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
-divides (nth_prime (max_prime_factor n)) n.
+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.
+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 prime_smallest_factor_n.assumption.
qed.
-theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to divides n m \to
+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 le_max_n.apply divides_to_le.assumption.assumption.
apply divides_to_divides_b_true.
cut prime (nth_prime (max_prime_factor n)).
apply lt_O_nth_prime_n.apply prime_nth_prime.
-cut divides (nth_prime (max_prime_factor n)) n.
+cut nth_prime (max_prime_factor n) \divides n.
apply transitive_divides ? n.
apply divides_max_prime_factor_n.
assumption.assumption.
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.
elim Hcut.assumption.
-absurd (divides (nth_prime (max_prime_factor n)) r).
+absurd nth_prime (max_prime_factor n) \divides r.
rewrite < H4.
apply divides_max_prime_factor_n.
assumption.
-change with (divides (nth_prime (max_prime_factor n)) r) \to False.
+change with nth_prime (max_prime_factor n) \divides r \to False.
intro.
-cut \not (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 divides_to_max_prime_factor.
assumption.assumption.
-apply witness r n (exp (nth_prime p) 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).
apply divides_to_max_prime_factor.assumption.assumption.
-apply witness r n (exp (nth_prime p) 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.
qed.
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)]].
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)))]]].
let rec defactorize_aux f i \def
match f with
- [ (nf_last n) \Rightarrow exp (nth_prime i) (S n)
+ [ (nf_last n) \Rightarrow (nth_prime i) \sup (S n)
| (nf_cons n g) \Rightarrow
- (exp (nth_prime i) n)*(defactorize_aux g (S i))].
+ (nth_prime i) \sup n *(defactorize_aux g (S i))].
definition defactorize : nat_fact_all \to nat \def
\lambda f : nat_fact_all.
| 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.
+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.
+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.
+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.
+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
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))).
+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*(exp (nth_prime n) q).
+cut n1 = r * (nth_prime n) \sup q.
rewrite > H.
simplify.rewrite < assoc_times.
rewrite < Hcut.reflexivity.
simplify. 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_lt_to_le.assumption.
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.
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)).
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))).
+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)) = (exp (nth_prime p) q)*r.
+cut (S(S m1)) = (nth_prime p) \sup q *r.
cut O <r.
rewrite > defactorize_aux_factorize_aux.
-change with r*(exp (nth_prime p) (S (pred q))) = (S(S m1)).
+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.
elim Hcut2.assumption.
-absurd divides (nth_prime p) (S (S m1)).
+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.
rewrite > Hcut3 in \vdash (? (? ? %)).
-change with divides (nth_prime p) 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.
assumption.
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.
+apply p_ord_to_lt_max_prime_factor1 (S(S m1)) ? q r.
simplify.apply le_S_S. apply le_O_n.
apply le_n.
assumption.assumption.
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 *)
| (nf_cons n g) \Rightarrow max_p_exponent g].
theorem divides_max_p_defactorize: \forall f:nat_fact.\forall i:nat.
-divides (nth_prime ((max_p f)+i)) (defactorize_aux f i).
+nth_prime ((max_p f)+i) \divides defactorize_aux f i.
intro.
-elim f.simplify.apply witness ? ? (exp (nth_prime i) n).
+elim f.simplify.apply witness ? ? ((nth_prime i) \sup n).
reflexivity.
change with
-divides (nth_prime (S(max_p n1)+i))
-((exp (nth_prime i) n)*(defactorize_aux n1 (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*(exp (nth_prime i) n)).
+apply witness ? ? (n2* (nth_prime i) \sup n).
reflexivity.
qed.
theorem divides_exp_to_divides:
\forall p,n,m:nat. prime p \to
-divides p (exp n m) \to divides p n.
+p \divides n \sup m \to p \divides n.
intros 3.elim m.simplify in H1.
apply transitive_divides p (S O).assumption.
apply divides_SO_n.
-cut divides p n \lor divides p (exp n n1).
+cut p \divides n \lor p \divides n \sup n1.
elim Hcut.assumption.
apply H.assumption.assumption.
apply divides_times_to_divides.assumption.
theorem divides_exp_to_eq:
\forall p,q,m:nat. prime p \to prime q \to
-divides p (exp q m) \to p = q.
+p \divides q \sup m \to p = q.
intros.
simplify in H1.
elim H1.apply H4.
qed.
theorem not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
-i < j \to \not divides (nth_prime i) (defactorize_aux f j).
+i < j \to nth_prime i \ndivides defactorize_aux f j.
intro.elim f.
change with
-divides (nth_prime i) (exp (nth_prime j) (S n)) \to False.
+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.
apply not_le_Sn_n i.rewrite > Hcut in \vdash (? ? %).assumption.
apply injective_nth_prime ? ? H2.
change with
-divides (nth_prime i) ((exp (nth_prime j) 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 divides (nth_prime i) (exp (nth_prime j) n)
-\lor divides (nth_prime i) (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.
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.rewrite < plus_n_O.
+intro.
+apply not_divides_defactorize_aux f i (S i) ?.
+simplify.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.
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 divides (nth_prime (S(max_p n2)+i)) (defactorize_aux (nf_cons n1 n2) i).
-apply divides_max_p_defactorize.
-rewrite < H2.
-change with
-(divides (nth_prime (S(max_p n2)+i)) (exp (nth_prime i) (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 divides (nth_prime (S(max_p n1)+i)) (defactorize_aux (nf_cons n n1) i).
-apply divides_max_p_defactorize.
-rewrite > H2.
-change with
-(divides (nth_prime (S(max_p n1)+i)) (exp (nth_prime i) (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.
-(exp (nth_prime i) n)*(defactorize_aux n1 (S i)) =
-(exp (nth_prime i) n2)*(defactorize_aux n3 (S i)) \to
+((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).
intro.
elim n4. apply eq_f.
rewrite > plus_n_O.
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 ? ? ((exp (nth_prime i) 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 ? ? ((exp (nth_prime i) 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.
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.
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.
projects / helm.git / blobdiff