set "baseuri" "cic:/matita/nat/factorization".
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 (n \mod (nth_prime p)) O)).
+theorem lt_SO_max_prime: \forall m. S O < m \to
+S O < max m (λi:nat.primeb i∧divides_b i m).
+intros.
+apply (lt_to_le_to_lt ? (smallest_factor m))
+ [apply lt_SO_smallest_factor.assumption
+ |apply f_m_to_le_max
+ [apply le_smallest_factor_n
+ |apply true_to_true_to_andb_true
+ [apply prime_to_primeb_true.
+ apply prime_smallest_factor_n.
+ assumption
+ |apply divides_to_divides_b_true
+ [apply lt_O_smallest_factor.apply lt_to_le.assumption
+ |apply divides_smallest_factor_n.
+ apply lt_to_le.assumption
+ ]
+ ]
+ ]
+ ]
+qed.
(* 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 (n \mod (nth_prime p)) O) n);
- cut (\exists i. nth_prime i = smallest_factor n);
+intros.
+apply divides_b_true_to_divides.
+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);
split;
| rewrite > H1;
apply le_smallest_factor_n; ]
| rewrite > H1;
- (*CSC: simplify here does something nasty! *)
change with (divides_b (smallest_factor n) n = true);
apply divides_to_divides_b_true;
[ apply (trans_lt ? (S O));
[ unfold lt; apply le_n;
| apply lt_SO_smallest_factor; assumption; ]
- | apply divides_smallest_factor_n;
+ | letin x \def le.autobatch new.
+ (*
+ apply divides_smallest_factor_n;
apply (trans_lt ? (S O));
[ unfold lt; apply le_n;
- | assumption; ] ] ]
- | apply prime_to_nth_prime;
+ | assumption; ] *) ] ]
+ | autobatch.
+ (*
+ apply prime_to_nth_prime;
apply prime_smallest_factor_n;
- assumption; ] ]
+ assumption; *) ]
qed.
theorem divides_to_max_prime_factor : \forall n,m. (S O) < n \to O < m \to n \divides m \to
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).
-apply divides_max_prime_factor_n.
-assumption.assumption.
-apply divides_b_true_to_divides.
-apply lt_O_nth_prime_n.
-apply divides_to_divides_b_true.
-apply lt_O_nth_prime_n.
-apply divides_max_prime_factor_n.
-assumption.
+autobatch.
+autobatch.
+(*
+ [ apply (transitive_divides ? n);
+ [ apply divides_max_prime_factor_n.
+ assumption.
+ | assumption.
+ ]
+ | apply divides_b_true_to_divides;
+ [ apply lt_O_nth_prime_n.
+ | apply divides_to_divides_b_true;
+ [ apply lt_O_nth_prime_n.
+ | apply divides_max_prime_factor_n.
+ assumption.
+ ]
+ ]
+ ]
+*)
+qed.
+
+theorem divides_to_max_prime_factor1 : \forall n,m. O < n \to O < m \to n \divides m \to
+max_prime_factor n \le max_prime_factor m.
+intros 3.
+elim (le_to_or_lt_eq ? ? H)
+ [apply divides_to_max_prime_factor
+ [assumption|assumption|assumption]
+ |rewrite < H1.
+ simplify.apply le_O_n.
+ ]
+qed.
+
+theorem max_prime_factor_to_not_p_ord_O : \forall n,p,r.
+ (S O) < n \to
+ p = max_prime_factor n \to
+ p_ord n (nth_prime p) \neq pair nat nat O r.
+intros.unfold Not.intro.
+apply (p_ord_O_to_not_divides ? ? ? ? H2)
+ [apply (trans_lt ? (S O))[apply lt_O_S|assumption]
+ |rewrite > H1.
+ apply divides_max_prime_factor_n.
+ assumption
+ ]
qed.
theorem p_ord_to_lt_max_prime_factor: \forall n,p,q,r. O < n \to
apply divides_max_prime_factor_n.
assumption.unfold Not.
intro.
-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 (p_ord_aux_to_not_mod_O n n ? q r).
-apply lt_SO_nth_prime_n.assumption.
-apply le_n.
-rewrite < H1.assumption.
+cut (r \mod (nth_prime (max_prime_factor n)) \neq O);
+ [unfold Not in Hcut1.autobatch new.
+ (*
+ apply Hcut1.apply divides_to_mod_O;
+ [ apply lt_O_nth_prime_n.
+ | assumption.
+ ]
+ *)
+ |letin z \def le.
+ cut(pair nat nat q r=p_ord_aux n n (nth_prime (max_prime_factor n)));
+ [2: rewrite < H1.assumption.|letin x \def le.autobatch width = 4 depth = 2]
+ (* CERCA COME MAI le_n non lo applica se lo trova come Const e non Rel *)
+ ].
+(*
+ 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 (le_to_or_lt_eq ? p H1).
qed.
+lemma lt_max_prime_factor_to_not_divides: \forall n,p:nat.
+O < n \to n=S O \lor max_prime_factor n < p \to
+(nth_prime p \ndivides n).
+intros.unfold Not.intro.
+elim H1
+ [rewrite > H3 in H2.
+ apply (le_to_not_lt (nth_prime p) (S O))
+ [apply divides_to_le[apply le_n|assumption]
+ |apply lt_SO_nth_prime_n
+ ]
+ |apply (not_le_Sn_n p).
+ change with (p < p).
+ apply (le_to_lt_to_lt ? ? ? ? H3).
+ unfold max_prime_factor.
+ apply f_m_to_le_max
+ [apply (trans_le ? (nth_prime p))
+ [apply lt_to_le.
+ apply lt_n_nth_prime_n
+ |apply divides_to_le;assumption
+ ]
+ |apply eq_to_eqb_true.
+ apply divides_to_mod_O
+ [apply lt_O_nth_prime_n|assumption]
+ ]
+ ]
+qed.
+
(* datatypes and functions *)
inductive nat_fact : Set \def
| 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.
+theorem lt_O_defactorize_aux:
+ \forall f:nat_fact.
+ \forall i:nat.
+ O < defactorize_aux f i.
+intro; elim f;
+[1,2:
+ simplify; unfold lt;
+ rewrite > times_n_SO;
+ apply le_times;
+ [ change with (O < nth_prime i);
+ apply lt_O_nth_prime_n;
+ |2,3:
+ 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.
left.split.assumption.reflexivity.
intro.right.rewrite > Hcut2.
simplify.unfold lt.apply le_S_S.apply le_O_n.
-cut (r \lt (S O) \or r=(S O)).
+cut (r < (S O) ∨ r=(S O)).
elim Hcut2.absurd (O=r).
apply le_n_O_to_eq.apply le_S_S_to_le.exact H5.
unfold Not.intro.
cut (O=n1).
apply (not_le_Sn_O O).
-rewrite > Hcut3 in \vdash (? ? %).
+rewrite > Hcut3 in ⊢ (? ? %).
assumption.rewrite > Hcut.
rewrite < H6.reflexivity.
assumption.
intro.
apply (nat_case n).reflexivity.
intro.apply (nat_case m).reflexivity.
-intro.(*CSC: simplify here does something really nasty *)
+intro.
change 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
cut ((S(S m1)) = (nth_prime p) \sup q *r).
cut (O<r).
rewrite > defactorize_aux_factorize_aux.
-(*CSC: simplify here does something really nasty *)
change with (r*(nth_prime p) \sup (S (pred q)) = (S(S m1))).
cut ((S (pred q)) = q).
rewrite > Hcut2.
unfold lt.apply le_S_S.apply le_S_S. apply le_O_n.
cut ((S(S m1)) = r).
rewrite > Hcut3 in \vdash (? (? ? %)).
-(*CSC: simplify here does something really nasty *)
change with (nth_prime p \divides r \to False).
intro.
apply (p_ord_aux_to_not_mod_O (S(S m1)) (S(S m1)) (nth_prime p) q r).
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 (p_ord_aux_to_exp (S(S m1))).
apply lt_O_nth_prime_n.
assumption.
reflexivity.
qed.
-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 divides_SO_n.
-cut (p \divides n \lor p \divides n \sup n1).
-elim Hcut.assumption.
-apply H.assumption.assumption.
-apply divides_times_to_divides.assumption.
-exact H2.
-qed.
-
-theorem divides_exp_to_eq:
-\forall p,q,m:nat. prime p \to prime q \to
-p \divides q \sup m \to p = q.
+lemma eq_p_max: \forall n,p,r:nat. O < n \to
+O < r \to
+r = (S O) \lor (max r (\lambda p:nat.eqb (r \mod (nth_prime p)) O)) < p \to
+p = max_prime_factor (r*(nth_prime p)\sup n).
intros.
-unfold prime in H1.
-elim H1.apply H4.
-apply (divides_exp_to_divides p q m).
-assumption.assumption.
-unfold prime in H.elim H.assumption.
+apply sym_eq.
+unfold max_prime_factor.
+apply max_spec_to_max.
+split
+ [split
+ [rewrite > times_n_SO in \vdash (? % ?).
+ rewrite > sym_times.
+ apply le_times
+ [assumption
+ |apply lt_to_le.
+ apply (lt_to_le_to_lt ? (nth_prime p))
+ [apply lt_n_nth_prime_n
+ |rewrite > exp_n_SO in \vdash (? % ?).
+ apply le_exp
+ [apply lt_O_nth_prime_n
+ |assumption
+ ]
+ ]
+ ]
+ |apply eq_to_eqb_true.
+ apply divides_to_mod_O
+ [apply lt_O_nth_prime_n
+ |apply (lt_O_n_elim ? H).
+ intro.
+ apply (witness ? ? (r*(nth_prime p \sup m))).
+ rewrite < assoc_times.
+ rewrite < sym_times in \vdash (? ? ? (? % ?)).
+ rewrite > exp_n_SO in \vdash (? ? ? (? (? ? %) ?)).
+ rewrite > assoc_times.
+ rewrite < exp_plus_times.
+ reflexivity
+ ]
+ ]
+ |intros.
+ apply not_eq_to_eqb_false.
+ unfold Not.intro.
+ lapply (mod_O_to_divides ? ? ? H5)
+ [apply lt_O_nth_prime_n
+ |cut (Not (divides (nth_prime i) ((nth_prime p)\sup n)))
+ [elim H2
+ [rewrite > H6 in Hletin.
+ simplify in Hletin.
+ rewrite < plus_n_O in Hletin.
+ apply Hcut.assumption
+ |elim (divides_times_to_divides ? ? ? ? Hletin)
+ [apply (lt_to_not_le ? ? H3).
+ apply lt_to_le.
+ apply (le_to_lt_to_lt ? ? ? ? H6).
+ apply f_m_to_le_max
+ [apply (trans_le ? (nth_prime i))
+ [apply lt_to_le.
+ apply lt_n_nth_prime_n
+ |apply divides_to_le[assumption|assumption]
+ ]
+ |apply eq_to_eqb_true.
+ apply divides_to_mod_O
+ [apply lt_O_nth_prime_n|assumption]
+ ]
+ |apply prime_nth_prime
+ |apply Hcut.assumption
+ ]
+ ]
+ |unfold Not.intro.
+ apply (lt_to_not_eq ? ? H3).
+ apply sym_eq.
+ elim (prime_nth_prime p).
+ apply injective_nth_prime.
+ apply H8
+ [apply (divides_exp_to_divides ? ? ? ? H6).
+ apply prime_nth_prime.
+ |apply lt_SO_nth_prime_n
+ ]
+ ]
+ ]
+ ]
qed.
theorem not_divides_defactorize_aux: \forall f:nat_fact. \forall i,j:nat.
qed.
theorem factorize_defactorize:
-\forall f,g: nat_fact_all. factorize (defactorize f) = f.
+\forall f: nat_fact_all. factorize (defactorize f) = f.
intros.
apply injective_defactorize.
apply defactorize_factorize.
qed.
-
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