include "arithmetics/sqrt.ma".
include "arithmetics/chebyshev/psi_bounds.ma".
-include "arithmetics/chebyshev/chebyshev_teta.ma".
+include "arithmetics/chebyshev/chebyshev_theta.ma".
definition bertrand ≝ λn. ∃p.n < p ∧ p ≤ 2*n ∧ prime p.
]
qed.
-theorem le_B1_teta:∀n.18 ≤ n → not_bertrand n →
- B1 (2*n) ≤ teta (2 * n / 3).
-#n #le18 #not_Bn >B1_def >teta_def
+theorem le_B1_theta:∀n.18 ≤ n → not_bertrand n →
+ B1 (2*n) ≤ theta (2 * n / 3).
+#n #le18 #not_Bn >B1_def >theta_def
@(transitive_le ? (∏_{p < S (2*n) | primeb p} (p\sup(bool_to_nat (eqb (k (2*n) p) 1)))))
[@le_pi #p #ltp #primebp @le_exp
[@prime_to_lt_O @primeb_true_to_prime //
∀n.exp 2 7 ≤ n → not_bertrand n →
B (2*n) ≤ (exp 2 (2*(2 * n / 3)))*(exp (2*n) (pred(sqrt(2*n)/2))).
#n #len #notB >eq_B_Bk >eq_Bk_B1_B2 @le_times
- [@(transitive_le ? (teta ((2*n)/3)))
- [@le_B1_teta [@(transitive_le … len) @leb_true_to_le //|//]
- |@le_teta
+ [@(transitive_le ? (theta ((2*n)/3)))
+ [@le_B1_theta [@(transitive_le … len) @leb_true_to_le //|//]
+ |@le_theta
]
|@le_B2_exp //
]
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "arithmetics/log.ma".
+include "arithmetics/sigma_pi.ma".
+
+(* (prim n) counts the number of prime numbers up to n included *)
+definition prim ≝ λn. ∑_{i < S n | primeb i} 1.
+
+lemma le_prim_n: ∀n. prim n ≤ n.
+#n elim n // -n #n #H
+whd in ⊢ (?%?); cases (primeb (S n)) whd in ⊢ (?%?);
+ [@le_S_S @H |@le_S @H]
+qed.
+
+lemma not_prime_times_2: ∀n. 1 < n → ¬prime (2*n).
+#n #ltn % * #H #H1 @(absurd (2 = 2*n))
+ [@H1 // %{n} //
+ |@lt_to_not_eq >(times_n_1 2) in ⊢ (?%?); @monotonic_lt_times_r //
+ ]
+qed.
+
+theorem eq_prim_prim_pred: ∀n. 1 < n →
+ prim (2*n) = prim (pred (2*n)).
+#n #ltn
+lapply (S_pred (2*n) ?) [>(times_n_1 0) in ⊢ (?%?); @lt_times //] #H2n
+lapply (not_prime_times_2 n ltn) #notp2n
+whd in ⊢ (??%?); >(not_prime_to_primeb_false … notp2n) whd in ⊢ (??%?);
+<H2n in ⊢ (??%?); %
+qed.
+
+theorem le_prim_n1: ∀n. 4 ≤ n →
+ prim (S(2*n)) ≤ n.
+#n #le4 elim le4 -le4
+ [@le_n
+ |#m #le4 cut (S (2*m) = pred (2*(S m))) [normalize //] #Heq >Heq
+ <eq_prim_prim_pred [2: @le_S_S @(transitive_le … le4) //]
+ #H whd in ⊢ (?%?); cases (primeb (S (2*S m)))
+ [@le_S_S @H |@le_S @H]
+ ]
+qed.
+
+(* usefull to kill a successor in bertrand *)
+theorem le_prim_n2: ∀n. 7 ≤ n → prim (S(2*n)) ≤ pred n.
+#n #le7 elim le7 -le7
+ [@le_n
+ |#m #le7 cut (S (2*m) = pred (2*(S m))) [normalize //] #Heq >Heq
+ <eq_prim_prim_pred [2: @le_S_S @(transitive_le … le7) //]
+ #H whd in ⊢ (?%?);
+ whd in ⊢ (??%); <(S_pred m) in ⊢ (??%); [2: @(transitive_le … le7) //]
+ cases (primeb (S (2*S m))) [@le_S_S @H |@le_S @H]
+ ]
+qed.
+
+lemma even_or_odd: ∀n.∃a.n=2*a ∨ n = S(2*a).
+#n elim n -n
+ [%{0} %1 %
+ |#n * #a * #Hn [%{a} %2 @eq_f @Hn | %{(S a)} %1 >Hn normalize //
+ ]
+qed.
+
+(* la prova potrebbe essere migliorata *)
+theorem le_prim_n3: ∀n. 15 ≤ n →
+ prim n ≤ pred (n/2).
+#n #len cases (even_or_odd (pred n)) #a * #Hpredn
+ [cut (n = S (2*a)) [<Hpredn @sym_eq @S_pred @(transitive_le … len) //] #Hn
+ >Hn @(transitive_le ? (pred a))
+ [@le_prim_n2 @(le_times_to_le 2) [//|@le_S_S_to_le <Hn @len]
+ |@monotonic_pred @le_times_to_le_div //
+ ]
+ |cut (n = (2*S a))
+ [normalize normalize in Hpredn:(???%); <plus_n_Sm <Hpredn @sym_eq @S_pred
+ @(transitive_le … len) //] #Hn
+ >Hn @(transitive_le ? (pred a))
+ [>eq_prim_prim_pred
+ [2:@(lt_times_n_to_lt_r 2) <Hn @(transitive_le … len) //]
+ <Hn >Hpredn @le_prim_n2 @le_S_S_to_le @(lt_times_n_to_lt_r 2) <Hn @len
+ |@monotonic_pred @le_times_to_le_div //
+ ]
+ ]
+qed.
+
+(* This is chebishev psi function *)
+definition Psi: nat → nat ≝
+ λn.∏_{p < S n | primeb p} (exp p (log p n)).
+
+definition psi_def : ∀n.
+ Psi n = ∏_{p < S n | primeb p} (exp p (log p n)).
+// qed.
+
+theorem le_Psil1: ∀n.
+ Psi n ≤ ∏_{p < S n | primeb p} n.
+#n cases n [@le_n |#m @le_pi #i #_ #_ @le_exp_log //]
+qed.
+
+theorem le_Psil: ∀n. Psi n ≤ exp n (prim n).
+#n <exp_sigma @le_Psil1
+qed.
+
+theorem lePsi_r2: ∀n.
+ exp n (prim n) ≤ Psi n * Psi n.
+#n elim (le_to_or_lt_eq ?? (le_O_n n)) #Hn
+ [<(exp_sigma (S n) n primeb) <times_pi
+ @le_pi #i #lti #primei
+ cut (1<n)
+ [@(lt_to_le_to_lt … (le_S_S_to_le … lti)) @prime_to_lt_SO
+ @primeb_true_to_prime //] #lt1n
+ <exp_plus_times
+ @(transitive_le ? (exp i (S(log i n))))
+ [@lt_to_le @lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
+ |@le_exp
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |>(plus_n_O (log i n)) in ⊢ (?%?); >plus_n_Sm
+ @monotonic_le_plus_r @lt_O_log //
+ @le_S_S_to_le //
+ ]
+ ]
+ |<Hn @le_n
+ ]
+qed.
+
+(* an equivalent formulation for psi *)
+definition Psi': nat → nat ≝
+λn. ∏_{p < S n | primeb p} (∏_{i < log p n} p).
+
+lemma Psidef: ∀n. Psi' n = ∏_{p < S n | primeb p} (∏_{i < log p n} p).
+// qed-.
+
+theorem eq_Psi_Psi': ∀n.Psi n = Psi' n.
+#n @same_bigop // #i #lti #primebi
+@(trans_eq ? ? (exp i (∑_{x < log i n} 1)))
+ [@eq_f @sym_eq @sigma_const
+ |@sym_eq @exp_sigma
+ ]
+qed.
include "arithmetics/binomial.ma".
include "arithmetics/gcd.ma".
-include "arithmetics/chebyshev/chebyshev.ma".
+include "arithmetics/chebyshev/chebyshev_psi.ma".
-(* This is chebishev teta function *)
+(* This is chebishev theta function *)
-definition teta: nat → nat ≝ λn.
+definition theta: nat → nat ≝ λn.
∏_{p < S n| primeb p} p.
-lemma teta_def: ∀n.teta n = ∏_{p < S n| primeb p} p.
+lemma theta_def: ∀n.theta n = ∏_{p < S n| primeb p} p.
// qed.
-theorem lt_O_teta: ∀n. O < teta n.
+theorem lt_O_theta: ∀n. O < theta n.
#n elim n
[@le_n
|#n1 #Hind cases (true_or_false (primeb (S n1))) #Hc
- [>teta_def >bigop_Strue
+ [>theta_def >bigop_Strue
[>(times_n_O O) @lt_times // | //]
- |>teta_def >bigop_Sfalse //
+ |>theta_def >bigop_Sfalse //
]
]
qed.
#m @divides_pi_p_M1 @le_n
qed.
-theorem teta_pi_p_teta: ∀m. teta (S (2*m))
-= (∏_{p < S (S (2*m)) | leb (S (S m)) p∧primeb p} p)*teta (S m).
-#m >teta_def >teta_def
+theorem theta_pi_p_theta: ∀m. theta (S (2*m))
+= (∏_{p < S (S (2*m)) | leb (S (S m)) p∧primeb p} p)*theta (S m).
+#m >theta_def >theta_def
<(bigop_I ???? timesA)
>(bigop_sumI 0 (S(S m)) (S(S(2*m))) (λp.primeb p) … timesA (λp.p))
[2:@le_S_S @le_S_S // |3:@le_O_n]
[>bigop_I_gen // |@(bigop_I … timesA)]
qed.
-theorem div_teta_teta: ∀m.
- teta (S(2*m))/teta (S m) =
+theorem div_theta_theta: ∀m.
+ theta (S(2*m))/theta (S m) =
∏_{p < S(S(2*m)) | leb (S(S m)) p ∧ primeb p} p.
#m @(div_mod_spec_to_eq ????? 0 (div_mod_spec_div_mod …))
- [@lt_O_teta
- |@div_mod_spec_intro [@lt_O_teta |<plus_n_O @teta_pi_p_teta]
+ [@lt_O_theta
+ |@div_mod_spec_intro [@lt_O_theta |<plus_n_O @theta_pi_p_theta]
]
qed.
-theorem le_teta_M_teta: ∀m.
- teta (S(2*m)) ≤ (M m)*teta (S m).
-#m >teta_pi_p_teta @le_times [2://] @divides_to_le
+theorem le_theta_M_theta: ∀m.
+ theta (S(2*m)) ≤ (M m)*theta (S m).
+#m >theta_pi_p_theta @le_times [2://] @divides_to_le
[@lt_O_bc @lt_to_le @le_S_S // | @divides_pi_p_M
]
qed.
-theorem lt_O_to_le_teta_exp_teta: ∀m. O < m→
- teta (S(2*m)) < exp 2 (2*m)*teta (S m).
-#m #posm @(le_to_lt_to_lt ? (M m*teta (S m)))
- [@le_teta_M_teta
- |@monotonic_lt_times_l [@lt_O_teta|@lt_M //]
+theorem lt_O_to_le_theta_exp_theta: ∀m. O < m→
+ theta (S(2*m)) < exp 2 (2*m)*theta (S m).
+#m #posm @(le_to_lt_to_lt ? (M m*theta (S m)))
+ [@le_theta_M_theta
+ |@monotonic_lt_times_l [@lt_O_theta|@lt_M //]
]
qed.
-theorem teta_pred: ∀n. 1 < n → teta (2*n) = teta (pred (2*n)).
-#n #lt1n >teta_def >teta_def >bigop_Sfalse
+theorem theta_pred: ∀n. 1 < n → theta (2*n) = theta (pred (2*n)).
+#n #lt1n >theta_def >theta_def >bigop_Sfalse
[>S_pred
[//
|>(times_n_O 2) in ⊢ (?%?); @monotonic_lt_times_r @lt_to_le //
]
qed.
-theorem le_teta: ∀m.teta m ≤ exp 2 (2*m).
+theorem le_theta: ∀m.theta m ≤ exp 2 (2*m).
#m @(nat_elim1 m) #m1 #Hind
cut (∀a. 2 *a = a+a) [//] #times2
cases (even_or_odd m1) #a * #Ha >Ha
[#_ @le_n
|#n cases n
[#_ @leb_true_to_le //
- |#n1 #Hn1 >teta_pred
+ |#n1 #Hn1 >theta_pred
[cut (pred (2*S(S n1)) = (S (2*S n1)))
[normalize >plus_n_Sm in ⊢ (???%); //] #Hcut >Hcut
- @(transitive_le ? (exp 2 (2*(S n1))*teta (S (S n1))))
- [@lt_to_le @lt_O_to_le_teta_exp_teta @lt_O_S
+ @(transitive_le ? (exp 2 (2*(S n1))*theta (S (S n1))))
+ [@lt_to_le @lt_O_to_le_theta_exp_theta @lt_O_S
|>times2 in ⊢ (??%);>exp_plus_times in ⊢ (??%); @le_times
[@le_exp [@lt_O_S |@monotonic_le_times_r @le_n_Sn]
|@Hind >Hn1 >times2 //
]
|lapply Ha cases a
[#_ @leb_true_to_le //
- |#n #Hn @(transitive_le ? (exp 2 (2*(S n))*teta (S (S n))))
- [@lt_to_le @lt_O_to_le_teta_exp_teta @lt_O_S
+ |#n #Hn @(transitive_le ? (exp 2 (2*(S n))*theta (S (S n))))
+ [@lt_to_le @lt_O_to_le_theta_exp_theta @lt_O_S
|>times2 in ⊢ (??%); <plus_n_Sm <commutative_plus >plus_n_Sm
>exp_plus_times in ⊢ (??%); @monotonic_le_times_r
cut (∀a. 2*(S a) = S(S(2*a))) [#a normalize <plus_n_Sm //] #timesSS
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "arithmetics/binomial.ma".
+include "arithmetics/gcd.ma".
+include "arithmetics/chebyshev/chebyshev_psi.ma".
+
+(* This is chebishev theta function *)
+
+definition theta: nat → nat ≝ λn.
+ ∏_{p < S n| primeb p} p.
+
+lemma theta_def: ∀n.theta n = ∏_{p < S n| primeb p} p.
+// qed.
+
+theorem lt_O_theta: ∀n. O < theta n.
+#n elim n
+ [@le_n
+ |#n1 #Hind cases (true_or_false (primeb (S n1))) #Hc
+ [>theta_def >bigop_Strue
+ [>(times_n_O O) @lt_times // | //]
+ |>theta_def >bigop_Sfalse //
+ ]
+ ]
+qed.
+
+theorem divides_fact_to_divides: ∀p,n. prime p → divides p n! →
+ ∃m.O < m ∧ m \le n ∧ divides p m.
+#p #n #primep elim n
+ [normalize in ⊢ (%→?); #H @False_ind @(absurd (p ≤1))
+ [@divides_to_le //|@lt_to_not_le @prime_to_lt_SO @primep]
+ |#n1 #Hind >factS #Hdiv
+ cases (divides_times_to_divides ??? primep Hdiv) #Hcase
+ [%{(S n1)} %[ % [@lt_O_S |@le_n] |@Hcase]
+ |cases(Hind Hcase) #a * * #posa #lea #diva
+ %{a} % [% [// |@le_S //] |//]
+ ]
+ ]
+qed.
+
+theorem divides_fact_to_le: ∀p,n. prime p → divides p n! → p ≤ n.
+#p #n #primep #divp cases (divides_fact_to_divides p n primep divp)
+#a * * #posa #lea #diva @(transitive_le ? a) [@divides_to_le // | //]
+qed.
+
+theorem prime_to_divides_M: ∀m,p.
+ prime p → S m < p → p ≤ S(2*m) → divides p (M m).
+#m #p #primep #lemp #lep >Mdef >bceq
+cases (bc2 (S(2*m)) m ?)
+ [#q #Hq >Hq >commutative_times >div_times
+ [cases (divides_times_to_divides p (m!*(S (2*m)-m)!) q primep ?)
+ [#Hdiv @False_ind
+ cases (divides_times_to_divides p (m!) (S (2*m)-m)! primep ?)
+ [-Hdiv #Hdiv @(absurd (p ≤ m))
+ [@divides_fact_to_le //
+ |@(lt_to_not_le ?? (lt_to_le ?? lemp))
+ ]
+ |-Hdiv #Hdiv @(absurd (p ≤S m))
+ [@(divides_fact_to_le … primep)
+ cut (S m = S(2*m)-m)
+ [normalize in ⊢ (???(?%?)); <plus_n_O
+ >plus_n_Sm >commutative_plus @minus_plus_m_m
+ ] #HSm
+ >HSm //
+ |@lt_to_not_le //
+ ]
+ |//
+ ]
+ |//
+ |<Hq @divides_fact [@prime_to_lt_O // |//]
+ ]
+ |>(times_n_O O) in ⊢ (?%?); @lt_times //
+ ]
+ |normalize in ⊢ (??(?%)); <plus_n_O //
+ ]
+qed.
+
+theorem divides_pi_p_M1: ∀m.∀i. i ≤ S(S(2*m)) →
+ ∏_{p < i | leb (S(S m)) p ∧ primeb p} p ∣ M m .
+#m #i elim i
+ [#_ @(quotient ?? (M m)) >commutative_times @times_n_1
+ |#n #Hind #len
+ cases (true_or_false (leb (S (S m)) n ∧ primeb n)) #Hc
+ [>bigop_Strue
+ [@divides_to_divides_times
+ [@primeb_true_to_prime @(andb_true_r ?? Hc)
+ |cut (∀p.prime p → n ≤ p → ¬p∣∏_{p < n | leb (S (S m)) p∧primeb p} p)
+ [2: #Hcut @(Hcut … (le_n ?)) @primeb_true_to_prime @(andb_true_r ?? Hc)]
+ #p #primep elim n
+ [#_ normalize @(not_to_not ? (p ≤ 1))
+ [@divides_to_le @lt_O_S|@lt_to_not_le @prime_to_lt_SO //]
+ |#n1 #Hind1 #Hn1 cases (true_or_false (leb (S (S m)) n1∧primeb n1)) #Hc1
+ [>bigop_Strue
+ [% #Habs cases(divides_times_to_divides ??? primep Habs)
+ [-Habs #Habs @(absurd … Hn1) @le_to_not_lt
+ @(divides_to_le … Habs) @prime_to_lt_O
+ @primeb_true_to_prime @(andb_true_r ?? Hc1)
+ |-Habs #Habs @(absurd … Habs) @Hind1 @lt_to_le //
+ ]
+ |@Hc1
+ ]
+ |>bigop_Sfalse // @Hind1 @lt_to_le //
+ ]
+ ]
+ |@prime_to_divides_M
+ [@primeb_true_to_prime @(andb_true_r ?? Hc)
+ |@leb_true_to_le @(andb_true_l ?? Hc)
+ |@le_S_S_to_le //
+ ]
+ |@Hind @lt_to_le //
+ ]
+ |@Hc
+ ]
+ |>bigop_Sfalse // @Hind @lt_to_le @len
+ ]
+ ]
+qed.
+
+theorem divides_pi_p_M:∀m.
+ ∏_{p < S(S(2*m)) | leb (S(S m)) p ∧ primeb p} p ∣ (M m).
+#m @divides_pi_p_M1 @le_n
+qed.
+
+theorem theta_pi_p_theta: ∀m. theta (S (2*m))
+= (∏_{p < S (S (2*m)) | leb (S (S m)) p∧primeb p} p)*theta (S m).
+#m >theta_def >theta_def
+<(bigop_I ???? timesA)
+>(bigop_sumI 0 (S(S m)) (S(S(2*m))) (λp.primeb p) … timesA (λp.p))
+ [2:@le_S_S @le_S_S // |3:@le_O_n]
+@eq_f2
+ [>bigop_I_gen // |@(bigop_I … timesA)]
+qed.
+
+theorem div_theta_theta: ∀m.
+ theta (S(2*m))/theta (S m) =
+ ∏_{p < S(S(2*m)) | leb (S(S m)) p ∧ primeb p} p.
+#m @(div_mod_spec_to_eq ????? 0 (div_mod_spec_div_mod …))
+ [@lt_O_theta
+ |@div_mod_spec_intro [@lt_O_theta |<plus_n_O @theta_pi_p_theta]
+ ]
+qed.
+
+theorem le_theta_M_theta: ∀m.
+ theta (S(2*m)) ≤ (M m)*theta (S m).
+#m >theta_pi_p_theta @le_times [2://] @divides_to_le
+ [@lt_O_bc @lt_to_le @le_S_S // | @divides_pi_p_M
+ ]
+qed.
+
+theorem lt_O_to_le_theta_exp_theta: ∀m. O < m→
+ theta (S(2*m)) < exp 2 (2*m)*theta (S m).
+#m #posm @(le_to_lt_to_lt ? (M m*theta (S m)))
+ [@le_theta_M_theta
+ |@monotonic_lt_times_l [@lt_O_theta|@lt_M //]
+ ]
+qed.
+
+theorem theta_pred: ∀n. 1 < n → theta (2*n) = theta (pred (2*n)).
+#n #lt1n >theta_def >theta_def >bigop_Sfalse
+ [>S_pred
+ [//
+ |>(times_n_O 2) in ⊢ (?%?); @monotonic_lt_times_r @lt_to_le //
+ ]
+ |@not_prime_to_primeb_false % * #lt2n #Hprime
+ @(absurd (2=2*n))
+ [@(Hprime … (le_n ?)) %{n} //
+ |@lt_to_not_eq >(times_n_1 2) in ⊢ (?%?); @monotonic_lt_times_r //
+ ]
+ ]
+qed.
+
+theorem le_theta: ∀m.theta m ≤ exp 2 (2*m).
+#m @(nat_elim1 m) #m1 #Hind
+cut (∀a. 2 *a = a+a) [//] #times2
+cases (even_or_odd m1) #a * #Ha >Ha
+ [lapply Ha cases a
+ [#_ @le_n
+ |#n cases n
+ [#_ @leb_true_to_le //
+ |#n1 #Hn1 >theta_pred
+ [cut (pred (2*S(S n1)) = (S (2*S n1)))
+ [normalize >plus_n_Sm in ⊢ (???%); //] #Hcut >Hcut
+ @(transitive_le ? (exp 2 (2*(S n1))*theta (S (S n1))))
+ [@lt_to_le @lt_O_to_le_theta_exp_theta @lt_O_S
+ |>times2 in ⊢ (??%);>exp_plus_times in ⊢ (??%); @le_times
+ [@le_exp [@lt_O_S |@monotonic_le_times_r @le_n_Sn]
+ |@Hind >Hn1 >times2 //
+ ]
+ ]
+ |@le_S_S @lt_O_S
+ ]
+ ]
+ ]
+ |lapply Ha cases a
+ [#_ @leb_true_to_le //
+ |#n #Hn @(transitive_le ? (exp 2 (2*(S n))*theta (S (S n))))
+ [@lt_to_le @lt_O_to_le_theta_exp_theta @lt_O_S
+ |>times2 in ⊢ (??%); <plus_n_Sm <commutative_plus >plus_n_Sm
+ >exp_plus_times in ⊢ (??%); @monotonic_le_times_r
+ cut (∀a. 2*(S a) = S(S(2*a))) [#a normalize <plus_n_Sm //] #timesSS
+ <timesSS @Hind >Hn @le_S_S >times2 //
+ ]
+ ]
+ ]
+qed.
+
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "arithmetics/log.ma".
+include "arithmetics/sigma_pi.ma".
+include "arithmetics/ord.ma".
+
+theorem eq_pi_p_primeb_divides_b: ∀n,m.
+∏_{p<n | primeb p ∧ dividesb p m} (exp p (ord m p))
+ = ∏_{p<n | primeb p} (exp p (ord m p)).
+#n #m elim n // #n1 #Hind cases (true_or_false (primeb n1)) #Hprime
+ [>bigop_Strue in ⊢ (???%); //
+ cases (true_or_false (dividesb n1 m)) #Hdivides
+ [>bigop_Strue [@eq_f @Hind| @true_to_andb_true //]
+ |>bigop_Sfalse
+ [>not_divides_to_ord_O
+ [whd in ⊢ (???(?%?)); //
+ |@dividesb_false_to_not_divides //
+ |@primeb_true_to_prime //
+ ]
+ |>Hprime >Hdivides %
+ ]
+ ]
+|>bigop_Sfalse [>bigop_Sfalse // |>Hprime %]
+]
+qed.
+
+lemma lt_1_max_prime: ∀n. 1 < n →
+ 1 < max (S n) (λi:nat.primeb i∧dividesb i n).
+#n #lt1n
+@(lt_to_le_to_lt ? (smallest_factor n))
+ [@lt_SO_smallest_factor //
+ |@true_to_le_max
+ [@le_S_S @le_smallest_factor_n
+ |@true_to_andb_true
+ [@prime_to_primeb_true @prime_smallest_factor_n //
+ |@divides_to_dividesb_true
+ [@lt_O_smallest_factor @lt_to_le //
+ |@divides_smallest_factor_n @lt_to_le //
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem lt_max_to_pi_p_primeb: ∀q,m. O<m → max (S m) (λi.primeb i ∧ dividesb i m)<q →
+ m = ∏_{p < q | primeb p ∧ dividesb p m} (exp p (ord m p)).
+#q elim q
+ [#m #posm #Hmax normalize @False_ind @(absurd … Hmax (not_le_Sn_O ?))
+ |#n #Hind #m #posm #Hmax
+ cases (true_or_false (primeb n∧dividesb n m)) #Hcase
+ [>bigop_Strue
+ [>(exp_ord n m … posm) in ⊢ (??%?);
+ [@eq_f >(Hind (ord_rem m n))
+ [@same_bigop
+ [#x #ltxn cases (true_or_false (primeb x)) #Hx >Hx
+ [cases (true_or_false (dividesb x (ord_rem m n))) #Hx1 >Hx1
+ [@sym_eq @divides_to_dividesb_true
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@(transitive_divides ? (ord_rem m n))
+ [@dividesb_true_to_divides //
+ |@(divides_ord_rem … posm) @(transitive_lt … ltxn)
+ @prime_to_lt_SO @primeb_true_to_prime //
+ ]
+ ]
+ |@sym_eq @not_divides_to_dividesb_false
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@(ord_O_to_not_divides … posm)
+ [@primeb_true_to_prime //
+ |<(ord_ord_rem n … posm … ltxn)
+ [@not_divides_to_ord_O
+ [@primeb_true_to_prime //
+ |@dividesb_false_to_not_divides //
+ ]
+ |@primeb_true_to_prime //
+ |@primeb_true_to_prime @(andb_true_l ?? Hcase)
+ ]
+ ]
+ ]
+ ]
+ |//
+ ]
+ |#x #ltxn #Hx @eq_f @ord_ord_rem //
+ [@primeb_true_to_prime @(andb_true_l ? ? Hcase)
+ |@primeb_true_to_prime @(andb_true_l ? ? Hx)
+ ]
+ ]
+ |@not_eq_to_le_to_lt
+ [elim (exists_max_forall_false (λi:nat.primeb i∧dividesb i (ord_rem m n)) (S(ord_rem m n)))
+ [* #Hex #Hord_rem cases Hex #x * #H6 #H7 % #H
+ >H in Hord_rem; #Hn @(absurd ?? (not_divides_ord_rem m n posm ?))
+ [@dividesb_true_to_divides @(andb_true_r ?? Hn)
+ |@prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hn)
+ ]
+ |* #Hall #Hmax >Hmax @lt_to_not_eq @prime_to_lt_O
+ @primeb_true_to_prime @(andb_true_l ?? Hcase)
+ ]
+ |@(transitive_le ? (max (S m) (λi:nat.primeb i∧dividesb i (ord_rem m n))))
+ [@le_to_le_max @le_S_S @(divides_to_le … posm) @(divides_ord_rem … posm)
+ @prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hcase)
+ |@(transitive_le ? (max (S m) (λi:nat.primeb i∧dividesb i m)))
+ [@le_max_f_max_g #i #ltim #Hi
+ cases (true_or_false (primeb i)) #Hprimei >Hprimei
+ [@divides_to_dividesb_true
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@(transitive_divides ? (ord_rem m n))
+ [@dividesb_true_to_divides @(andb_true_r ?? Hi)
+ |@(divides_ord_rem … posm)
+ @prime_to_lt_SO @primeb_true_to_prime
+ @(andb_true_l ?? Hcase)
+ ]
+ ]
+ |>Hprimei in Hi; #Hi @Hi
+ ]
+ |@le_S_S_to_le //
+ ]
+ ]
+ ]
+ |@(lt_O_ord_rem … posm) @prime_to_lt_SO
+ @primeb_true_to_prime @(andb_true_l ?? Hcase)
+ ]
+ |@prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hcase)
+ ]
+ |//
+ ]
+ |cases (le_to_or_lt_eq ?? posm) #Hm
+ [>bigop_Sfalse
+ [@(Hind … posm) @false_to_lt_max
+ [@(lt_to_le_to_lt ? (max (S m) (λi:nat.primeb i∧dividesb i m)))
+ [@lt_to_le @lt_1_max_prime //
+ |@le_S_S_to_le //
+ ]
+ |//
+ |@le_S_S_to_le //
+ ]
+ |@Hcase
+ ]
+ |<Hm
+ <(bigop_false (S n) ? 1 times (λp:nat.p\sup(ord 1 p))) in ⊢ (??%?);
+ @same_bigop
+ [#i #lein cases (true_or_false (primeb i)) #primei >primei //
+ @sym_eq @not_divides_to_dividesb_false
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |% #divi @(absurd ?? (le_to_not_lt i 1 ?))
+ [@prime_to_lt_SO @(primeb_true_to_prime ? primei)
+ |@divides_to_le //
+ ]
+ ]
+ |//
+ ]
+ ]
+ ]
+ ]
+qed.
+
+(* factorization of n into primes *)
+theorem pi_p_primeb_dividesb: ∀n. O < n →
+ n = ∏_{ p < S n | primeb p ∧ dividesb p n} (exp p (ord n p)).
+#n #posn @lt_max_to_pi_p_primeb // @lt_max_n @le_S @posn
+qed.
+
+theorem pi_p_primeb: ∀n. O < n →
+ n = ∏_{ p < (S n) | primeb p}(exp p (ord n p)).
+#n #posn <eq_pi_p_primeb_divides_b @pi_p_primeb_dividesb //
+qed.
+
+(* more result on order functions *)
+theorem le_ord_log: ∀n,p. O < n → 1 < p →
+ ord n p ≤ log p n.
+#n #p #posn #lt1p >(exp_ord p ? lt1p posn) in ⊢ (??(??%));
+>log_exp // @lt_O_ord_rem //
+qed.
+
+theorem sigma_p_dividesb:
+∀m,n,p. O < n → prime p → p ∤ n →
+m = ∑_{ i < m | dividesb (p\sup (S i)) ((exp p m)*n)} 1.
+#m elim m // -m #m #Hind #n #p #posn #primep #ndivp
+>bigop_Strue
+ [>commutative_plus <plus_n_Sm @eq_f <plus_n_O
+ >(Hind n p posn primep ndivp) in ⊢ (? ? % ?);
+ @same_bigop
+ [#i #ltim cases (true_or_false (dividesb (p\sup(S i)) (p\sup m*n))) #Hc >Hc
+ [@sym_eq @divides_to_dividesb_true
+ [@lt_O_exp @prime_to_lt_O //
+ |%{((exp p (m - i))*n)} <associative_times
+ <exp_plus_times @eq_f2 // @eq_f normalize @eq_f >commutative_plus
+ @plus_minus_m_m @lt_to_le //
+ ]
+ |@False_ind @(absurd ?? (dividesb_false_to_not_divides ? ? Hc))
+ %{((exp p (m - S i))*n)} <associative_times <exp_plus_times @eq_f2
+ [@eq_f >commutative_plus @plus_minus_m_m //
+ assumption
+ |%
+ ]
+ ]
+ |//
+ ]
+ |@divides_to_dividesb_true
+ [@lt_O_exp @prime_to_lt_O // | %{n} //]
+ ]
+qed.
+
+theorem sigma_p_dividesb1:
+∀m,n,p,k. O < n → prime p → p ∤ n → m ≤ k →
+ m = ∑_{i < k | dividesb (p\sup (S i)) ((exp p m)*n)} 1.
+#m #n #p #k #posn #primep #ndivp #lemk
+lapply (prime_to_lt_SO ? primep) #lt1p
+lapply (prime_to_lt_O ? primep) #posp
+>(sigma_p_dividesb m n p posn primep ndivp) in ⊢ (??%?);
+>(pad_bigop k m) // @same_bigop
+ [#i #ltik cases (true_or_false (leb m i)) #Him > Him
+ [whd in ⊢(??%?); @sym_eq
+ @not_divides_to_dividesb_false
+ [@lt_O_exp //
+ |lapply (leb_true_to_le … Him) -Him #Him
+ % #Hdiv @(absurd ?? (le_to_not_lt ?? Him))
+ (* <(ord_exp p m lt1p) *) >(plus_n_O m)
+ <(not_divides_to_ord_O ?? primep ndivp)
+ <(ord_exp p m lt1p)
+ <ord_times //
+ [whd <(ord_exp p (S i) lt1p)
+ @divides_to_le_ord //
+ [@lt_O_exp //
+ |>(times_n_O O) @lt_times // @lt_O_exp //
+ ]
+ |@lt_O_exp //
+ ]
+ ]
+ |%
+ ]
+ |//
+ ]
+qed.
+
+theorem eq_ord_sigma_p:
+∀n,m,x. O < n → prime x →
+exp x m ≤ n → n < exp x (S m) →
+ord n x= ∑_{i < m | dividesb (x\sup (S i)) n} 1.
+#n #m #x #posn #primex #Hexp #ltn
+lapply (prime_to_lt_SO ? primex) #lt1x
+>(exp_ord x n) in ⊢ (???%); // @sigma_p_dividesb1
+ [@lt_O_ord_rem //
+ |//
+ |@not_divides_ord_rem //
+ |@le_S_S_to_le @(le_to_lt_to_lt ? (log x n))
+ [@le_ord_log //
+ |@(lt_exp_to_lt x)
+ [@lt_to_le //
+ |@(le_to_lt_to_lt ? n ? ? ltn) @le_exp_log //
+ ]
+ ]
+ ]
+qed.
+
+theorem pi_p_primeb1: ∀n. O < n →
+n = ∏_{p < S n| primeb p}
+ (∏_{i < log p n| dividesb (exp p (S i)) n} p).
+#n #posn >(pi_p_primeb n posn) in ⊢ (??%?);
+@same_bigop
+ [//
+ |#p #ltp #primep >exp_sigma @eq_f @eq_ord_sigma_p
+ [//
+ |@primeb_true_to_prime //
+ |@le_exp_log //
+ |@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
+ ]
+ ]
+qed.
+
+(* the factorial function *)
+theorem eq_fact_pi_p:∀n.
+ fact n = ∏_{i < S n | leb 1 i} i.
+#n elim n // #n1 #Hind whd in ⊢ (??%?); >commutative_times >bigop_Strue
+ [@eq_f // | % ]
+qed.
+
+theorem eq_sigma_p_div: ∀n,q.O < q →
+ ∑_{ m < S n | leb (S O) m ∧ dividesb q m} 1 =n/q.
+#n #q #posq
+@(div_mod_spec_to_eq n q ? (n \mod q) ? (n \mod q))
+ [@div_mod_spec_intro
+ [@lt_mod_m_m //
+ |elim n
+ [normalize cases q //
+ |#n1 #Hind cases (or_div_mod1 n1 q posq)
+ [* #divq #eqn1 >divides_to_mod_O //
+ <plus_n_O >bigop_Strue
+ [>eqn1 in ⊢ (??%?); @eq_f2
+ [<commutative_plus <plus_n_Sm <plus_n_O @eq_f
+ @(div_mod_spec_to_eq n1 q (div n1 q) (mod n1 q) ? (mod n1 q))
+ [@div_mod_spec_div_mod //
+ |@div_mod_spec_intro [@lt_mod_m_m // | //]
+ ]
+ |%
+ ]
+ |@true_to_andb_true [//|@divides_to_dividesb_true //]
+ ]
+ |* #ndiv #eqn1 >bigop_Sfalse
+ [>(mod_S … posq)
+ [< plus_n_Sm @eq_f //
+ |cases (le_to_or_lt_eq (S (mod n1 q)) q ?)
+ [//
+ |#eqq @False_ind cases ndiv #Hdiv @Hdiv
+ %{(S(div n1 q))} <times_n_Sm <commutative_plus //
+ |@lt_mod_m_m //
+ ]
+ ]
+ |>not_divides_to_dividesb_false //
+ ]
+ ]
+ ]
+ ]
+ |@div_mod_spec_div_mod //
+ ]
+qed.
+
+lemma timesACdef: ∀n,m. timesAC n m = n * m.
+// qed-.
+
+(* still another characterization of the factorial *)
+theorem fact_pi_p: ∀n.
+fact n = ∏_{ p < S n | primeb p}
+ (∏_{i < log p n} (exp p (n /(exp p (S i))))).
+#n >eq_fact_pi_p
+@(trans_eq ??
+ (∏_{m < S n | leb 1 m}
+ (∏_{p < S m | primeb p}
+ (∏_{i < log p m | dividesb (exp p (S i)) m} p))))
+ [@same_bigop [// |#x #Hx1 #Hx2 @pi_p_primeb1 @leb_true_to_le //]
+ |@(trans_eq ??
+ (∏_{m < S n | leb 1 m}
+ (∏_{p < S m | primeb p ∧ leb p m}
+ (∏_{ i < log p m | dividesb ((p)\sup(S i)) m} p))))
+ [@same_bigop
+ [//
+ |#x #Hx1 #Hx2 @same_bigop
+ [#p #ltp >(le_to_leb_true … (le_S_S_to_le …ltp))
+ cases (primeb p) //
+ |//
+ ]
+ ]
+ |@(trans_eq ??
+ (∏_{m < S n | leb 1 m}
+ (∏_{p < S n | primeb p ∧ leb p m}
+ (∏_{i < log p m |dividesb ((p)\sup(S i)) m} p))))
+ [@same_bigop
+ [//
+ |#p #Hp1 #Hp2 @pad_bigop1
+ [@Hp1
+ |#i #lti #upi >lt_to_leb_false
+ [cases (primeb i) //|@lti]
+ ]
+ ]
+ |(* make a general theorem *)
+ @(trans_eq ??
+ (∏_{p < S n | primeb p}
+ (∏_{m < S n | leb p m}
+ (∏_{i < log p m | dividesb (p^(S i)) m} p))))
+ [@(bigop_commute … timesAC … (lt_O_S n) (lt_O_S n))
+ #i #j #lti #ltj
+ cases (true_or_false (primeb j ∧ leb j i)) #Hc >Hc
+ [>(le_to_leb_true 1 i)
+ [//
+ |@(transitive_le ? j)
+ [@prime_to_lt_O @primeb_true_to_prime @(andb_true_l ? ? Hc)
+ |@leb_true_to_le @(andb_true_r ?? Hc)
+ ]
+ ]
+ |cases(leb 1 i) //
+ ]
+ |@same_bigop
+ [//
+ |#p #Hp1 #Hp2
+ @(trans_eq ??
+ (∏_{m < S n | leb p m}
+ (∏_{i < log p n | dividesb (p\sup(S i)) m} p)))
+ [@same_bigop
+ [//
+ |#x #Hx1 #Hx2 @sym_eq
+ @sym_eq @pad_bigop1
+ [@le_log
+ [@prime_to_lt_SO @primeb_true_to_prime //
+ |@le_S_S_to_le //
+ ]
+ |#i #Hi1 #Hi2 @not_divides_to_dividesb_false
+ [@lt_O_exp @prime_to_lt_O @primeb_true_to_prime //
+ |@(not_to_not … (lt_to_not_le x (exp p (S i)) ?))
+ [#H @divides_to_le // @(lt_to_le_to_lt ? p)
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@leb_true_to_le //
+ ]
+ |@(lt_to_le_to_lt ? (exp p (S(log p x))))
+ [@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
+ |@le_exp
+ [@ prime_to_lt_O @primeb_true_to_prime //
+ |@le_S_S //
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ |@
+ (trans_eq ? ?
+ (∏_{i < log p n}
+ (∏_{m < S n | leb p m ∧ dividesb (p\sup(S i)) m} p)))
+ [@(bigop_commute ?????? nat 1 timesAC (λm,i.p) ???) //
+ cut (p ≤ n) [@le_S_S_to_le //] #lepn
+ @(lt_O_log … lepn) @(lt_to_le_to_lt … lepn) @prime_to_lt_SO
+ @primeb_true_to_prime //
+ |@same_bigop
+ [//
+ |#m #ltm #_ >exp_sigma @eq_f
+ @(trans_eq ??
+ (∑_{i < S n |leb 1 i∧dividesb (p\sup(S m)) i} 1))
+ [@same_bigop
+ [#i #lti
+ cases (true_or_false (dividesb (p\sup(S m)) i)) #Hc >Hc
+ [cases (true_or_false (leb p i)) #Hp3 >Hp3
+ [>le_to_leb_true
+ [//
+ |@(transitive_le ? p)
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@leb_true_to_le //
+ ]
+ ]
+ |>lt_to_leb_false
+ [//
+ |@not_le_to_lt
+ @(not_to_not ??? (leb_false_to_not_le ?? Hp3)) #posi
+ @(transitive_le ? (exp p (S m)))
+ [>(exp_n_1 p) in ⊢ (?%?);
+ @le_exp
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@le_S_S @le_O_n
+ ]
+ |@(divides_to_le … posi)
+ @dividesb_true_to_divides //
+ ]
+ ]
+ ]
+ |cases (leb p i) cases (leb 1 i) //
+ ]
+ |//
+ ]
+ |@eq_sigma_p_div @lt_O_exp
+ @prime_to_lt_O @primeb_true_to_prime //
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem fact_pi_p2: ∀n. fact (2*n) =
+∏_{p < S (2*n) | primeb p}
+ (∏_{i < log p (2*n)}
+ (exp p (2*(n /(exp p (S i))))*(exp p (mod (2*n /(exp p (S i))) 2)))).
+#n >fact_pi_p @same_bigop
+ [//
+ |#p #ltp #primep @same_bigop
+ [//
+ |#i #lti #_ <exp_plus_times @eq_f
+ >commutative_times in ⊢ (???(?%?));
+ cut (0 < p ^ (S i))
+ [@lt_O_exp @prime_to_lt_O @primeb_true_to_prime //]
+ generalize in match (p ^(S i)); #a #posa
+ >(div_times_times n a 2) // >(commutative_times n 2)
+ <eq_div_div_div_times //
+ ]
+ ]
+qed.
+
+theorem fact_pi_p3: ∀n. fact (2*n) =
+∏_{p < S (2*n) | primeb p}
+ (∏_{i < log p (2*n)}(exp p (2*(n /(exp p (S i)))))) *
+∏_{p < S (2*n) | primeb p}
+ (∏_{i < log p (2*n)}(exp p (mod (2*n /(exp p (S i))) 2))).
+#n <times_pi >fact_pi_p2 @same_bigop
+ [//
+ |#p #ltp #primep @times_pi
+ ]
+qed.
+
+theorem pi_p_primeb4: ∀n. 1 < n →
+∏_{p < S (2*n) | primeb p}
+ (∏_{i < log p (2*n)}(exp p (2*(n /(exp p (S i))))))
+=
+∏_{p < S n | primeb p}
+ (∏_{i < log p (2*n)}(exp p (2*(n /(exp p (S i)))))).
+#n #lt1n
+@sym_eq @(pad_bigop_nil … timesAC)
+ [@le_S_S /2 by /
+ |#i #ltn #lti %2
+ >log_i_2n //
+ [>bigop_Strue // whd in ⊢ (??(??%)?); <times_n_1
+ <exp_n_1 >eq_div_O //
+ |@le_S_S_to_le //
+ ]
+ ]
+qed.
+
+theorem pi_p_primeb5: ∀n. 1 < n →
+∏_{p < S (2*n) | primeb p}
+ (∏_{i < log p (2*n)} (exp p (2*(n /(exp p (S i))))))
+=
+∏_{p < S n | primeb p}
+ (∏_{i < log p n} (exp p (2*(n /(exp p (S i)))))).
+#n #lt1n >(pi_p_primeb4 ? lt1n) @same_bigop
+ [//
+ |#p #lepn #primebp @sym_eq @(pad_bigop_nil … timesAC)
+ [@le_log
+ [@prime_to_lt_SO @primeb_true_to_prime //
+ |//
+ ]
+ |#i #lelog #lti %2 >eq_div_O //
+ @(lt_to_le_to_lt ? (exp p (S(log p n))))
+ [@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
+ |@le_exp
+ [@prime_to_lt_O @primeb_true_to_prime // |@le_S_S //]
+ ]
+ ]
+ ]
+qed.
+
+theorem exp_fact_2: ∀n.
+exp (fact n) 2 =
+ ∏_{p < S n| primeb p}
+ (∏_{i < log p n} (exp p (2*(n /(exp p (S i)))))).
+#n >fact_pi_p <exp_pi @same_bigop
+ [//
+ |#p #ltp #primebp @sym_eq
+ @(trans_eq ?? (∏_{x < log p n} (exp (exp p (n/(exp p (S x)))) 2)))
+ [@same_bigop
+ [//
+ |#x #ltx #_ @sym_eq >commutative_times @exp_exp_times
+ ]
+ |@exp_pi
+ ]
+qed.
+
+definition B ≝ λn.
+∏_{p < S n | primeb p}
+ (∏_{i < log p n} (exp p (mod (n /(exp p (S i))) 2))).
+
+lemma Bdef : ∀n.B n =
+ ∏_{p < S n | primeb p}
+ (∏_{i < log p n} (exp p (mod (n /(exp p (S i))) 2))).
+// qed-.
+
+example B_SSSO: B 3 = 6. //
+qed.
+
+example B_SSSSO: B 4 = 6. //
+qed.
+
+example B_SSSSSO: B 5 = 30. //
+qed.
+
+example B_SSSSSSO: B 6 = 20. //
+qed.
+
+example B_SSSSSSSO: B 7 = 140. //
+qed.
+
+example B_SSSSSSSSO: B 8 = 70. //
+qed.
+
+theorem eq_fact_B:∀n. 1 < n →
+ (2*n)! = exp (n!) 2 * B(2*n).
+#n #lt1n >fact_pi_p3 @eq_f2
+ [@sym_eq >pi_p_primeb5 [@exp_fact_2|//] |// ]
+qed.
+
+theorem lt_SO_to_le_B_exp: ∀n. 1 < n →
+ B (2*n) ≤ exp 2 (pred (2*n)).
+#n #lt1n @(le_times_to_le (exp (fact n) 2))
+ [@lt_O_exp //
+ |<(eq_fact_B … lt1n) <commutative_times in ⊢ (??%);
+ >exp_2 <associative_times @fact_to_exp
+ ]
+qed.
+
+theorem le_B_exp: ∀n.
+ B (2*n) ≤ exp 2 (pred (2*n)).
+#n cases n
+ [@le_n|#n1 cases n1 [@le_n |#n2 @lt_SO_to_le_B_exp @le_S_S @lt_O_S]]
+qed.
+
+theorem lt_4_to_le_B_exp: ∀n.4 < n →
+ B (2*n) \le exp 2 ((2*n)-2).
+#n #lt4n @(le_times_to_le (exp (fact n) 2))
+ [@lt_O_exp //
+ |<eq_fact_B
+ [<commutative_times in ⊢ (??%); >exp_2 <associative_times
+ @lt_4_to_fact //
+ |@lt_to_le @lt_to_le @lt_to_le //
+ ]
+ ]
+qed.
+
+theorem lt_1_to_le_exp_B: ∀n. 1 < n →
+ exp 2 (2*n) ≤ 2 * n * B (2*n).
+#n #lt1n
+@(le_times_to_le (exp (fact n) 2))
+ [@lt_O_exp @le_1_fact
+ |<associative_times in ⊢ (??%); >commutative_times in ⊢ (??(?%?));
+ >associative_times in ⊢ (??%); <(eq_fact_B … lt1n)
+ <commutative_times; @exp_to_fact2 @lt_to_le //
+ ]
+qed.
+
+theorem le_exp_B: ∀n. O < n →
+ exp 2 (2*n) ≤ 2 * n * B (2*n).
+#n #posn cases posn
+ [@le_n |#m #lt1m @lt_1_to_le_exp_B @le_S_S // ]
+qed.
\ No newline at end of file
--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A||
+ \ / This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ V_____________________________________________________________*)
+
+include "arithmetics/chebyshev/chebyshev_psi.ma".
+include "arithmetics/chebyshev/factorization.ma".
+
+theorem le_B_Psi: ∀n. B n ≤ Psi n.
+#n >eq_Psi_Psi' @le_pi #p #ltp #primep
+@le_pi #i #lti #_ >(exp_n_1 p) in ⊢ (??%); @le_exp
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@le_S_S_to_le @lt_mod_m_m @lt_O_S
+ ]
+qed.
+
+theorem le_B_Psi4: ∀n. O < n → 2 * B (4*n) ≤ Psi (4*n).
+#n #posn >eq_Psi_Psi'
+cut (2 < (S (4*n)))
+ [@le_S_S >(times_n_1 2) in ⊢ (?%?); @le_times //] #H2
+cut (O<log 2 (4*n))
+ [@lt_O_log [@le_S_S_to_le @H2 |@le_S_S_to_le @H2]] #Hlog
+>Bdef >(bigop_diff ??? timesAC ? 2 ? H2) [2:%]
+>Psidef >(bigop_diff ??? timesAC ? 2 ? H2) in ⊢ (??%); [2:%]
+<associative_times @le_times
+ [>(bigop_diff ??? timesAC ? 0 ? Hlog) [2://]
+ >(bigop_diff ??? timesAC ? 0 ? Hlog) in ⊢ (??%); [2:%]
+ <associative_times >timesACdef @le_times
+ [<exp_n_1 cut (4=2*2) [//] #H4 >H4 >associative_times
+ >commutative_times in ⊢ (?(??(??(?(?%?)?)))?);
+ >div_times [2://] >divides_to_mod_O
+ [@le_n |%{n} // |@lt_O_S]
+ |@le_pi #i #lti #H >(exp_n_1 2) in ⊢ (??%);
+ @le_exp [@lt_O_S |@le_S_S_to_le @lt_mod_m_m @lt_O_S]
+ ]
+ |@le_pi #p #ltp #Hp @le_pi #i #lti #H
+ >(exp_n_1 p) in ⊢ (??%); @le_exp
+ [@prime_to_lt_O @primeb_true_to_prime @(andb_true_r ?? Hp)
+ |@le_S_S_to_le @lt_mod_m_m @lt_O_S
+ ]
+ ]
+qed.
+
+let rec bool_to_nat b ≝
+ match b with [true ⇒ 1 | false ⇒ 0].
+
+theorem eq_Psi_2_n: ∀n.O < n →
+Psi(2*n) =
+ ∏_{p <S (2*n) | primeb p}
+ (∏_{i <log p (2*n)} (exp p (bool_to_nat (leb (S n) (exp p (S i)))))) *Psi n.
+#n #posn >eq_Psi_Psi' > eq_Psi_Psi'
+cut (
+ ∏_{p < S n | primeb p} (∏_{i <log p n} p)
+ = ∏_{p < S (2*n) | primeb p}
+ (∏_{i <log p (2*n)} (p\sup(bool_to_nat (¬ (leb (S n) (exp p (S i))))))))
+ [2: #Hcut >Psidef in ⊢ (???%); >Hcut
+ <times_pi >Psidef @same_bigop
+ [//
+ |#p #lt1p #primep <times_pi @same_bigop
+ [//
+ |#i #lt1i #_ cases (true_or_false (leb (S n) (exp p (S i)))) #Hc >Hc
+ [normalize <times_n_1 >plus_n_O //
+ |normalize <plus_n_O <plus_n_O //
+ ]
+ ]
+ ]
+ |@(trans_eq ??
+ (∏_{p < S n | primeb p}
+ (∏_{i < log p n} (p \sup(bool_to_nat (¬leb (S n) (exp p (S i))))))))
+ [@same_bigop
+ [//
+ |#p #lt1p #primep @same_bigop
+ [//
+ |#i #lti#_ >lt_to_leb_false
+ [normalize @plus_n_O
+ |@le_S_S @(transitive_le ? (exp p (log p n)))
+ [@le_exp [@prime_to_lt_O @primeb_true_to_prime //|//]
+ |@le_exp_log //
+ ]
+ ]
+ ]
+ ]
+ |@(trans_eq ??
+ (∏_{p < S (2*n) | primeb p}
+ (∏_{i <log p n} (p \sup(bool_to_nat (¬leb (S n) (p \sup(S i))))))))
+ [@(pad_bigop_nil … timesA)
+ [@le_S_S //|#i #lti #upi %2 >lt_to_log_O //]
+ |@same_bigop
+ [//
+ |#p #ltp #primep @(pad_bigop_nil … timesA)
+ [@le_log
+ [@prime_to_lt_SO @primeb_true_to_prime //|//]
+ |#i #lei #iup %2 >le_to_leb_true
+ [%
+ |@(lt_to_le_to_lt ? (exp p (S (log p n))))
+ [@lt_exp_log @prime_to_lt_SO @primeb_true_to_prime //
+ |@le_exp
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@le_S_S //
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem le_Psi_BPsi1: ∀n. O < n →
+ Psi(2*n) ≤ B(2*n)*Psi n.
+#n #posn >(eq_Psi_2_n … posn) @le_times [2:@le_n]
+>Bdef @le_pi #p #ltp #primep @le_pi #i #lti #_ @le_exp
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |cases (true_or_false (leb (S n) (exp p (S i)))) #Hc >Hc
+ [whd in ⊢ (?%?);
+ cut (2*n/p\sup(S i) = 1) [2: #Hcut >Hcut @le_n]
+ @(div_mod_spec_to_eq (2*n) (exp p (S i))
+ ? (mod (2*n) (exp p (S i))) ? (minus (2*n) (exp p (S i))) )
+ [@div_mod_spec_div_mod @lt_O_exp @prime_to_lt_O @primeb_true_to_prime //
+ |cut (p\sup(S i)≤2*n)
+ [@(transitive_le ? (exp p (log p (2*n))))
+ [@le_exp [@prime_to_lt_O @primeb_true_to_prime // | //]
+ |@le_exp_log >(times_n_O O) in ⊢ (?%?); @lt_times //
+ ]
+ ] #Hcut
+ @div_mod_spec_intro
+ [@lt_plus_to_minus
+ [// |normalize in ⊢ (? % ?); < plus_n_O @lt_plus @leb_true_to_le //]
+ |>commutative_plus >commutative_times in ⊢ (???(??%));
+ < times_n_1 @plus_minus_m_m //
+ ]
+ ]
+ |@le_O_n
+ ]
+ ]
+qed.
+
+theorem le_Psi_BPsi: ∀n. Psi(2*n) \le B(2*n)*Psi n.
+#n cases n [@le_n |#m @le_Psi_BPsi1 @lt_O_S]
+qed.
+
+theorem le_Psi_exp: ∀n. Psi(2*n) ≤ (exp 2 (pred (2*n)))*Psi n.
+#n @(transitive_le ? (B(2*n)*Psi n))
+ [@le_Psi_BPsi |@le_times [@le_B_exp |//]]
+qed.
+
+theorem lt_4_to_le_Psi_exp: ∀n. 4 < n →
+ Psi(2*n) ≤ exp 2 ((2*n)-2)*Psi n.
+#n #lt4n @(transitive_le ? (B(2*n)*Psi n))
+ [@le_Psi_BPsi|@le_times [@(lt_4_to_le_B_exp … lt4n) |@le_n]]
+qed.
+
+(* two technical lemmas *)
+lemma times_2_pred: ∀n. 2*(pred n) \le pred (2*n).
+#n cases n
+ [@le_n|#m @monotonic_le_plus_r @le_n_Sn]
+qed.
+
+lemma le_S_times_2: ∀n. O < n → S n ≤ 2*n.
+#n #posn elim posn
+ [@le_n
+ |#m #posm #Hind
+ cut (2*(S m) = S(S(2*m))) [normalize <plus_n_Sm //] #Hcut >Hcut
+ @le_S_S @(transitive_le … Hind) @le_n_Sn
+ ]
+qed.
+
+theorem le_Psi_exp1: ∀n.
+ Psi(exp 2 n) ≤ exp 2 ((2*(exp 2 n)-(S(S n)))).
+#n elim n
+ [@le_n
+ |#n1 #Hind whd in ⊢ (?(?%)?); >commutative_times
+ @(transitive_le ??? (le_Psi_exp ?))
+ @(transitive_le ? (2\sup(pred (2*2^n1))*2^(2*2^n1-(S(S n1)))))
+ [@monotonic_le_times_r //
+ |<exp_plus_times @(le_exp … (lt_O_S ?))
+ cut (S(S n1) ≤ 2*(exp 2 n1))
+ [elim n1
+ [@le_n
+ |#n2 >commutative_times in ⊢ (%→?); #Hind1 @(transitive_le ? (2*(S(S n2))))
+ [@le_S_times_2 @lt_O_S |@monotonic_le_times_r //]
+ ]
+ ] #Hcut
+ @le_S_S_to_le cut(∀a,b. S a + b = S (a+b)) [//] #Hplus <Hplus >S_pred
+ [>eq_minus_S_pred in ⊢ (??%); >S_pred
+ [>plus_minus_commutative
+ [@monotonic_le_minus_l
+ cut (∀a. 2*a = a + a) [//] #times2 <times2
+ @monotonic_le_times_r >commutative_times @le_n
+ |@Hcut
+ ]
+ |@lt_plus_to_minus_r whd in ⊢ (?%?);
+ @(lt_to_le_to_lt ? (2*(S(S n1))))
+ [>(times_n_1 (S(S n1))) in ⊢ (?%?); >commutative_times
+ @monotonic_lt_times_l [@lt_O_S |@le_n]
+ |@monotonic_le_times_r whd in ⊢ (??%); //
+ ]
+ ]
+ |whd >(times_n_1 1) in ⊢ (?%?); @le_times
+ [@le_n_Sn |@lt_O_exp @lt_O_S]
+ ]
+ ]
+ ]
+qed.
+
+theorem monotonic_Psi: monotonic nat le Psi.
+#n #m #lenm elim lenm
+ [@le_n
+ |#n1 #len #Hind @(transitive_le … Hind)
+ cut (∏_{p < S n1 | primeb p}(p^(log p n1))
+ ≤∏_{p < S n1 | primeb p}(p^(log p (S n1))))
+ [@le_pi #p #ltp #primep @le_exp
+ [@prime_to_lt_O @primeb_true_to_prime //
+ |@le_log [@prime_to_lt_SO @primeb_true_to_prime // | //]
+ ]
+ ] #Hcut
+ >psi_def in ⊢ (??%); cases (true_or_false (primeb (S n1))) #Hc
+ [>bigop_Strue in ⊢ (??%); [2://]
+ >(times_n_1 (Psi n1)) >commutative_times @le_times
+ [@lt_O_exp @lt_O_S |@Hcut]
+ |>bigop_Sfalse in ⊢ (??%); //
+ ]
+ ]
+qed.
+
+(* example *)
+example Psi_1: Psi 1 = 1. // qed.
+
+example Psi_2: Psi 2 = 2. // qed.
+
+example Psi_3: Psi 3 = 6. // qed.
+
+example Psi_4: Psi 4 = 12. // qed.
+
+theorem le_Psi_exp4: ∀n. 1 < n →
+ Psi(n) ≤ (pred n)*exp 2 ((2 * n) -3).
+#n @(nat_elim1 n)
+#m #Hind cases (even_or_odd m)
+#a * #Hm >Hm #Hlt
+ [cut (0<a)
+ [cases a in Hlt;
+ [whd in ⊢ (??%→?); #lt10 @False_ind @(absurd ? lt10 (not_le_Sn_O 1))
+ |#b #_ //]
+ ] #Hcut
+ cases (le_to_or_lt_eq … Hcut) #Ha
+ [@(transitive_le ? (exp 2 (pred(2*a))*Psi a))
+ [@le_Psi_exp
+ |@(transitive_le ? (2\sup(pred(2*a))*((pred a)*2\sup((2*a)-3))))
+ [@monotonic_le_times_r @(Hind ?? Ha)
+ >Hm >(times_n_1 a) in ⊢ (?%?); >commutative_times
+ @monotonic_lt_times_l [@lt_to_le // |@le_n]
+ |<Hm <associative_times >commutative_times in ⊢ (?(?%?)?);
+ >associative_times; @le_times
+ [>Hm cases a[@le_n|#b normalize @le_plus_n_r]
+ |<exp_plus_times @le_exp
+ [@lt_O_S
+ |@(transitive_le ? (m+(m-3)))
+ [@monotonic_le_plus_l //
+ |normalize <plus_n_O >plus_minus_commutative
+ [@le_n
+ |>Hm @(transitive_le ? (2*2) ? (le_n_Sn 3))
+ @monotonic_le_times_r //
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ |<Ha normalize @le_n
+ ]
+ |cases (le_to_or_lt_eq O a (le_O_n ?)) #Ha
+ [@(transitive_le ? (Psi (2*(S a))))
+ [@monotonic_Psi >Hm normalize <plus_n_Sm @le_n_Sn
+ |@(transitive_le … (le_Psi_exp ?) )
+ @(transitive_le ? ((2\sup(pred (2*S a)))*(a*(exp 2 ((2*(S a))-3)))))
+ [@monotonic_le_times_r @Hind
+ [>Hm @le_S_S >(times_n_1 a) in ⊢ (?%?); >commutative_times
+ @monotonic_lt_times_l //
+ |@le_S_S //
+ ]
+ |cut (pred (S (2*a)) = 2*a) [//] #Spred >Spred
+ cut (pred (2*(S a)) = S (2 * a)) [normalize //] #Spred1 >Spred1
+ cut (2*(S a) = S(S(2*a))) [normalize <plus_n_Sm //] #times2
+ cut (exp 2 (2*S a -3) = 2*(exp 2 (S(2*a) -3)))
+ [>(commutative_times 2) in ⊢ (???%); >times2 >minus_Sn_m [%]
+ @le_S_S >(times_n_1 2) in ⊢ (?%?); @monotonic_le_times_r @Ha
+ ] #Hcut >Hcut
+ <associative_times in ⊢ (? (? ? %) ?); <associative_times
+ >commutative_times in ⊢ (? (? % ?) ?);
+ >commutative_times in ⊢ (? (? (? % ?) ?) ?);
+ >associative_times @monotonic_le_times_r
+ <exp_plus_times @(le_exp … (lt_O_S ?))
+ >plus_minus_commutative
+ [normalize >(plus_n_O (a+(a+0))) in ⊢ (?(?(??%)?)?); @le_n
+ |@le_S_S >(times_n_1 2) in ⊢ (?%?); @monotonic_le_times_r @Ha
+ ]
+ ]
+ ]
+ |@False_ind <Ha in Hlt; normalize #Hfalse @(absurd ? Hfalse) //
+ ]
+ ]
+qed.
+
+theorem le_n_8_to_le_Psi_exp: ∀n. n ≤ 8 →
+ Psi(n) ≤ exp 2 ((2 * n) -3).
+#n cases n
+ [#_ @le_n
+ |#n1 cases n1
+ [#_ @le_n
+ |#n2 cases n2
+ [#_ @le_n
+ |#n3 cases n3
+ [#_ @leb_true_to_le //
+ |#n4 cases n4
+ [#_ @leb_true_to_le //
+ |#n5 cases n5
+ [#_ @leb_true_to_le //
+ |#n6 cases n6
+ [#_ @leb_true_to_le //
+ |#n7 cases n7
+ [#_ @leb_true_to_le //
+ |#n8 cases n8
+ [#_ @leb_true_to_le //
+ |#n9 #H @False_ind @(absurd ?? (lt_to_not_le ?? H))
+ @leb_true_to_le //
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
+
+theorem le_Psi_exp5: ∀n. Psi(n) ≤ exp 2 ((2 * n) -3).
+#n @(nat_elim1 n) #m #Hind
+cases (decidable_le 9 m)
+ [#lem cases (even_or_odd m) #a * #Hm
+ [>Hm in ⊢ (?%?); @(transitive_le … (le_Psi_exp ?))
+ @(transitive_le ? (2\sup(pred(2*a))*(2\sup((2*a)-3))))
+ [@monotonic_le_times_r @Hind >Hm >(times_n_1 a) in ⊢ (?%?);
+ >commutative_times @(monotonic_lt_times_l … (le_n ?))
+ @(transitive_lt ? 3)
+ [@lt_O_S |@(le_times_to_le 2) [@lt_O_S |<Hm @lt_to_le @lem]]
+ |<Hm <exp_plus_times @(le_exp … (lt_O_S ?))
+ whd in match (times 2 m); >commutative_times <times_n_1
+ <plus_minus_commutative
+ [@monotonic_le_plus_l @le_pred_n
+ |@(transitive_le … lem) @leb_true_to_le //
+ ]
+ ]
+ |@(transitive_le ? (Psi (2*(S a))))
+ [@monotonic_Psi >Hm normalize <plus_n_Sm @le_n_Sn
+ |@(transitive_le ? ((exp 2 ((2*(S a))-2))*Psi (S a)))
+ [@lt_4_to_le_Psi_exp @le_S_S
+ @(le_times_to_le 2)[@le_n_Sn|@le_S_S_to_le <Hm @lem]
+ |@(transitive_le ? ((2\sup((2*S a)-2))*(exp 2 ((2*(S a))-3))))
+ [@monotonic_le_times_r @Hind >Hm @le_S_S
+ >(times_n_1 a) in ⊢ (?%?);
+ >commutative_times @(monotonic_lt_times_l … (le_n ?))
+ @(transitive_lt ? 3)
+ [@lt_O_S |@(le_times_to_le 2) [@lt_O_S |@le_S_S_to_le <Hm @lem]]
+ |cut (∀a. 2*(S a) = S(S(2*a))) [normalize #a <plus_n_Sm //] #times2
+ >times2 <Hm <exp_plus_times @(le_exp … (lt_O_S ?))
+ cases m
+ [@le_n
+ |#n1 cases n1
+ [@le_n
+ |#n2 normalize <minus_n_O <plus_n_O <plus_n_Sm
+ normalize <minus_n_O <plus_n_Sm @le_n
+ ]
+ ]
+ ]
+ ]
+ ]
+ ]
+ |#H @le_n_8_to_le_Psi_exp @le_S_S_to_le @not_le_to_lt //
+ ]
+qed.
+
+theorem le_exp_Psil:∀n. O < n → exp 2 n ≤ Psi (2 * n).
+#n #posn @(transitive_le ? ((exp 2 (2*n))/(2*n)))
+ [@le_times_to_le_div
+ [>(times_n_O O) in ⊢ (?%?); @lt_times [@lt_O_S|//]
+ |normalize in ⊢ (??(??%)); < plus_n_O >exp_plus_times
+ @le_times [2://] elim posn [//]
+ #m #le1m #Hind whd in ⊢ (??%); >commutative_times in ⊢ (??%);
+ @monotonic_le_times_r @(transitive_le … Hind)
+ >(times_n_1 m) in ⊢ (?%?); >commutative_times
+ @(monotonic_lt_times_l … (le_n ?)) @le1m
+ ]
+ |@le_times_to_le_div2
+ [>(times_n_O O) in ⊢ (?%?); @lt_times [@lt_O_S|//]
+ |@(transitive_le ? ((B (2*n)*(2*n))))
+ [<commutative_times in ⊢ (??%); @le_exp_B //
+ |@le_times [@le_B_Psi|@le_n]
+ ]
+ ]
+ ]
+qed.
+
+theorem le_exp_Psi2:∀n. 1 < n → exp 2 (n / 2) \le Psi n.
+#n #lt1n @(transitive_le ? (Psi(n/2*2)))
+ [>commutative_times @le_exp_Psil
+ cases (le_to_or_lt_eq ? ? (le_O_n (n/2))) [//]
+ #Heq @False_ind @(absurd ?? (lt_to_not_le ?? lt1n))
+ >(div_mod n 2) <Heq whd in ⊢ (?%?);
+ @le_S_S_to_le @(lt_mod_m_m n 2) @lt_O_S
+ |@monotonic_Psi >(div_mod n 2) in ⊢ (??%); @le_plus_n_r
+ ]
+qed.
+
+theorem eq_sigma_pi_SO_n: ∀n.∑_{i < n} 1 = n.
+#n elim n //
+qed.
+
+theorem lePsi_prim: ∀n.
+ exp n (prim n) \le Psi n * ∏_{p < S n | primeb p} p.
+#n <(exp_sigma (S n) n primeb) <times_pi @le_pi
+#p #ltp #primep @lt_to_le @lt_exp_log
+@prime_to_lt_SO @primeb_true_to_prime //
+qed.
+
+theorem le_prim_log : ∀n,b. 1 < b →
+ log b (Psi n) ≤ prim n * (S (log b n)).
+#n #b #lt1b @(transitive_le … (log_exp1 …)) [@le_log // | //]
+qed.
+
+(*********************** the inequalities ***********************)
+lemma exp_Sn: ∀b,n. exp b (S n) = b * (exp b n).
+normalize //
+qed.
+
+theorem le_exp_priml: ∀n. O < n →
+ exp 2 (2*n) ≤ exp (2*n) (S(prim (2*n))).
+#n #posn @(transitive_le ? (((2*n*(B (2*n))))))
+ [@le_exp_B //
+ |>exp_Sn @monotonic_le_times_r @(transitive_le ? (Psi (2*n)))
+ [@le_B_Psi |@le_Psil]
+ ]
+qed.
+
+theorem le_exp_prim4l: ∀n. O < n →
+ exp 2 (S(4*n)) ≤ exp (4*n) (S(prim (4*n))).
+#n #posn @(transitive_le ? (2*(4*n*(B (4*n)))))
+ [>exp_Sn @monotonic_le_times_r
+ cut (4*n = 2*(2*n)) [<associative_times //] #Hcut
+ >Hcut @le_exp_B @lt_to_le whd >(times_n_1 2) in ⊢ (?%?);
+ @monotonic_le_times_r //
+ |>exp_Sn <associative_times >commutative_times in ⊢ (?(?%?)?);
+ >associative_times @monotonic_le_times_r @(transitive_le ? (Psi (4*n)))
+ [@le_B_Psi4 // |@le_Psil]
+ ]
+qed.
+
+theorem le_priml: ∀n. O < n →
+ 2*n ≤ (S (log 2 (2*n)))*S(prim (2*n)).
+#n #posn <(eq_log_exp 2 (2*n) ?) in ⊢ (?%?);
+ [@(transitive_le ? ((log 2) (exp (2*n) (S(prim (2*n))))))
+ [@le_log [@le_n |@le_exp_priml //]
+ |>commutative_times in ⊢ (??%); @log_exp1 @le_n
+ ]
+ |@le_n
+ ]
+qed.
+
+theorem le_exp_primr: ∀n.
+ exp n (prim n) ≤ exp 2 (2*(2*n-3)).
+#n @(transitive_le ? (exp (Psi n) 2))
+ [>exp_Sn >exp_Sn whd in match (exp ? 0); <times_n_1 @lePsi_r2
+ |>commutative_times <exp_exp_times @le_exp1 [@lt_O_S |@le_Psi_exp5]
+ ]
+qed.
+
+(* bounds *)
+theorem le_primr: ∀n. 1 < n → prim n \le 2*(2*n-3)/log 2 n.
+#n #lt1n @le_times_to_le_div
+ [@lt_O_log //
+ |@(transitive_le ? (log 2 (exp n (prim n))))
+ [>commutative_times @log_exp2
+ [@le_n |@lt_to_le //]
+ |<(eq_log_exp 2 (2*(2*n-3))) in ⊢ (??%);
+ [@le_log [@le_n |@le_exp_primr]
+ |@le_n
+ ]
+ ]
+ ]
+qed.
+
+theorem le_priml1: ∀n. O < n →
+ 2*n/((log 2 n)+2) - 1 ≤ prim (2*n).
+#n #posn @le_plus_to_minus @le_times_to_le_div2
+ [>commutative_plus @lt_O_S
+ |>commutative_times in ⊢ (??%); <plus_n_Sm <plus_n_Sm in ⊢ (??(??%));
+ <plus_n_O <commutative_plus <log_exp
+ [@le_priml // | //| @le_n]
+ ]
+qed.
+
+
+
+
projects / helm.git / commitdiff