|apply le_S_S;assumption]]]]]]]]
qed.
-(*
-
-theorem le_log_C2_stima : \forall n,b. S O < b \to
+theorem le_log_C2_stima : \forall n,b. S O < b \to b*b < n \to
log b (C2 n) \leq
-(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +
+(*(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +*)
+((S (((S (S (S (S O))))*(S (log b (pred n)))) +
+ ((S (S (S (S O))))*n)))/(log b n)) +
(((S (((S (S (S (S O))))*(S (log b (pred n)))) +
((S (S (S (S O))))*n)))/(log b n)) +
(((sigma_p n (\lambda x.leb (S n) (x*x))
- (\lambda i.((S (((S (S (S (S O))))*(S (log b (pred n)))) +
- ((S (S (S (S O))))*n)))/(log b n))* S (n!/i)))
- *(S (log b 3)))/n!)).
-elim daemon.
-
-theorem le_log_C2_sigma_p : \forall n,b. S O < b \to
-log b (C2 n) \leq
-(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +
-(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x)))
- (\lambda p.(sigma_p (n+p) (\lambda i.leb p i)
- (\lambda i.S ((n+p)!/i*(S (log b 3)))))/(n+p)!)).
-intros;unfold C2;
-apply (trans_le ? (sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x)) (λx:nat.1)
-+sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x))
- (λi.log b (S (n/i)))))
- [apply log_pi_p;assumption
+ (\lambda i.((S (((S (S (S (S O))))*(S (log b (pred i)))) +
+ ((S (S (S (S O))))*i)))/(log b i))* S (n!/i)))
+ *(S (log b (S (S (S O))))))/n!)).intros.
+apply (trans_le ? ? ? (le_log_C2_sigma_p ? ? ?))
+ [assumption
|apply le_plus
- [apply le_n
- |apply le_sigma_p;intros;cut (S (n/i) = (n+i)/i)
- [rewrite > Hcut;apply le_log_div_sigma_p
+ [apply (trans_le ? ? ? ? (le_prim_n_stima ? ? ? ?));
+ [unfold prim;apply le_sigma_p1;intros;
+ do 2 rewrite < times_n_SO;elim (primeb i)
+ [elim (leb (S n) (i*i));simplify [apply le_n|apply le_O_n]
+ |simplify;apply le_n]
+ |assumption
+ |apply (trans_le ? ? ? ? H1);apply le_S;apply le_times_n;
+ apply lt_to_le;assumption]
+ |apply le_plus
+ [apply le_prim_n_stima;
+ [assumption
+ |apply (trans_le ? (b*b))
+ [apply le_times_n;apply lt_to_le;assumption
+ |apply lt_to_le;assumption]]
+ |apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_l;apply le_sigma_p;intros;apply le_times_l;
+ apply le_prim_n_stima
[assumption
- |apply prime_to_lt_O;apply primeb_true_to_prime;
- elim (and_true ? ? H2);assumption
- |apply le_plus_n
- |apply le_n]
- |lapply (prime_to_lt_O i (primeb_true_to_prime ? (andb_true_true ? ? H2))) as H3;
- apply (div_mod_spec_to_eq (n+i) i (S (n/i)) (n \mod i) ? ((n+i) \mod i))
- [constructor 1
- [apply lt_mod_m_m;assumption
- |simplify;rewrite > assoc_plus;rewrite < div_mod;
- [apply sym_plus
+ |apply (le_exp_to_le1 ? ? (S (S O)));
+ [apply le_S;apply le_n
+ |do 2 rewrite > exp_SSO;apply (trans_le ? n)
+ [apply lt_to_le;assumption
+ |apply lt_to_le;apply leb_true_to_le;assumption]]]]]]]
+qed.
+
+lemma log_interval : \forall b,k,n. S O < b \to exp b k \leq n \to n < exp b (S k) \to
+ log b n = k.
+intros 2.elim k
+ [simplify in H2;rewrite < times_n_SO in H2;apply lt_to_log_O;assumption
+ |cut (log b n1 < S (S n))
+ [cut (n < log b n1)
+ [apply antisymmetric_le
+ [apply le_S_S_to_le;assumption
+ |assumption]
+ |apply (trans_le ? (log b (exp b (S n))))
+ [rewrite > eq_log_exp
+ [apply le_n
+ |assumption]
+ |apply le_log;assumption]]
+ |apply le_S_S;apply (trans_le ? (log b (pred (exp b (S (S n))))))
+ [apply le_log
+ [assumption
+ |apply le_S_S_to_le;apply (trans_le ? ? ? H3);
+ rewrite > minus_n_O in \vdash (? ? (? (? %)));
+ rewrite < (eq_minus_S_pred (exp b (S (S n))) O);
+ rewrite > minus_n_O in \vdash (? % ?);
+ apply minus_le_S_minus_S]
+ |unfold log;apply f_false_to_le_max;
+ [apply (ex_intro ? ? (S n));split
+ [apply (trans_le ? (exp b (S n)));
+ [apply lt_to_le;apply lt_m_exp_nm;assumption
+ |rewrite > minus_n_O in ⊢ (? ? (? %));
+ rewrite < eq_minus_S_pred;apply le_plus_to_minus_r;
+ rewrite > sym_plus;
+ change in \vdash (? % ?) with (S (O + exp b (S n)));
+ apply lt_minus_to_plus;
+ change in ⊢ (? ? (? % ?)) with (b * (exp b (S n)));
+ rewrite > times_n_SO in \vdash (? ? (? ? %));
+ rewrite > sym_times in \vdash (? ? (? % ?));
+ rewrite < distributive_times_minus;unfold lt;
+ rewrite > times_n_SO in \vdash (? % ?);apply le_times
+ [apply lt_O_exp;apply (trans_le ? ? ? ? H1);
+ apply le_S;apply le_n
+ |apply le_plus_to_minus_r;simplify;assumption]]
+ |apply le_to_leb_true;
+ rewrite > minus_n_O in \vdash (? ? (? %));
+ rewrite < eq_minus_S_pred;apply le_plus_to_minus_r;
+ rewrite > sym_plus;change in \vdash (? % ?) with (S (exp b (S n)));
+ apply lt_exp;
+ [assumption
+ |apply le_n]]
+ |intros;apply lt_to_leb_false;unfold lt;
+ rewrite > minus_n_O in \vdash (? (? (? %)) ?);
+ rewrite < eq_minus_S_pred;rewrite < minus_Sn_m
+ [rewrite > minus_S_S;rewrite < minus_n_O;apply le_exp;
+ [apply (trans_le ? ? ? ? H1);apply le_S;apply le_n
+ |assumption]
+ |apply lt_O_exp;apply (trans_le ? ? ? ? H1);apply le_S;apply le_n]]]]]
+qed.
+
+lemma log_strano : \forall b,i.S O < b \to S O < i \to
+ ((S (S (S (S O)))) * log b (pred i)) + (S (S (S (S (S O))))) \leq
+ (S (S (S O)))*i.
+alias num (instance 0) = "natural number".
+cut (\forall b,i,k.S O < b \to S O < i \to
+ (exp b k) \leq i-1 \to i-1 < (exp b (S k)) \to
+ ((S (S (S (S O)))) * log b (pred i)) + (S (S (S (S (S O))))) \leq
+ (S (S (S O)))*i)
+ [intros;apply (Hcut ? ? (log b (i-1)) H H1);
+ [apply le_exp_log;rewrite > (minus_n_n 1) in \vdash (? % ?);
+ apply lt_plus_to_lt_minus;
+ [apply le_n
+ |rewrite < eq_minus_plus_plus_minus
+ [rewrite > sym_plus;rewrite > eq_minus_plus_plus_minus;
+ [rewrite < minus_n_n;rewrite < plus_n_O;assumption
+ |apply le_n]
+ |apply lt_to_le;assumption]]
+ |apply lt_exp_log;assumption]
+ |intros;rewrite > minus_n_O in ⊢ (? (? (? ? (? ? (? %))) ?) ?);
+ rewrite < eq_minus_S_pred;rewrite > (log_interval ? k)
+ [apply (trans_le ? (3*(exp b k) + 3))
+ [change in \vdash (? (? ? %) ?) with (2+3);
+ rewrite < assoc_plus;apply le_plus_l;
+ elim k
+ [simplify;apply le_S;apply le_n
+ |elim (decidable_eq_nat O n)
+ [rewrite < H5;apply (trans_le ? (3*(exp 2 1)));
+ [simplify;apply le_n
+ |apply le_times_r;apply monotonic_exp1;assumption]
+ |rewrite < times_n_Sm;apply (trans_le ? (3*(exp b n) + 4))
+ [rewrite > assoc_plus;rewrite > sym_plus;apply le_plus_l;
+ assumption
+ |rewrite < sym_plus;change in \vdash (? % ?) with (S (3 + (3*(exp b n))));
+ apply lt_minus_to_plus;
+ change in ⊢ (? ? (? (? ? %) ?)) with (b*(exp b n));
+ rewrite > sym_times in \vdash (? ? (? (? ? %) ?));
+ rewrite < assoc_times;
+ rewrite > times_n_SO in ⊢ (? ? (? ? (? ? %)));
+ rewrite < assoc_times;rewrite < distr_times_minus;
+ apply (trans_le ? (3*2*1))
+ [simplify;apply le_S;apply le_S;apply le_n
+ |apply le_times
+ [apply le_times_r;apply (trans_le ? (exp 2 n))
+ [rewrite > exp_n_SO in \vdash (? % ?);apply le_exp
+ [apply le_S;apply le_n
+ |generalize in match H5;cases n
+ [intro;elim H6;reflexivity
+ |intro;apply le_S_S;apply le_O_n]]
+ |apply monotonic_exp1;assumption]
+ |apply le_S_S_to_le;rewrite < minus_Sn_m;
+ [simplify;rewrite < minus_n_O;assumption
+ |apply lt_to_le;assumption]]]]]]
+ |rewrite > times_n_SO in \vdash (? (? ? %) ?);
+ rewrite < distr_times_plus;apply le_times_r;
+ rewrite < plus_n_SO;apply (trans_le ? (S (i-1)))
+ [apply le_S_S;assumption
+ |rewrite < minus_Sn_m
+ [simplify;rewrite < minus_n_O;apply le_n
+ |apply lt_to_le;assumption]]]
+ |assumption
+ |assumption
+ |assumption]]
+qed.
+
+lemma eq_div_div_times : \forall x,y,z.O < z \to O < y \to x/y = (z*x)/(z*y).
+intros.rewrite > (div_mod x y) in \vdash (? ? ? %);
+ [rewrite > distr_times_plus;rewrite > sym_times;rewrite > assoc_times;
+ rewrite > sym_times in ⊢ (? ? ? (? (? (? ? %) ?) ?));
+ rewrite > div_plus_times
+ [reflexivity
+ |generalize in match H;cases z;intros
+ [elim (not_le_Sn_O ? H2)
+ |apply lt_times_r;apply lt_mod_m_m;assumption]]
+ |assumption]
+qed.
+
+alias num (instance 0) = "natural number".
+lemma le_sigma_p_lemma1 : \forall n,b. S O < b \to b*b < n \to
+ (sigma_p n (\lambda x.leb (S n) (x*x))
+ (\lambda i.((S (((S (S (S (S O))))*(S (log b (pred i)))) +
+ ((S (S (S (S O))))*i)))/(log b i))* S (n!/i)))
+ \leq ((28 * n * n!)/(pred (log b n))).
+intros.apply (trans_le ? (sigma_p n (\lambda x.leb (S n) (x*x)) (\lambda i. ((7*i)/(log b i))*(S (n!/i)))))
+ [apply le_sigma_p;intros;cut (b \leq i)
+ [cut (1 < i) [|apply (trans_le ? ? ? H Hcut)]
+ apply le_times_l;apply monotonic_div
+ [apply lt_O_log
+ [generalize in match H3;cases i
+ [simplify;intros;destruct H4
+ |intro;apply le_S_S;apply le_O_n]
+ |assumption]
+ |rewrite > sym_times;simplify in ⊢ (? (? (? % ?)) ?);
+ change in ⊢ (? % ?) with (1 + ((4 + (log b (pred i) * 4)) + 4*i));
+ rewrite > assoc_plus;rewrite < assoc_plus;
+ simplify in \vdash (? (? % ?) ?);rewrite < assoc_plus;
+ apply (trans_le ? (3*i + 4*i))
+ [apply le_minus_to_plus;rewrite > eq_minus_plus_plus_minus
+ [rewrite < minus_n_n;rewrite < plus_n_O;
+ rewrite > sym_plus;rewrite > sym_times;apply log_strano
+ [assumption
+ |lapply (leb_true_to_le ? ? H3);apply (trans_le ? ? ? H);
+ apply (le_exp_to_le1 ? ? 2);
+ [apply le_S_S;apply le_O_n
+ |apply lt_to_le;do 2 rewrite > exp_SSO;apply (trans_le ? ? ? H1);
+ apply lt_to_le;assumption]]
+ |apply le_n]
+ |rewrite > sym_times in \vdash (? (? % ?) ?);
+ rewrite > sym_times in \vdash (? (? ? %) ?);
+ rewrite < distr_times_plus;rewrite > sym_times;apply le_n]]
+ |apply (le_exp_to_le1 ? ? 2);
+ [apply le_S_S;apply le_O_n
+ |apply lt_to_le;do 2 rewrite > exp_SSO;apply (trans_le ? ? ? H1);
+ apply (trans_le ? ? ? ? (leb_true_to_le ? ? H3));apply le_S;
+ apply le_n]]
+ |apply (trans_le ? (sigma_p n (λx:nat.leb (S n) (x*x)) (λi:nat.7*i/log b i*((2*(n!))/i))))
+ [apply le_sigma_p;intros;apply le_times_r;apply (trans_le ? (n!/i + n!/i))
+ [change in \vdash (? % ?) with (1 + n!/i);apply le_plus_l;
+ apply le_times_to_le_div
+ [generalize in match H3;cases i;simplify;intro
+ [destruct H4
+ |apply le_S_S;apply le_O_n]
+ |generalize in match H2;cases n;intro
+ [elim (not_le_Sn_O ? H4)
+ |change in \vdash (? ? %) with ((S n1)*(n1!));apply le_times
+ [apply lt_to_le;assumption
+ |apply lt_O_fact]]]
+ |rewrite > plus_n_O in \vdash (? (? ? %) ?);
+ change in \vdash (? % ?) with (2 * (n!/i));
+ apply le_times_div_div_times;
+ generalize in match H3;cases i;simplify;intro
+ [destruct H4
+ |apply le_S_S;apply le_O_n]]
+ |apply (trans_le ? (sigma_p n (\lambda x:nat.leb (S n) (x*x)) (\lambda i.((14*(n!))/log b i))))
+ [apply le_sigma_p;intros;
+ cut (b \leq i)
+ [|apply (le_exp_to_le1 ? ? 2);
+ [apply le_S_S;apply le_O_n
+ |apply lt_to_le;do 2 rewrite > exp_SSO;apply (trans_le ? ? ? H1);
+ apply (trans_le ? ? ? ? (leb_true_to_le ? ? H3));apply le_S;
+ apply le_n]]
+ cut (1 < i)
+ [|apply (trans_le ? ? ? H Hcut)]
+ change in ⊢ (? ? (? % ?)) with ((7*2)*(n!));
+ rewrite > assoc_times in \vdash (? ? (? % ?));
+ apply (trans_le ? ? ? (le_times_div_div_times ? ? ? ?));
+ [apply lt_to_le;assumption
+ |rewrite > (eq_div_div_times ? ? 7)
+ [rewrite > sym_times in ⊢ (? (? (? ? %) ?) ?);
+ rewrite < assoc_times in \vdash (? (? % ?) ?);
+ apply (trans_le ? ? ? (le_div_times_m ? ? ? ? ?))
+ [apply lt_O_log
+ [apply lt_to_le;assumption
+ |assumption]
+ |unfold lt;rewrite > times_n_SO in \vdash (? % ?);
+ apply le_times;
+ [apply le_S_S;apply le_O_n
+ |apply lt_to_le;assumption]
+ |apply le_n]
+ |apply le_S_S;apply le_O_n
+ |apply lt_to_le;assumption]]
+ |apply (trans_le ? (sigma_p (S n) (\lambda x.leb (S n) (x*x))
+ (\lambda i.14*n!/log b i)))
+ [apply (bool_elim ? (leb (S n) (n*n)));intro
+ [rewrite > true_to_sigma_p_Sn
+ [rewrite > sym_plus;rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n
+ |assumption]
+ |rewrite > false_to_sigma_p_Sn
+ [apply le_n
|assumption]]
- |apply div_mod_spec_div_mod;assumption]]]]
+ |apply (trans_le ? ? ? (le_sigma_p_div_log_div_pred_log ? ? ? ? ?));
+ [assumption
+ |apply lt_to_le;assumption
+ |rewrite < assoc_times;
+ rewrite > sym_times in ⊢ (? (? (? % ?) ?) ?);
+ rewrite < assoc_times;apply le_n]]]]]
+qed.
+
+theorem le_log_C2_stima2 : \forall n,b. S O < b \to b*b < n \to
+log b (C2 n) \leq
+(14*n/(log b n)) +
+ ((((28*n*n!)/pred (log b n))
+ *(S (log b (S (S (S O))))))/n!).intros.
+apply (trans_le ? ? ? (le_log_C2_stima ? ? H H1));
+rewrite < assoc_plus in \vdash (? % ?);apply le_plus
+ [rewrite > times_SSO_n in \vdash (? % ?);
+ apply (trans_le ? ? ? (le_times_div_div_times ? ? ? ?))
+ [apply lt_O_log
+ [apply (trans_le ? (b*b))
+ [rewrite > times_n_SO;apply le_times;apply lt_to_le;assumption
+ |apply lt_to_le;assumption]
+ |apply (trans_le ? (b*b))
+ [rewrite > times_n_SO in \vdash (? % ?);apply le_times
+ [apply le_n|apply lt_to_le;assumption]
+ |apply lt_to_le;assumption]]
+ |change in \vdash (? ? (? (? % ?) ?)) with (2*7);
+ apply monotonic_div;
+ [apply lt_O_log
+ [apply (trans_le ? (b*b))
+ [rewrite > times_n_SO;apply le_times;apply lt_to_le;assumption
+ |apply lt_to_le;assumption]
+ |apply (trans_le ? (b*b))
+ [rewrite > times_n_SO in \vdash (? % ?);apply le_times
+ [apply le_n|apply lt_to_le;assumption]
+ |apply lt_to_le;assumption]]
+ |rewrite > assoc_times;apply le_times_r;
+ change in \vdash (? (? (? (? ? %) ?)) ?) with (1 + (log b (pred n)));
+ rewrite > distr_times_plus;
+ change in \vdash (? % ?) with (1 + (4*1+4*log b (pred n)+4*n));
+ do 2 rewrite < assoc_plus;simplify in ⊢ (? (? (? % ?) ?) ?);
+ change in ⊢ (? ? %) with ((3+4)*n);rewrite > sym_times in \vdash (? ? %);
+ rewrite > distr_times_plus;
+ rewrite > sym_times in \vdash (? ? (? % ?));
+ rewrite > sym_times in \vdash (? ? (? ? %));
+ apply le_plus_l;rewrite > sym_plus;apply log_strano
+ [assumption
+ |apply (trans_le ? ? ? H);apply (trans_le ? (b*b))
+ [rewrite > times_n_SO in \vdash (? % ?);
+ apply le_times_r;apply lt_to_le;assumption
+ |apply lt_to_le;assumption]]]]
+ |apply monotonic_div
+ [apply lt_O_fact
+ |apply le_times_l;apply (le_sigma_p_lemma1 ? ? H H1)]]
+qed.
+
+theorem le_log_C2_stima3 : \forall n,b. S O < b \to b*b < n \to
+log b (C2 n) \leq
+(14*n/(log b n)) +
+ ((28*n)*(S (log b (S (S (S O)))))/pred (log b n)).intros.
+apply (trans_le ? ? ? (le_log_C2_stima2 ? ? H H1));apply le_plus_r;
+(*apply (trans_le ? ((28*n)*(28*n*n!/pred (log b n)*S (log b 3)/(28*n*n!))))
+ [*)
+rewrite > (eq_div_div_times ? ? (28*n)) in \vdash (? % ?)
+ [rewrite > sym_times in ⊢ (? (? (? ? %) ?) ?);
+ rewrite < assoc_times;
+ apply le_div_times_m;
+ [apply (trans_le ? (pred (log b (b*b))))
+ [rewrite < exp_SSO;rewrite > eq_log_exp;
+ [apply le_n
+ |assumption]
+ |rewrite < exp_SSO;rewrite > eq_log_exp;
+ [simplify;rewrite > minus_n_O in \vdash (? ? (? %));
+ rewrite < eq_minus_S_pred;
+ apply le_plus_to_minus_r;simplify;
+ rewrite < (eq_log_exp b 2);
+ [apply le_log
+ [assumption
+ |rewrite > exp_SSO;apply lt_to_le;assumption]
+ |assumption]
+ |assumption]]
+ |unfold lt;rewrite > times_n_SO in \vdash (? % ?);apply le_times
+ [rewrite > times_n_SO in \vdash (? % ?);apply le_times
+ [apply le_S_S;apply le_O_n
+ |apply (trans_le ? ? ? ? H1);apply le_S_S;
+ rewrite > times_n_SO;apply le_times
+ [apply le_O_n
+ |apply lt_to_le;assumption]]
+ |apply lt_O_fact]]
+ |unfold lt;rewrite > times_n_SO in \vdash (? % ?);apply le_times
+ [apply le_S_S;apply le_O_n
+ |apply (trans_le ? ? ? ? H1);apply le_S_S;
+ rewrite > times_n_SO;apply le_times
+ [apply le_O_n
+ |apply lt_to_le;assumption]]
+ |apply lt_O_fact]
+qed.
+
+lemma le_prim_log1: \forall n,b. S O < b \to O < n \to
+ (prim n)*(log b n) \leq
+ log b (A n) + log b (C1 n) + log b (C2 n) + 2.
+intros.change in \vdash (? ? (? ? %)) with (1+1).
+rewrite < assoc_plus;rewrite > assoc_plus in ⊢ (? ? (? (? % ?) ?)).
+rewrite > assoc_plus in ⊢ (? ? (? % ?));
+apply (trans_le ? (log b (A n) + (log b (C1 n * C2 n)) + 1));
+ [apply (trans_le ? (log b (A n * (C1 n * C2 n))))
+ [apply (trans_le ? (log b (exp n (prim n))))
+ [apply log_exp2;assumption
+ |apply le_log
+ [assumption
+ |rewrite < jj;apply asdasd]]
+ |rewrite > sym_plus;simplify;apply log_times;assumption]
+ |apply le_plus_l;apply le_plus_r;rewrite > sym_plus;simplify;apply log_times;
+ assumption]
+qed.
+
+lemma le_log_A1 : \forall n,b. S O < b \to S O < n \to
+ log b (A n) \leq 2*(S (log b (pred n))) + (2*n)*(S (log b 2)) + 1.
+intros.apply (trans_le ? (log b ((exp (pred n) 2)*(exp 2 (2*n)))))
+ [apply le_log
+ [assumption
+ |simplify in ⊢ (? ? (? (? % ?) ?));apply le_A_exp3;assumption]
+ |rewrite < sym_plus;apply (trans_le ? (1 + ((log b (exp (pred n) 2)) + (log b (exp 2 (2*n))))));
+ [change in \vdash (? ? %) with (S (log b ((pred n)\sup(2))+log b ((2)\sup(2*n))));
+ apply log_times;assumption
+ |apply le_plus_r;apply le_plus;apply log_exp1;assumption]]
+qed.
+
+lemma le_theta_pi_A : \forall n.theta_pi n \leq A n.
+intro.unfold theta_pi.unfold A.apply le_pi_p.intros.
+rewrite > exp_n_SO in \vdash (? % ?).
+cut (O < i)
+ [apply le_exp
+ [assumption
+ |apply lt_O_log
+ [apply (trans_le ? ? ? Hcut);apply le_S_S_to_le;assumption
+ |apply le_S_S_to_le;assumption]]
+ |apply prime_to_lt_O;apply primeb_true_to_prime;assumption]
qed.
-*)
definition sqrt \def
\lambda n.max n (\lambda x.leb (x*x) n).
[apply le_O_n
|simplify;reflexivity]]
qed.
+
+theorem lt_sqrt_to_le_times_l : \forall m,n.n < sqrt m \to n*n < m.
+intros;apply (trans_le ? (sqrt m * sqrt m))
+ [apply (trans_le ? (S n * S n))
+ [simplify in \vdash (? ? %);apply le_S_S;apply (trans_le ? (n * S n))
+ [apply le_times_r;apply le_S;apply le_n
+ |rewrite > sym_plus;rewrite > plus_n_O in \vdash (? % ?);
+ apply le_plus_r;apply le_O_n]
+ |apply le_times;assumption]
+ |apply le_sqrt_to_le_times_l;apply le_n]
+qed.
theorem le_sqrt_to_le_times_r : \forall m,n.sqrt m < n \to m < n*n.
intros;apply not_le_to_lt;intro;
|apply le_sqrt_to_le_times_r;assumption]
|reflexivity]]
qed.
+
+lemma le_sqrt_n_n : \forall n.sqrt n \leq n.
+intro.unfold sqrt.apply le_max_n.
+qed.
+
+lemma le_prim_log_stima: \forall n,b. S O < b \to b < sqrt n \to
+ (prim n)*(log b n) \leq
+ 2*S (log b (pred n))+2*n*S (log b 2)
+ +(2*S (log b (pred (sqrt n)))+2*sqrt n*S (log b 2))
+ +(14*n/log b n+28*n*S (log b 3)/pred (log b n))
+ +4.
+intros.cut (1 < n)
+ [apply (trans_le ? ((2*(S (log b (pred n))) + (2*n)*(S (log b 2)) + 1) +
+ (2*(S (log b (pred (sqrt n)))) + (2*(sqrt n))*(S (log b 2)) + 1) +
+ ((14*n/(log b n)) + ((28*n)*(S (log b (S (S (S O)))))/pred (log b n))) + 2))
+ [apply (trans_le ? ? ? (le_prim_log1 ? ? H ?))
+ [apply lt_to_le;assumption
+ |apply le_plus_l;apply le_plus
+ [apply le_plus
+ [apply le_log_A1;assumption
+ |rewrite < eq_theta_pi_sqrt_C1;apply (trans_le ? (log b (A (sqrt n))))
+ [apply le_log
+ [assumption
+ |apply le_theta_pi_A]
+ |apply le_log_A1
+ [assumption
+ |apply (trans_le ? ? ? H);apply lt_to_le;assumption]]]
+ |apply le_log_C2_stima3;
+ [assumption
+ |apply lt_sqrt_to_le_times_l;assumption]]]
+ |rewrite > assoc_plus in ⊢ (? (? % ?) ?);
+ rewrite > sym_plus in ⊢ (? (? (? ? %) ?) ?);
+ rewrite > assoc_plus in \vdash (? % ?);
+ rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite > assoc_plus in ⊢ (? (? % ?) ?);
+ rewrite > assoc_plus in \vdash (? % ?);
+ rewrite < assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite > assoc_plus in ⊢ (? (? ? (? % ?)) ?);
+ rewrite > sym_plus in ⊢ (? (? ? (? (? ? %) ?)) ?);
+ rewrite < assoc_plus in ⊢ (? (? ? (? % ?)) ?);
+ rewrite < assoc_plus in \vdash (? % ?);
+ rewrite < assoc_plus in ⊢ (? (? % ?) ?);
+ rewrite > assoc_plus in \vdash (? % ?);
+ rewrite > sym_plus in ⊢ (? (? ? %) ?);
+ rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite > assoc_plus in ⊢ (? (? ? (? % ?)) ?);
+ rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite > assoc_plus in ⊢ (? (? ? (? ? %)) ?);
+ simplify in ⊢ (? (? ? (? ? (? ? %))) ?);
+ rewrite < assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite < assoc_plus in ⊢ (? % ?);apply le_plus_l;
+ rewrite > assoc_plus in \vdash (? % ?);
+ rewrite > assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite > sym_plus in ⊢ (? (? ? (? ? %)) ?);
+ rewrite < assoc_plus in ⊢ (? (? ? %) ?);
+ rewrite < assoc_plus in \vdash (? % ?);apply le_plus_l;
+ rewrite > assoc_plus in \vdash (? ? %);apply le_n]
+ |apply (trans_le ? ? ? H);apply lt_to_le;apply (trans_le ? ? ? H1);
+ apply le_sqrt_n_n]
+qed.
+
+(*intros;apply lt_to_le;lapply (lt_div_S (((S (S (S (S O))))* log b (pred i)) + (S (S (S (S (S O)))))) i)
+ [apply (trans_le ? ? ? Hletin);apply le_times_l;apply le_S_S;
+ elim H1
+ [rewrite > log_SO;
+ [simplify;apply le_n
+ |assumption]
+ |
+ apply le_times_to_le_div2;
+ |*)
+
+(*
+theorem le_log_C2_sigma_p : \forall n,b. S O < b \to
+log b (C2 n) \leq
+(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x))) (\lambda x.S O)) +
+(sigma_p (S n) (\lambda x.(primeb x) \land (leb (S n) (x*x)))
+ (\lambda p.(sigma_p (n+p) (\lambda i.leb p i)
+ (\lambda i.S ((n+p)!/i*(S (log b 3)))))/(n+p)!)).
+intros;unfold C2;
+apply (trans_le ? (sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x)) (λx:nat.1)
++sigma_p (S n) (λx:nat.primeb x∧leb (S n) (x*x))
+ (λi.log b (S (n/i)))))
+ [apply log_pi_p;assumption
+ |apply le_plus
+ [apply le_n
+ |apply le_sigma_p;intros;cut (S (n/i) = (n+i)/i)
+ [rewrite > Hcut;apply le_log_div_sigma_p
+ [assumption
+ |apply prime_to_lt_O;apply primeb_true_to_prime;
+ elim (and_true ? ? H2);assumption
+ |apply le_plus_n
+ |apply le_n]
+ |lapply (prime_to_lt_O i (primeb_true_to_prime ? (andb_true_true ? ? H2))) as H3;
+ apply (div_mod_spec_to_eq (n+i) i (S (n/i)) (n \mod i) ? ((n+i) \mod i))
+ [constructor 1
+ [apply lt_mod_m_m;assumption
+ |simplify;rewrite > assoc_plus;rewrite < div_mod;
+ [apply sym_plus
+ |assumption]]
+ |apply div_mod_spec_div_mod;assumption]]]]
+qed.
+*)
+
projects / helm.git / commitdiff