X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fchebyshev.ma;h=ff7db61cc241eaeaebcd241f06ceceafecb08ceb;hb=9e291b4d0a99118cd0a1c5540ef00c25ca37a56d;hp=317a82d15f2ebd3ae4b6648970cb3825ec3c7717;hpb=7265c6e8e0d548f6f97885727653b24c209d83f5;p=helm.git diff --git a/helm/software/matita/library/nat/chebyshev.ma b/helm/software/matita/library/nat/chebyshev.ma index 317a82d15..ff7db61cc 100644 --- a/helm/software/matita/library/nat/chebyshev.ma +++ b/helm/software/matita/library/nat/chebyshev.ma @@ -20,6 +20,203 @@ include "nat/factorial2.ma". definition prim: nat \to nat \def \lambda n. sigma_p (S n) primeb (\lambda p.(S O)). +theorem le_prim_n: \forall n. prim n \le n. +intros.unfold prim. elim n + [apply le_n + |apply (bool_elim ? (primeb (S n1)));intro + [rewrite > true_to_sigma_p_Sn + [rewrite > sym_plus. + rewrite < plus_n_Sm. + rewrite < plus_n_O. + apply le_S_S.assumption + |assumption + ] + |rewrite > false_to_sigma_p_Sn + [apply le_S.assumption + |assumption + ] + ] + ] +qed. + +theorem not_prime_times_SSO: \forall n. 1 < n \to \lnot prime (2*n). +intros.intro.elim H1. +absurd (2 = 2*n) + [apply H3 + [apply (witness ? ? n).reflexivity + |apply le_n + ] + |apply lt_to_not_eq. + rewrite > times_n_SO in ⊢ (? % ?). + apply lt_times_r. + assumption + ] +qed. + +theorem eq_prim_prim_pred: \forall n. 1 < n \to +(prim (2*n)) = (prim (pred (2*n))). +intros.unfold prim. +rewrite < S_pred + [rewrite > false_to_sigma_p_Sn + [reflexivity + |apply not_prime_to_primeb_false. + apply not_prime_times_SSO. + assumption + ] + |apply (trans_lt ? (2*1)) + [simplify.apply lt_O_S + |apply lt_times_r. + assumption + ] + ] +qed. + +theorem le_prim_n1: \forall n. 4 \le n \to prim (S(2*n)) \le n. +intros.unfold prim. elim H + [simplify.apply le_n + |cut (sigma_p (2*S n1) primeb (λp:nat.1) = sigma_p (S (2*S n1)) primeb (λp:nat.1)) + [apply (bool_elim ? (primeb (S(2*(S n1)))));intro + [rewrite > true_to_sigma_p_Sn + [rewrite > sym_plus. + rewrite < plus_n_Sm. + rewrite < plus_n_O. + apply le_S_S. + rewrite < Hcut. + rewrite > times_SSO. + assumption + |assumption + ] + |rewrite > false_to_sigma_p_Sn + [apply le_S. + rewrite < Hcut. + rewrite > times_SSO. + assumption + |assumption + ] + ] + |apply sym_eq.apply (eq_prim_prim_pred (S n1)). + apply le_S_S.apply (trans_le ? 4) + [apply leb_true_to_le.reflexivity + |assumption + ] + ] + ] +qed. + +(* usefull to kill a successor in bertrand *) +theorem le_prim_n2: \forall n. 7 \le n \to prim (S(2*n)) \le pred n. +intros.unfold prim. elim H + [apply leb_true_to_le.reflexivity. + |cut (sigma_p (2*S n1) primeb (λp:nat.1) = sigma_p (S (2*S n1)) primeb (λp:nat.1)) + [apply (bool_elim ? (primeb (S(2*(S n1)))));intro + [rewrite > true_to_sigma_p_Sn + [rewrite > sym_plus. + rewrite < plus_n_Sm. + rewrite < plus_n_O. + simplify in ⊢ (? ? %). + rewrite > S_pred in ⊢ (? ? %) + [apply le_S_S. + rewrite < Hcut. + rewrite > times_SSO. + assumption + |apply (ltn_to_ltO ? ? H1) + ] + |assumption + ] + |rewrite > false_to_sigma_p_Sn + [simplify in ⊢ (? ? %). + apply (trans_le ? ? ? ? (le_pred_n n1)). + rewrite < Hcut. + rewrite > times_SSO. + assumption + |assumption + ] + ] + |apply sym_eq.apply (eq_prim_prim_pred (S n1)). + apply le_S_S.apply (trans_le ? 4) + [apply leb_true_to_le.reflexivity + |apply (trans_le ? ? ? ? H1). + apply leb_true_to_le.reflexivity + ] + ] + ] +qed. + +(* da spostare *) +theorem le_pred: \forall n,m. n \le m \to pred n \le pred m. +apply nat_elim2;intros + [apply le_O_n + |apply False_ind.apply (le_to_not_lt ? ? H). + apply lt_O_S + |simplify.apply le_S_S_to_le.assumption + ] +qed. + +(* si dovrebbe poter migliorare *) +theorem le_prim_n3: \forall n. 15 \le n \to +prim n \le pred (n/2). +intros. +elim (or_eq_eq_S (pred n)). +elim H1 + [cut (n = S (2*a)) + [rewrite > Hcut. + apply (trans_le ? (pred a)) + [apply le_prim_n2. + apply (le_times_to_le 2) + [apply le_n_Sn + |apply le_S_S_to_le. + rewrite < Hcut. + assumption + ] + |apply le_pred. + apply le_times_to_le_div + [apply lt_O_S + |apply le_n_Sn + ] + ] + |rewrite < H2. + apply S_pred. + apply (ltn_to_ltO ? ? H) + ] + |cut (n=2*(S a)) + [rewrite > Hcut. + rewrite > eq_prim_prim_pred + [rewrite > times_SSO in ⊢ (? % ?). + change in ⊢ (? (? %) ?) with (S (2*a)). + rewrite > sym_times in ⊢ (? ? (? (? % ?))). + rewrite > lt_O_to_div_times + [apply (trans_le ? (pred a)) + [apply le_prim_n2. + apply le_S_S_to_le. + apply (lt_times_to_lt 2) + [apply le_n_Sn + |apply le_S_S_to_le. + rewrite < Hcut. + apply le_S_S. + assumption + ] + |apply le_pred. + apply le_n_Sn + ] + |apply lt_O_S + ] + |apply le_S_S. + apply not_lt_to_le.intro. + apply (le_to_not_lt ? ? H). + rewrite > Hcut. + lapply (le_S_S_to_le ? ? H3) as H4. + apply (le_n_O_elim ? H4). + apply leb_true_to_le.reflexivity + ] + |rewrite > times_SSO. + rewrite > S_pred + [apply eq_f.assumption + |apply (ltn_to_ltO ? ? H) + ] + ] + ] +qed. + (* This is chebishev psi function *) definition A: nat \to nat \def \lambda n. pi_p (S n) primeb (\lambda p.exp p (log p n)). @@ -1511,23 +1708,6 @@ theorem A_SSSSO: A (S(S(S(S O)))) = S(S(S(S(S(S(S(S(S(S(S(S O))))))))))). reflexivity. qed. -(* da spostare *) -theorem or_eq_eq_S: \forall n.\exists m. -n = (S(S O))*m \lor n = S ((S(S O))*m). -intro.elim n - [apply (ex_intro ? ? O). - left.reflexivity - |elim H.elim H1 - [apply (ex_intro ? ? a). - right.apply eq_f.assumption - |apply (ex_intro ? ? (S a)). - left.rewrite > H2. - apply sym_eq. - apply times_SSO - ] - ] -qed. - (* (* a better result *) theorem le_A_exp3: \forall n. S O < n \to @@ -1987,7 +2167,7 @@ apply (trans_le ? (2*(4*n*(B (4*n))))) qed. theorem le_priml: \forall n. O < n \to -(S(S O))*n \le (S (log (S(S O)) ((S(S O))*n)))*S(prim ((S(S O))*n)). +2*n \le (S (log 2 (2*n)))*S(prim (2*n)). intros. rewrite < (eq_log_exp (S(S O))) in ⊢ (? % ?) [apply (trans_le ? ((log (S(S O)) (exp ((S(S O))*n) (S(prim ((S(S O))*n))))))) @@ -2017,3 +2197,68 @@ apply (trans_le ? (exp (A n) 2)) ] qed. +(* bounds *) +theorem le_primr: \forall n. 1 < n \to prim n \le 2*(2*n-3)/log 2 n. +intros. +apply le_times_to_le_div + [apply lt_O_log + [apply lt_to_le.assumption + |assumption + ] + |apply (trans_le ? (log 2 (exp n (prim n)))) + [rewrite > sym_times. + apply log_exp2 + [apply le_n + |apply lt_to_le.assumption + ] + |rewrite < (eq_log_exp 2) in ⊢ (? ? %) + [apply le_log + [apply le_n + |apply le_exp_primr + ] + |apply le_n + ] + ] + ] +qed. + +theorem le_priml1: \forall n. O < n \to +2*n/((log 2 n)+2) - 1 \le prim (2*n). +intros. +apply le_plus_to_minus. +apply le_times_to_le_div2 + [rewrite > sym_plus. + simplify.apply lt_O_S + |rewrite > sym_times in ⊢ (? ? %). + rewrite < plus_n_Sm. + rewrite < plus_n_Sm in ⊢ (? ? (? ? %)). + rewrite < plus_n_O. + rewrite < sym_plus. + rewrite < log_exp + [simplify in ⊢ (? ? (? (? (? ? (? % ?))) ?)). + apply le_priml. + assumption + |apply le_n + |assumption + ] + ] +qed. + +(* +theorem prim_SSSSSSO: \forall n.30\le n \to O < prim (8*n) - prim n. +intros. +apply lt_to_lt_O_minus. +change in ⊢ (? ? (? (? % ?))) with (2*4). +rewrite > assoc_times. +apply (le_to_lt_to_lt ? (2*(2*n-3)/log 2 n)) + [apply le_primr.apply (trans_lt ? ? ? ? H). + apply leb_true_to_le.reflexivity + |apply (lt_to_le_to_lt ? (2*(4*n)/((log 2 (4*n))+2) - 1)) + [elim H + [ +normalize in ⊢ (%);simplify. + |apply le_priml1. +*) + + +