (n1=n3).
 rewrite > Hcut1.rewrite > Hcut2.reflexivity.
 change with 
-match nf_cons n4 n1 with
+(match nf_cons n4 n1 with
 [ (nf_last m) \Rightarrow n1
-| (nf_cons m g) \Rightarrow g ] = n3.
+| (nf_cons m g) \Rightarrow g ] = n3).
 rewrite > Hcut.simplify.reflexivity.
 change with 
-match nf_cons n4 n1 with
+(match nf_cons n4 n1 with
 [ (nf_last m) \Rightarrow m
-| (nf_cons m g) \Rightarrow m ] = m.
+| (nf_cons m g) \Rightarrow m ] = m).
 rewrite > Hcut.simplify.reflexivity.
 apply H4.simplify in H5.
-apply inj_times_r1 (nth_prime i).
+apply (inj_times_r1 (nth_prime i)).
 apply lt_O_nth_prime_n.
 rewrite < assoc_times.rewrite < assoc_times.assumption.
 qed.
 
 theorem injective_defactorize_aux: \forall i:nat.
 injective nat_fact nat (\lambda f.defactorize_aux f i).
-change with \forall i:nat.\forall f,g:nat_fact.
-defactorize_aux f i = defactorize_aux g i \to f = g.
+change with (\forall i:nat.\forall f,g:nat_fact.
+defactorize_aux f i = defactorize_aux g i \to f = g).
 intros.
-apply eq_defactorize_aux_to_eq f g i H.
+apply (eq_defactorize_aux_to_eq f g i H).
 qed.
 
-(*
 theorem injective_defactorize: 
 injective nat_fact_all nat defactorize.
-change with \forall f,g:nat_fact_all.
-defactorize f = defactorize g \to f = g.
+change with (\forall f,g:nat_fact_all.
+defactorize f = defactorize g \to f = g).
 intro.elim f.
 generalize in match H.elim g.
-reflexivity.simplify in H1.
-*)
+(* zero - zero *)
+reflexivity.
+(* zero - one *)
+simplify in H1.
+apply False_ind.
+apply (not_eq_O_S O H1).
+(* zero - proper *)
+simplify in H1.
+apply False_ind.
+apply (not_le_Sn_n O).
+rewrite > H1 in \vdash (? ? %).
+change with (O < defactorize_aux n O).
+apply lt_O_defactorize_aux.
+generalize in match H.
+elim g.
+(* one - zero *)
+simplify in H1.
+apply False_ind.
+apply (not_eq_O_S O).apply sym_eq. assumption.
+(* one - one *)
+reflexivity.
+(* one - proper *)
+simplify in H1.
+apply False_ind.
+apply (not_le_Sn_n (S O)).
+rewrite > H1 in \vdash (? ? %).
+change with ((S O) < defactorize_aux n O).
+apply lt_SO_defactorize_aux.
+generalize in match H.elim g.
+(* proper - zero *)
+simplify in H1.
+apply False_ind.
+apply (not_le_Sn_n O).
+rewrite < H1 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 (? ? %).
+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.
+qed.
+
+theorem factorize_defactorize: 
+\forall f,g: nat_fact_all. factorize (defactorize f) = f.
+intros.
+apply injective_defactorize.
+(* uffa: perche' semplifica ??? *)
+change with (defactorize(factorize (defactorize f)) = (defactorize f)).
+apply defactorize_factorize.
+qed.