X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fchebyshev.ma;h=6ef8624e4983922e9d8276633f72a6d0163d42fc;hb=c99a38b6539be1eb667cced1eed2db3fc75e3162;hp=e4ca6587a2bb8ad4b6d53919fc6619ab774545cb;hpb=173965d294635f861850a850769b1530c73b835f;p=helm.git diff --git a/helm/software/matita/library/nat/chebyshev.ma b/helm/software/matita/library/nat/chebyshev.ma index e4ca6587a..6ef8624e4 100644 --- a/helm/software/matita/library/nat/chebyshev.ma +++ b/helm/software/matita/library/nat/chebyshev.ma @@ -12,15 +12,210 @@ (* *) (**************************************************************************) -set "baseuri" "cic:/matita/nat/chebishev". - include "nat/log.ma". include "nat/pi_p.ma". include "nat/factorization.ma". include "nat/factorial2.ma". definition prim: nat \to nat \def -\lambda n. sigma_p (S n) primeb (\lambda p.(S O)). +\lambda n. sigma_p (S n) primeb (\lambda p.1). + +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. + +(* la prova potrebbe essere migliorata *) +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 @@ -136,7 +331,7 @@ elim n ] ] qed. - + theorem lt_max_to_pi_p_primeb: \forall q,m. O < m \to max m (\lambda i.primeb i \land divides_b i m) < q \to m = pi_p q (\lambda i.primeb i \land divides_b i m) (\lambda p.exp p (ord m p)). intro.elim q @@ -349,7 +544,7 @@ apply pi_p_primeb_divides_b. assumption. qed. -theorem le_ord_log: \forall n,p. O < n \to S O < p \to +theorem le_ord_log: \forall n,p. O < n \to 1 < p \to ord n p \le log p n. intros. rewrite > (exp_ord p) in ⊢ (? ? (? ? %)) @@ -460,7 +655,7 @@ qed. theorem eq_ord_sigma_p: \forall n,m,x. O < n \to prime x \to exp x m \le n \to n < exp x (S m) \to -ord n x=sigma_p m (λi:nat.divides_b (x\sup (S i)) n) (λx:nat.S O). +ord n x=sigma_p m (λi:nat.divides_b (x\sup (S i)) n) (λx:nat.1). intros. lapply (prime_to_lt_SO ? H1). rewrite > (exp_ord x n) in ⊢ (? ? ? (? ? (λi:?.? ? %) ?)) @@ -602,20 +797,20 @@ pi_p (S n) primeb intros. rewrite > eq_fact_pi_p. apply (trans_eq ? ? - (pi_p (S n) (λi:nat.leb (S O) i) - (λn.pi_p (S n) primeb - (\lambda p.(pi_p (log p n) - (\lambda i.divides_b (exp p (S i)) n) (\lambda i.p)))))) + (pi_p (S n) (λm:nat.leb (S O) m) + (λm.pi_p (S m) primeb + (\lambda p.(pi_p (log p m) + (\lambda i.divides_b (exp p (S i)) m) (\lambda i.p)))))) [apply eq_pi_p1;intros [reflexivity |apply pi_p_primeb1. apply leb_true_to_le.assumption ] |apply (trans_eq ? ? - (pi_p (S n) (λi:nat.leb (S O) i) - (λn:nat - .pi_p (S n) (\lambda p.primeb p\land leb p n) - (λp:nat.pi_p (log p n) (λi:nat.divides_b ((p)\sup(S i)) n) (λi:nat.p))))) + (pi_p (S n) (λm:nat.leb (S O) m) + (λm:nat + .pi_p (S m) (\lambda p.primeb p\land leb p m) + (λp:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p))))) [apply eq_pi_p1 [intros.reflexivity |intros.apply eq_pi_p1 @@ -630,7 +825,7 @@ apply (trans_eq ? ? ] ] |apply (trans_eq ? ? - (pi_p (S n) (λi:nat.leb (S O) i) + (pi_p (S n) (λm:nat.leb (S O) m) (λm:nat .pi_p (S n) (λp:nat.primeb p∧leb p m) (λp:nat.pi_p (log p m) (λi:nat.divides_b ((p)\sup(S i)) m) (λi:nat.p))))) @@ -814,10 +1009,10 @@ intro.elim n | *) -theorem fact_pi_p2: \forall n. fact ((S(S O))*n) = -pi_p (S ((S(S O))*n)) primeb - (\lambda p.(pi_p (log p ((S(S O))*n)) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))*(exp p (mod ((S(S O))*n /(exp p (S i))) (S(S O)))))))). +theorem fact_pi_p2: \forall n. fact (2*n) = +pi_p (S (2*n)) primeb + (\lambda p.(pi_p (log p (2*n)) + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))*(exp p (mod (2*n /(exp p (S i))) 2)))))). intros.rewrite > fact_pi_p. apply eq_pi_p1 [intros.reflexivity @@ -836,13 +1031,13 @@ apply eq_pi_p1 ] qed. -theorem fact_pi_p3: \forall n. fact ((S(S O))*n) = -pi_p (S ((S(S O))*n)) primeb - (\lambda p.(pi_p (log p ((S(S O))*n)) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i))))))))* -pi_p (S ((S(S O))*n)) primeb - (\lambda p.(pi_p (log p ((S(S O))*n)) - (\lambda i.true) (\lambda i.(exp p (mod ((S(S O))*n /(exp p (S i))) (S(S O))))))). +theorem fact_pi_p3: \forall n. fact (2*n) = +pi_p (S (2*n)) primeb + (\lambda p.(pi_p (log p (2*n)) + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i))))))))* +pi_p (S (2*n)) primeb + (\lambda p.(pi_p (log p (2*n)) + (\lambda i.true) (\lambda i.(exp p (mod (2*n /(exp p (S i))) 2))))). intros. rewrite < times_pi_p. rewrite > fact_pi_p2. @@ -852,14 +1047,14 @@ apply eq_pi_p;intros ] qed. -theorem pi_p_primeb4: \forall n. S O < n \to -pi_p (S ((S(S O))*n)) primeb - (\lambda p.(pi_p (log p ((S(S O))*n)) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))) +theorem pi_p_primeb4: \forall n. 1 < n \to +pi_p (S (2*n)) primeb + (\lambda p.(pi_p (log p (2*n)) + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))) = pi_p (S n) primeb - (\lambda p.(pi_p (log p (S(S O)*n)) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))). + (\lambda p.(pi_p (log p (2*n)) + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))). intros. apply or_false_eq_SO_to_eq_pi_p [apply le_S_S. @@ -881,14 +1076,14 @@ apply or_false_eq_SO_to_eq_pi_p ] qed. -theorem pi_p_primeb5: \forall n. S O < n \to -pi_p (S ((S(S O))*n)) primeb +theorem pi_p_primeb5: \forall n. 1 < n \to +pi_p (S (2*n)) primeb (\lambda p.(pi_p (log p ((S(S O))*n)) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))) + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))) = pi_p (S n) primeb (\lambda p.(pi_p (log p n) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))). + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))). intros. rewrite > (pi_p_primeb4 ? H). apply eq_pi_p1;intros @@ -923,11 +1118,11 @@ apply eq_pi_p1;intros qed. theorem exp_fact_SSO: \forall n. -exp (fact n) (S(S O)) +exp (fact n) 2 = pi_p (S n) primeb (\lambda p.(pi_p (log p n) - (\lambda i.true) (\lambda i.(exp p ((S(S O))*(n /(exp p (S i)))))))). + (\lambda i.true) (\lambda i.(exp p (2*(n /(exp p (S i)))))))). intros. rewrite > fact_pi_p. rewrite < exp_pi_p. @@ -944,8 +1139,32 @@ pi_p (S n) primeb (\lambda p.(pi_p (log p n) (\lambda i.true) (\lambda i.(exp p (mod (n /(exp p (S i))) (S(S O))))))). +theorem B_SSSO: B 3 = 6. +reflexivity. +qed. + +theorem B_SSSSO: B 4 = 6. +reflexivity. +qed. + +theorem B_SSSSSO: B 5 = 30. +reflexivity. +qed. + +theorem B_SSSSSSO: B 6 = 20. +reflexivity. +qed. + +theorem B_SSSSSSSO: B 7 = 140. +reflexivity. +qed. + +theorem B_SSSSSSSSO: B 8 = 70. +reflexivity. +qed. + theorem eq_fact_B:\forall n.S O < n \to -fact ((S(S O))*n) = exp (fact n) (S(S O)) * B((S(S O))*n). +fact (2*n) = exp (fact n) 2 * B(2*n). intros. unfold B. rewrite > fact_pi_p3. apply eq_f2 @@ -1056,7 +1275,8 @@ cut ((S(S O)) < (S ((S(S(S(S O))))*n))) ] ] qed. - + +(* not usefull theorem le_fact_A:\forall n.S O < n \to fact (2*n) \le exp (fact n) 2 * A (2*n). intros. @@ -1065,7 +1285,7 @@ rewrite > eq_fact_B apply le_B_A |assumption ] -qed. +qed. *) theorem lt_SO_to_le_B_exp: \forall n.S O < n \to B (2*n) \le exp 2 (pred (2*n)). @@ -1095,8 +1315,25 @@ intro.cases n ] qed. -theorem lt_SO_to_le_exp_B: \forall n. S O < n \to -exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n). +theorem lt_SSSSO_to_le_B_exp: \forall n.4 < n \to +B (2*n) \le exp 2 ((2*n)-2). +intros. +apply (le_times_to_le (exp (fact n) (S(S O)))) + [apply lt_O_exp. + apply lt_O_fact + |rewrite < eq_fact_B + [rewrite < sym_times in ⊢ (? ? %). + rewrite > exp_SSO. + rewrite < assoc_times. + apply lt_SSSSO_to_fact.assumption + |apply lt_to_le.apply lt_to_le. + apply lt_to_le.assumption + ] + ] +qed. + +theorem lt_SO_to_le_exp_B: \forall n. 1 < n \to +exp 2 (2*n) \le 2 * n * B (2*n). intros. apply (le_times_to_le (exp (fact n) (S(S O)))) [apply lt_O_exp. @@ -1114,7 +1351,7 @@ apply (le_times_to_le (exp (fact n) (S(S O)))) qed. theorem le_exp_B: \forall n. O < n \to -exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n). +exp 2 (2*n) \le 2 * n * B (2*n). intros. elim H [apply le_n @@ -1124,9 +1361,9 @@ elim H qed. theorem eq_A_SSO_n: \forall n.O < n \to -A((S(S O))*n) = - pi_p (S ((S(S O))*n)) primeb - (\lambda p.(pi_p (log p ((S(S O))*n) ) +A(2*n) = + pi_p (S (2*n)) primeb + (\lambda p.(pi_p (log p (2*n) ) (\lambda i.true) (\lambda i.(exp p (bool_to_nat (leb (S n) (exp p (S i)))))))) *A n. intro. @@ -1222,7 +1459,7 @@ cut ( qed. theorem le_A_BA1: \forall n. O < n \to -A((S(S O))*n) \le B((S(S O))*n)*A n. +A(2*n) \le B(2*n)*A n. intros. rewrite > eq_A_SSO_n [apply le_times_l. @@ -1304,6 +1541,16 @@ apply (trans_le ? (B(2*n)*A n)) ] qed. +theorem lt_SSSSO_to_le_A_exp: \forall n. 4 < n \to +A(2*n) \le exp 2 ((2*n)-2)*A n. +intros. +apply (trans_le ? (B(2*n)*A n)) + [apply le_A_BA + |apply le_times_l. + apply lt_SSSSO_to_le_B_exp.assumption + ] +qed. + theorem times_SSO_pred: \forall n. 2*(pred n) \le pred (2*n). intro.cases n [apply le_n @@ -1462,23 +1709,6 @@ 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 @@ -1723,6 +1953,196 @@ elim H2 ] qed. +theorem le_n_SSSSSSSSO_to_le_A_exp: +\forall n. n \le 8 \to A(n) \le exp 2 ((2 * n) -3). +intro.cases n + [intro.apply le_n + |cases n1 + [intro.apply le_n + |cases n2 + [intro.apply le_n + |cases n3 + [intro.apply leb_true_to_le.reflexivity + |cases n4 + [intro.apply leb_true_to_le.reflexivity + |cases n5 + [intro.apply leb_true_to_le.reflexivity + |cases n6 + [intro.apply leb_true_to_le.reflexivity + |cases n7 + [intro.apply leb_true_to_le.reflexivity + |cases n8 + [intro.apply leb_true_to_le.reflexivity + |intro.apply False_ind. + apply (lt_to_not_le ? ? H). + apply leb_true_to_le.reflexivity + ] + ] + ] + ] + ] + ] + ] + ] + ] +qed. + +theorem le_A_exp5: \forall n. A(n) \le exp 2 ((2 * n) -3). +intro. +apply (nat_elim1 n). +intros. +elim (decidable_le 9 m) + [elim (or_eq_eq_S m). + elim H2 + [rewrite > H3 in ⊢ (? % ?). + apply (trans_le ? (exp 2 (pred(2*a))*A a)) + [apply le_A_exp + |apply (trans_le ? (2\sup(pred(2*a))*(2\sup((2*a)-3)))) + [apply le_times_r. + apply H. + rewrite > H3. + apply lt_m_nm + [apply (trans_lt ? 4) + [apply lt_O_S + |apply (lt_times_to_lt 2) + [apply lt_O_S + |rewrite < H3.assumption + ] + ] + |apply le_n + ] + |rewrite < H3. + rewrite < exp_plus_times. + apply le_exp + [apply lt_O_S + |simplify.rewrite < plus_n_O. + rewrite > eq_minus_plus_plus_minus + [apply le_plus_l. + apply le_pred_n + |apply (trans_le ? 9) + [apply leb_true_to_le. reflexivity + |assumption + ] + ] + ] + ] + ] + |apply (trans_le ? (A (2*(S a)))) + [apply monotonic_A. + rewrite > H3. + rewrite > times_SSO. + apply le_n_Sn + |apply (trans_le ? ((exp 2 ((2*(S a))-2))*A (S a))) + [apply lt_SSSSO_to_le_A_exp. + apply le_S_S. + apply (le_times_to_le 2) + [apply le_n_Sn. + |apply le_S_S_to_le.rewrite < H3.assumption + ] + |apply (trans_le ? ((2\sup((2*S a)-2))*(exp 2 ((2*(S a))-3)))) + [apply le_times_r. + apply H. + rewrite > H3. + apply le_S_S. + apply lt_m_nm + [apply (lt_to_le_to_lt ? 4) + [apply lt_O_S + |apply (le_times_to_le 2) + [apply lt_O_S + |apply le_S_S_to_le. + rewrite < H3.assumption + ] + ] + |apply le_n + ] + |rewrite > times_SSO. + rewrite < H3. + rewrite < exp_plus_times. + apply le_exp + [apply lt_O_S + |cases m + [apply le_n + |cases n1 + [apply le_n + |simplify. + rewrite < minus_n_O. + rewrite < plus_n_O. + rewrite < plus_n_Sm. + simplify.rewrite < minus_n_O. + rewrite < plus_n_Sm. + apply le_n + ] + ] + ] + ] + ] + ] + ] + |apply le_n_SSSSSSSSO_to_le_A_exp. + apply le_S_S_to_le. + apply not_le_to_lt. + assumption + ] +qed. + +theorem le_exp_Al:\forall n. O < n \to exp 2 n \le A (2 * n). +intros. +apply (trans_le ? ((exp 2 (2*n))/(2*n))) + [apply le_times_to_le_div + [rewrite > (times_n_O O) in ⊢ (? % ?). + apply lt_times + [apply lt_O_S + |assumption + ] + |simplify in ⊢ (? ? (? ? %)). + rewrite < plus_n_O. + rewrite > exp_plus_times. + apply le_times_l. + alias id "le_times_SSO_n_exp_SSO_n" = "cic:/matita/nat/o/le_times_SSO_n_exp_SSO_n.con". + apply le_times_SSO_n_exp_SSO_n + ] + |apply le_times_to_le_div2 + [rewrite > (times_n_O O) in ⊢ (? % ?). + apply lt_times + [apply lt_O_S + |assumption + ] + |apply (trans_le ? ((B (2*n)*(2*n)))) + [rewrite < sym_times in ⊢ (? ? %). + apply le_exp_B. + assumption + |apply le_times_l. + apply le_B_A + ] + ] + ] +qed. + +theorem le_exp_A2:\forall n. 1 < n \to exp 2 (n / 2) \le A n. +intros. +apply (trans_le ? (A(n/2*2))) + [rewrite > sym_times. + apply le_exp_Al. + elim (le_to_or_lt_eq ? ? (le_O_n (n/2))) + [assumption + |apply False_ind. + apply (lt_to_not_le ? ? H). + rewrite > (div_mod n 2) + [rewrite < H1. + change in ⊢ (? % ?) with (n\mod 2). + apply le_S_S_to_le. + apply lt_mod_m_m. + apply lt_O_S + |apply lt_O_S + ] + ] + |apply monotonic_A. + rewrite > (div_mod n 2) in ⊢ (? ? %). + apply le_plus_n_r. + apply lt_O_S + ] +qed. + theorem eq_sigma_pi_SO_n: \forall n. sigma_p n (\lambda i.true) (\lambda i.S O) = n. intro.elim n @@ -1765,21 +2185,19 @@ qed. (* the inequalities *) theorem le_exp_priml: \forall n. O < n \to -exp (S(S O)) ((S(S O))*n) \le exp ((S(S O))*n) (S(prim ((S(S O))*n))). +exp 2 (2*n) \le exp (2*n) (S(prim (2*n))). intros. -apply (trans_le ? ((((S(S O))*n*(B ((S(S O))*n)))))) +apply (trans_le ? (((2*n*(B (2*n)))))) [apply le_exp_B.assumption - |change in ⊢ (? ? %) with ((((S(S O))*n))*(((S(S O))*n))\sup (prim ((S(S O))*n))). + |change in ⊢ (? ? %) with (((2*n))*((2*n))\sup (prim (2*n))). apply le_times_r. - apply (trans_le ? (A ((S(S O))*n))) + apply (trans_le ? (A (2*n))) [apply le_B_A |apply le_Al ] ] qed. -alias num (instance 0) = "natural number". - theorem le_exp_prim4l: \forall n. O < n \to exp 2 (S(4*n)) \le exp (4*n) (S(prim (4*n))). intros. @@ -1808,7 +2226,7 @@ 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))))))) @@ -1824,23 +2242,82 @@ rewrite < (eq_log_exp (S(S O))) in ⊢ (? % ?) ] qed. -theorem le_exp_primr: \forall n. S O < n \to -exp n (prim n) \le exp (pred n) 2*(exp 2 (2*(2*n-3))). +theorem le_exp_primr: \forall n. +exp n (prim n) \le exp 2 (2*(2*n-3)). intros. apply (trans_le ? (exp (A n) 2)) [change in ⊢ (? ? %) with ((A n)*((A n)*(S O))). rewrite < times_n_SO. apply leA_r2 - |apply (trans_le ? (exp ((pred n)*(exp 2 (2*n - 3))) 2)) - [apply monotonic_exp1. - apply le_A_exp4. + |rewrite > sym_times. + rewrite < exp_exp_times. + apply monotonic_exp1. + apply le_A_exp5 + ] +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 - |rewrite < times_exp. - rewrite > exp_exp_times. - apply le_times_r. - rewrite > sym_times. - apply le_n + |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. +*) + + +