(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/primes".
-
include "nat/div_and_mod.ma".
include "nat/minimization.ma".
include "nat/sigma_and_pi.ma".
simplify.exact (not_le_Sn_n O).
qed.
+(*a variant of or_div_mod *)
+theorem or_div_mod1: \forall n,q. O < q \to
+(divides q (S n)) \land S n = (S (div n q)) * q \lor
+(\lnot (divides q (S n)) \land S n= (div n q) * q + S (n\mod q)).
+intros.elim (or_div_mod n q H);elim H1
+ [left.split
+ [apply (witness ? ? (S (n/q))).
+ rewrite > sym_times.assumption
+ |assumption
+ ]
+ |right.split
+ [intro.
+ apply (not_eq_O_S (n \mod q)).
+ (* come faccio a fare unfold nelleipotesi ? *)
+ cut ((S n) \mod q = O)
+ [rewrite < Hcut.
+ apply (div_mod_spec_to_eq2 (S n) q (div (S n) q) (mod (S n) q) (div n q) (S (mod n q)))
+ [apply div_mod_spec_div_mod.
+ assumption
+ |apply div_mod_spec_intro;assumption
+ ]
+ |apply divides_to_mod_O;assumption
+ ]
+ |assumption
+ ]
+ ]
+qed.
+
+theorem divides_to_div: \forall n,m.divides n m \to m/n*n = m.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+ [rewrite > plus_n_O.
+ rewrite < (divides_to_mod_O ? ? H H1).
+ apply sym_eq.
+ apply div_mod.
+ assumption
+ |elim H1.
+ generalize in match H2.
+ rewrite < H.
+ simplify.
+ intro.
+ rewrite > H3.
+ reflexivity
+ ]
+qed.
+
+theorem divides_div: \forall d,n. divides d n \to divides (n/d) n.
+intros.
+apply (witness ? ? d).
+apply sym_eq.
+apply divides_to_div.
+assumption.
+qed.
+
+theorem div_div: \forall n,d:nat. O < n \to divides d n \to
+n/(n/d) = d.
+intros.
+apply (inj_times_l1 (n/d))
+ [apply (lt_times_n_to_lt d)
+ [apply (divides_to_lt_O ? ? H H1).
+ |rewrite > divides_to_div;assumption
+ ]
+ |rewrite > divides_to_div
+ [rewrite > sym_times.
+ rewrite > divides_to_div
+ [reflexivity
+ |assumption
+ ]
+ |apply (witness ? ? d).
+ apply sym_eq.
+ apply divides_to_div.
+ assumption
+ ]
+ ]
+qed.
+
+theorem divides_to_eq_times_div_div_times: \forall a,b,c:nat.
+O \lt b \to c \divides b \to a * (b /c) = (a*b)/c.
+intros.
+elim H1.
+rewrite > H2.
+rewrite > (sym_times c n2).
+cut(O \lt c)
+[ rewrite > (lt_O_to_div_times n2 c)
+ [ rewrite < assoc_times.
+ rewrite > (lt_O_to_div_times (a *n2) c)
+ [ reflexivity
+ | assumption
+ ]
+ | assumption
+ ]
+| apply (divides_to_lt_O c b);
+ assumption.
+]
+qed.
+
+theorem eq_div_plus: \forall n,m,d. O < d \to
+divides d n \to divides d m \to
+(n + m ) / d = n/d + m/d.
+intros.
+elim H1.
+elim H2.
+rewrite > H3.rewrite > H4.
+rewrite < distr_times_plus.
+rewrite > sym_times.
+rewrite > sym_times in ⊢ (? ? ? (? (? % ?) ?)).
+rewrite > sym_times in ⊢ (? ? ? (? ? (? % ?))).
+rewrite > lt_O_to_div_times
+ [rewrite > lt_O_to_div_times
+ [rewrite > lt_O_to_div_times
+ [reflexivity
+ |assumption
+ ]
+ |assumption
+ ]
+ |assumption
+ ]
+qed.
+
(* boolean divides *)
definition divides_b : nat \to nat \to bool \def
\lambda n,m :nat. (eqb (m \mod n) O).
[apply (nat_case m)
[intro.apply divides_n_n
|simplify.intros.apply False_ind.
- apply not_eq_true_false.apply sym_eq.assumption
+ apply not_eq_true_false.apply sym_eq.
+ assumption
]
|intros.
apply divides_b_true_to_divides1
absurd (n \divides m).assumption.assumption.
qed.
+theorem divides_to_divides_b_true1 : \forall n,m:nat.
+O < m \to n \divides m \to divides_b n m = true.
+intro.
+elim (le_to_or_lt_eq O n (le_O_n n))
+ [apply divides_to_divides_b_true
+ [assumption|assumption]
+ |apply False_ind.
+ rewrite < H in H2.
+ elim H2.
+ simplify in H3.
+ apply (not_le_Sn_O O).
+ rewrite > H3 in H1.
+ assumption
+ ]
+qed.
+
theorem not_divides_to_divides_b_false: \forall n,m:nat. O < n \to
\lnot(n \divides m) \to (divides_b n m) = false.
intros.
reflexivity.
qed.
+theorem divides_b_div_true:
+\forall d,n. O < n \to
+ divides_b d n = true \to divides_b (n/d) n = true.
+intros.
+apply divides_to_divides_b_true1
+ [assumption
+ |apply divides_div.
+ apply divides_b_true_to_divides.
+ assumption
+ ]
+qed.
+
theorem divides_b_true_to_lt_O: \forall n,m. O < n \to divides_b m n = true \to O < m.
intros.
elim (le_to_or_lt_eq ? ? (le_O_n m))
intros.elim H.apply lt_to_le.assumption.
qed.
+theorem prime_to_lt_SO: \forall p. prime p \to S O < p.
+intros.elim H.
+assumption.
+qed.
+
(* smallest factor *)
definition smallest_factor : nat \to nat \def
\lambda n:nat.