include "nat/pi_p.ma".
include "nat/factorization.ma".
include "nat/factorial2.ma".
+include "nat/o.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
]
]
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
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 ⊢ (? ? (? ? %))
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:?.? ? %) ?))
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
]
]
|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)))))
|
*)
-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
]
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.
]
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.
]
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
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.
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
]
]
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.
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)).
]
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_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.
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
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.
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.
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_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.
+ 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
(* 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.
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.
+(* 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.
+*)
+
+
+
projects / helm.git / blobdiff