]
qed.
-theorem sigma_p_log_div: \forall n,b. S O < b \to
+theorem sigma_p_log_div1: \forall n,b. S O < b \to
sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
\le sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)*S(log b (S(S(S O)))))))/n!
).
]
qed.
-theorem sigma_p_log_div1: \forall n,b. S O < b \to
+theorem sigma_p_log_div2: \forall n,b. S O < b \to
sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
\le
(sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)))))*S(log b (S(S(S O))))/n!).
intros.
apply (trans_le ? (sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)*S(log b (S(S(S O)))))))/n!
)))
- [apply sigma_P_log_div.assumption
+ [apply sigma_p_log_div1.assumption
|apply le_times_to_le_div
[apply lt_O_fact
|rewrite > distributive_times_plus_sigma_p.
]
]
qed.
-
-(*
+
theorem sigma_p_log_div: \forall n,b. S O < b \to
sigma_p (S n) (\lambda p.(primeb p \land (leb (S n) (p*p)))) (\lambda p.(log b (n/p)))
-\le (sigma_p (S n) (\lambda i.leb (S n) (i*i)) (\lambda i.(prim i)*n!/i))*S(log b (S(S(S O)))/n!
-).
+\le (sigma_p n (\lambda i.leb (S n) (i*i)) (\lambda i.(prim i)*S(n!/i)))*S(log b (S(S(S O))))/n!
+.
intros.
apply (trans_le ? (sigma_p (S n) (\lambda p.primeb p \land (leb (S n) (p*p))) (\lambda p.(sigma_p n (\lambda i.leb p i) (\lambda i.S((n!/i)))))*S(log b (S(S(S O))))/n!))
- [apply sigma_p_log_div1
- |unfold prim.
-*)
+ [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.
(* theorem le_log_exp_Sn_log_exp_n: \forall n,m,a,p. O < m \to S O < p \to
divides n m \to
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