(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/nat/chebishev".
-
include "nat/log.ma".
include "nat/pi_p.ma".
include "nat/factorization.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)).
(\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).
intros. unfold B.
]
qed.
+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. S O < n \to
exp (S(S O)) ((S(S O))*n) \le (S(S O)) * n * B ((S(S O))*n).
intros.
]
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
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
]
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 eq_sigma_pi_SO_n: \forall n.
sigma_p n (\lambda i.true) (\lambda i.S O) = n.
intro.elim n
]
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.
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)))))))
]
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.
+*)
+
+
+
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