+ [apply sigma_p_log_div2.assumption
+ |apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_l.
+ unfold prim.
+ cut
+ (sigma_p (S n) (λp:nat.primeb p∧leb (S n) (p*p))
+ (λp:nat.sigma_p n (λi:nat.leb p i) (λi:nat.S (n!/i)))
+ = sigma_p n (λi:nat.leb (S n) (i*i))
+ (λi:nat.sigma_p (S n) (\lambda p.primeb p \land leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i))))
+ [rewrite > Hcut.
+ apply le_sigma_p.intros.
+ rewrite < sym_times.
+ rewrite > distributive_times_plus_sigma_p.
+ rewrite < times_n_SO.
+ cut
+ (sigma_p (S n) (λp:nat.primeb p∧leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i))
+ = sigma_p (S i) (\lambda p.primeb p \land leb (S n) (p*p) \land leb p i) (λp:nat.S (n!/i)))
+ [rewrite > Hcut1.
+ apply le_sigma_p1.intros.
+ rewrite < andb_sym.
+ rewrite < andb_sym in ⊢ (? (? (? (? ? %)) ?) ?).
+ apply (bool_elim ? (leb i1 i));intros
+ [apply (bool_elim ? (leb (S n) (i1*i1)));intros
+ [apply le_n
+ |apply le_O_n
+ ]
+ |apply le_O_n
+ ]
+ |apply or_false_to_eq_sigma_p
+ [apply le_S.assumption
+ |intros.
+ left.rewrite > (lt_to_leb_false i1 i)
+ [rewrite > andb_sym.reflexivity
+ |assumption
+ ]
+ ]
+ ]
+ |apply sigma_p_sigma_p.intros.
+ apply (bool_elim ? (leb x y));intros
+ [apply (bool_elim ? (leb (S n) (x*x)));intros
+ [rewrite > le_to_leb_true in ⊢ (? ? ? (? % ?))
+ [reflexivity
+ |apply (trans_le ? (x*x))
+ [apply leb_true_to_le.assumption
+ |apply le_times;apply leb_true_to_le;assumption
+ ]
+ ]
+ |rewrite < andb_sym in ⊢ (? ? (? % ?) ?).
+ rewrite < andb_sym in ⊢ (? ? ? (? ? (? % ?))).
+ rewrite < andb_sym in ⊢ (? ? ? %).
+ reflexivity
+ ]
+ |rewrite < andb_sym.
+ rewrite > andb_assoc in ⊢ (? ? ? %).
+ rewrite < andb_sym in ⊢ (? ? ? (? % ?)).
+ reflexivity
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem le_sigma_p_div_log_div_pred_log : \forall n,b,m. S O < b \to b*b \leq n \to
+sigma_p (S n) (\lambda i.leb (S n) (i*i)) (\lambda i.m/(log b i))
+\leq ((S (S O)) * n * m)/(pred (log b n)).
+intros.
+apply (trans_le ? (sigma_p (S n)
+ (\lambda i.leb (S n) (i*i)) (\lambda i.(S (S O))*m/(pred (log b n)))))
+ [apply le_sigma_p;intros;apply le_times_to_le_div
+ [rewrite > minus_n_O in ⊢ (? ? (? %));rewrite < eq_minus_S_pred;
+ apply le_plus_to_minus_r;simplify;
+ rewrite < (eq_log_exp b ? H);
+ apply le_log;
+ [assumption
+ |simplify;rewrite < times_n_SO;assumption]
+ |apply (trans_le ? ((pred (log b n) * m)/log b i))
+ [apply le_times_div_div_times;apply lt_O_log
+ [elim (le_to_or_lt_eq ? ? (le_O_n i))
+ [assumption
+ |apply False_ind;apply not_eq_true_false;rewrite < H3;rewrite < H4;
+ reflexivity]
+ |apply (le_exp_to_le1 ? ? (S (S O)))
+ [apply lt_O_S;
+ |apply (trans_le ? (S n))
+ [apply le_S;simplify;rewrite < times_n_SO;assumption
+ |rewrite > exp_SSO;apply leb_true_to_le;assumption]]]
+ |apply le_times_to_le_div2
+ [apply lt_O_log
+ [elim (le_to_or_lt_eq ? ? (le_O_n i))
+ [assumption
+ |apply False_ind;apply not_eq_true_false;rewrite < H3;rewrite < H4;
+ reflexivity]
+ |apply (le_exp_to_le1 ? ? (S (S O)))
+ [apply lt_O_S;
+ |apply (trans_le ? (S n))
+ [apply le_S;simplify;rewrite < times_n_SO;assumption
+ |rewrite > exp_SSO;apply leb_true_to_le;assumption]]]
+ |rewrite > sym_times in \vdash (? ? %);rewrite < assoc_times;
+ apply le_times_l;rewrite > sym_times;
+ rewrite > minus_n_O in \vdash (? (? %) ?);
+ rewrite < eq_minus_S_pred;apply le_plus_to_minus;
+ simplify;rewrite < plus_n_O;apply (trans_le ? (log b (i*i)))
+ [apply le_log
+ [assumption
+ |apply lt_to_le;apply leb_true_to_le;assumption]
+ |rewrite > sym_plus;simplify;apply log_times;assumption]]]]
+ |rewrite > times_n_SO in \vdash (? (? ? ? (\lambda i:?.%)) ?);
+ rewrite < distributive_times_plus_sigma_p;
+ apply (trans_le ? ((((S (S O))*m)/(pred (log b n)))*n))
+ [apply le_times_r;apply (trans_le ? (sigma_p (S n) (\lambda i:nat.leb (S O) (i*i)) (\lambda Hbeta1:nat.S O)))
+ [apply le_sigma_p1;intros;do 2 rewrite < times_n_SO;
+ apply (bool_elim ? (leb (S n) (i*i)))
+ [intro;cut (leb (S O) (i*i) = true)
+ [rewrite > Hcut;apply le_n
+ |apply le_to_leb_true;apply (trans_le ? (S n))
+ [apply le_S_S;apply le_O_n
+ |apply leb_true_to_le;assumption]]
+ |intro;simplify in \vdash (? % ?);apply le_O_n]
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (leb (S O) ((S n1)*(S n1))));intro
+ [rewrite > true_to_sigma_p_Sn
+ [change in \vdash (? % ?) with (S (sigma_p (S n1) (\lambda i:nat.leb (S O) (i*i)) (\lambda Hbeta1:nat.S O)));
+ apply le_S_S;assumption
+ |assumption]
+ |rewrite > false_to_sigma_p_Sn
+ [apply le_S;assumption
+ |assumption]]]]
+ |rewrite > sym_times in \vdash (? % ?);
+ rewrite > sym_times in \vdash (? ? (? (? % ?) ?));
+ rewrite > assoc_times;
+ apply le_times_div_div_times;
+ rewrite > minus_n_O in ⊢ (? ? (? %));rewrite < eq_minus_S_pred;
+ apply le_plus_to_minus_r;simplify;
+ rewrite < (eq_log_exp b ? H);
+ apply le_log;
+ [assumption
+ |simplify;rewrite < times_n_SO;assumption]]]
+qed.
+
+lemma neper_sigma_p1 : \forall n,a.n \divides a \to
+exp (a * S n) n =
+sigma_p (S n) (\lambda x.true) (\lambda k.(bc n k)*(exp n (n-k))*(exp a n)).
+intros;rewrite < times_exp;rewrite > exp_S_sigma_p;
+rewrite > distributive_times_plus_sigma_p;
+apply eq_sigma_p;intros;
+ [reflexivity
+ |rewrite > sym_times;reflexivity;]
+qed.
+
+lemma eq_exp_pi_p : \forall a,n.(exp a n) = pi_p n (\lambda x.true) (\lambda x.a).
+intros;elim n
+ [simplify;reflexivity
+ |change in \vdash (? ? % ?) with (a*exp a n1);rewrite > true_to_pi_p_Sn
+ [apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]
+qed.
+
+lemma eq_fact_pi_p : \forall n.n! = pi_p n (\lambda x.true) (\lambda x.S x).
+intros;elim n
+ [simplify;reflexivity
+ |rewrite > true_to_pi_p_Sn
+ [change in \vdash (? ? % ?) with (S n1*n1!);apply eq_f2
+ [reflexivity
+ |assumption]
+ |reflexivity]]
+qed.
+
+lemma divides_pi_p : \forall m,n,p,f.m \leq n \to pi_p m p f \divides pi_p n p f.
+intros;elim H
+ [apply divides_n_n
+ |apply (bool_elim ? (p n1));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > sym_times;rewrite > times_n_SO;apply divides_times
+ [assumption
+ |apply divides_SO_n]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]
+qed.
+
+lemma divides_fact_fact : \forall m,n.m \leq n \to m! \divides n!.
+intros;do 2 rewrite > eq_fact_pi_p;apply divides_pi_p;assumption.
+qed.
+
+lemma divides_times_to_eq : \forall a,b,c.O < c \to c \divides a \to a*b/c = a/c*b.
+intros;elim H1;rewrite > H2;cases H;rewrite > assoc_times;do 2 rewrite > div_times;
+reflexivity;
+qed.
+
+lemma divides_pi_p_to_eq : \forall k,p,f,g.(\forall x.p x = true \to O < g x \land (g x \divides f x)) \to
+pi_p k p f/pi_p k p g = pi_p k p (\lambda x.(f x)/(g x)).
+intros;
+cut (\forall k1.(pi_p k1 p g \divides pi_p k1 p f))
+ [|intro;elim k1
+ [simplify;apply divides_n_n
+ |apply (bool_elim ? (p n));intro;
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > true_to_pi_p_Sn
+ [elim (H n)
+ [elim H4;elim H1;rewrite > H5;rewrite > H6;
+ rewrite < assoc_times;rewrite > assoc_times in ⊢ (? ? (? % ?));
+ rewrite > sym_times in ⊢ (? ? (? (? ? %) ?));
+ rewrite > assoc_times;rewrite > assoc_times;
+ apply divides_times
+ [apply divides_n_n
+ |rewrite > times_n_SO in \vdash (? % ?);apply divides_times
+ [apply divides_n_n
+ |apply divides_SO_n]]
+ |assumption]
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn
+ [rewrite > false_to_pi_p_Sn
+ [assumption
+ |assumption]
+ |assumption]]]]
+elim k
+ [simplify;reflexivity
+ |apply (bool_elim ? (p n))
+ [intro;rewrite > true_to_pi_p_Sn;
+ [rewrite > true_to_pi_p_Sn;
+ [rewrite > true_to_pi_p_Sn;
+ [elim (H n);
+ [elim H4;rewrite > H5;rewrite < eq_div_div_div_times;
+ [cases H3
+ [rewrite > assoc_times;do 2 rewrite > div_times;
+ elim (Hcut n);rewrite > H6;rewrite < assoc_times;
+ rewrite < sym_times in \vdash (? ? (? (? % ?) ?) ?);
+ cut (O < pi_p n p g)
+ [rewrite < H1;rewrite > H6;cases Hcut1;
+ rewrite > assoc_times;do 2 rewrite > div_times;reflexivity
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n3));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n3);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]
+ |rewrite > assoc_times;do 2 rewrite > div_times;
+ elim (Hcut n);rewrite > H7;rewrite < assoc_times;
+ rewrite < sym_times in \vdash (? ? (? (? % ?) ?) ?);
+ cut (O < pi_p n p g)
+ [rewrite < H1;rewrite > H7;cases Hcut1;
+ rewrite > assoc_times;do 2 rewrite > div_times;reflexivity
+ |elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n3));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n3);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]]
+ |assumption
+ |(*già usata 2 volte: fattorizzare*)
+ elim n
+ [simplify;apply le_n
+ |apply (bool_elim ? (p n1));intro
+ [rewrite > true_to_pi_p_Sn
+ [rewrite > (times_n_O O);apply lt_times
+ [elim (H n1);assumption
+ |assumption]
+ |assumption]
+ |rewrite > false_to_pi_p_Sn;assumption]]]
+ |assumption]
+ |assumption]
+ |assumption]
+ |assumption]
+ |intro;rewrite > (false_to_pi_p_Sn ? ? ? H2);
+ rewrite > (false_to_pi_p_Sn ? ? ? H2);rewrite > (false_to_pi_p_Sn ? ? ? H2);
+ assumption]]
+qed.
+
+lemma divides_times_to_divides_div : \forall a,b,c.O < b \to
+ a*b \divides c \to a \divides c/b.
+intros;elim H1;rewrite > H2;rewrite > sym_times in \vdash (? ? (? (? % ?) ?));
+rewrite > assoc_times;cases H;rewrite > div_times;rewrite > times_n_SO in \vdash (? % ?);
+apply divides_times
+ [1,3:apply divides_n_n
+ |*:apply divides_SO_n]
+qed.