+++ /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/big_pi.ma".
-include "arithmetics/ord.ma".
-
-(* include "nat/factorization.ma".
-include "nat/factorial2.ma".
-include "nat/o.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.
-
-(* axiom daemon : ∀P:Prop.P. *)
-
-(* 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 A: nat → nat ≝
- λn.∏_{p < S n | primeb p} (exp p (log p n)).
-
-definition psi_def : ∀n.
- A n = ∏_{p < S n | primeb p} (exp p (log p n)).
-// qed.
-
-theorem le_Al1: ∀n.
- A n ≤ ∏_{p < S n | primeb p} n.
-#n cases n [@le_n |#m @le_pi #i #_ #_ @le_exp_log //]
-qed.
-
-theorem le_Al: ∀n. A n ≤ exp n (prim n).
-#n <exp_sigma @le_Al1
-qed.
-
-theorem leA_r2: ∀n.
- exp n (prim n) ≤ A n * A 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 A': nat → nat ≝
-λn. ∏_{p < S n | primeb p} (∏_{i < log p n} p).
-
-lemma Adef: ∀n. A' n = ∏_{p < S n | primeb p} (∏_{i < log p n} p).
-// qed-.
-
-theorem eq_A_A': ∀n.A n = A' 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.
-
-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.
-
-(* integrations to minimization *)
-theorem false_to_lt_max: ∀f,n,m.O < n →
- f n = false → max m f ≤ n → max m f < n.
-#f #n #m #posn #Hfn #Hmax cases (le_to_or_lt_eq ?? Hmax) -Hmax #Hmax
- [//
- |cases (exists_max_forall_false f m)
- [* #_ #Hfmax @False_ind @(absurd ?? not_eq_true_false) //
- |* //
- ]
- ]
-qed.
-
-(* boh ...
-theorem lt_max_to_false : ∀f,n,m.
- max n f < m → m ≤ n → f m = false.
-#f #n elim n
- [#m #H1 #H2 @False_ind @(absurd ? H2) @lt_to_not_le //
- |#n1 #Hind #m whd in ⊢ (?%?→?); #Hmax #ltm
-elim (max_S_max f n1); in H1 ⊢ %.
-elim H1.
-absurd (m \le S n1).assumption.
-apply lt_to_not_le.rewrite < H5.assumption.
-elim H1.
-apply (le_n_Sm_elim m n1 H2).
-intro.
-apply H.rewrite < H5.assumption.
-apply le_S_S_to_le.assumption.
-intro.rewrite > H6.assumption.
-qed. *)
-
-(* integrations to minimization
-lemma lt_1_max_prime: ∀n. 1 < n →
- 1 < max 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_smallest_factor_n
- |apply true_to_true_to_andb_true
- [apply prime_to_primeb_true.
- apply prime_smallest_factor_n.
- assumption
- |apply divides_to_divides_b_true
- [apply lt_O_smallest_factor.apply lt_to_le.assumption
- |apply divides_smallest_factor_n.
- apply lt_to_le.assumption
- ]
- ]
- ]
- ]
-qed. *)
-
-theorem lt_max_to_pi_p_primeb: ∀q,m. O<m → max 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)) (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 m (λi:nat.primeb i∧dividesb i (ord_rem m n))))
- [@le_to_le_max @(divides_to_le … posm) @(divides_ord_rem … posm)
- @prime_to_lt_SO @primeb_true_to_prime @(andb_true_l ?? Hcase)
- |@(transitive_le ? (max 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 m (λi:nat.primeb i∧dividesb i m)))
- [@lt_to_le @dae (* portare @lt_SO_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 // @le_S_S @le_max_n
-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.
-
-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 //
- ]
- |(* @sym_eq *)
- @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.
-
-definition Atimes ≝ mk_Aop nat 1 times ???.
- [#a normalize <plus_n_O %
- |#a @sym_eq @times_n_1
- |#a #b #c @sym_eq @associative_times
- ]
-qed.
-
-definition ACtimes ≝ mk_ACop nat 1 Atimes commutative_times.
-
-lemma ACtimesdef: ∀n,m. ACtimes 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 @sym_eq (* COMPLETARE
- apply false_to_eq_pi_p
- [assumption
- |intros.rewrite > lt_to_leb_false
- [elim primeb;reflexivity|assumption]
- ] *)
- @daemon
- ]
- |(* make a general theorem *)
- @(trans_eq ??
- (∏_{p < S n | primeb p}
- (∏_{m < S n | leb p m}
- (∏_{i < log p m | dividesb ((p)\sup(S i)) m} p))))
- [@daemon
- (* PORTARE
- @pi_p_pi_p1.
- intros.
- apply (bool_elim ? (primeb y \landy x));intros
- [rewrite > (le_to_leb_true (S O) x)
- [reflexivity
- |apply (trans_le ? y)
- [apply prime_to_lt_O.
- apply primeb_true_to_prime.
- apply (andb_true_true ? ? H2)
- |apply leb_true_to_le.
- apply (andb_true_true_r ? ? H2)
- ]
- ]
- |elim (leb (S O) x);reflexivity
- ] *)
- |@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 ACtimes (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 … ACtimes)
- [@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 … ACtimes)
- [@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 le_B_A: ∀n. B n ≤ A n.
-#n >eq_A_A' @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_A4: ∀n. O < n → 2 * B (4*n) ≤ A (4*n).
-#n #posn >eq_A_A'
-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 ??? ACtimes ? 2 ? H2) [2:%]
->Adef >(bigop_diff ??? ACtimes ? 2 ? H2) in ⊢ (??%); [2:%]
-<associative_times @le_times
- [>(bigop_diff ??? ACtimes ? 0 ? Hlog) [2://]
- >(bigop_diff ??? ACtimes ? 0 ? Hlog) in ⊢ (??%); [2:%]
- <associative_times >ACtimesdef @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.
-
-(* not usefull
-theorem le_fact_A:\forall n.S O < n \to
-fact (2*n) \le exp (fact n) 2 * A (2*n).
-intros.
-rewrite > eq_fact_B
- [apply le_times_r.
- apply le_B_A
- |assumption
- ]
-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.
-
-let rec bool_to_nat b ≝
- match b with [true ⇒ 1 | false ⇒ 0].
-
-theorem eq_A_2_n: ∀n.O < n →
-A(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)))))) *A n.
-#n #posn >eq_A_A' > eq_A_A'
-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 >Adef in ⊢ (???%); >Hcut
- <times_pi >Adef @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 … Atimes)
- [@le_S_S //|#i #lti #upi %2 >lt_to_log_O //]
- |@same_bigop
- [//
- |#p #ltp #primep @(pad_bigop_nil … Atimes)
- [@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_A_BA1: ∀n. O < n →
- A(2*n) ≤ B(2*n)*A n.
-#n #posn >(eq_A_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_A_BA: ∀n. A(2*n) \le B(2*n)*A n.
-#n cases n [@le_n |#m @le_A_BA1 @lt_O_S]
-qed.
-
-theorem le_A_exp: ∀n. A(2*n) ≤ (exp 2 (pred (2*n)))*A n.
-#n @(transitive_le ? (B(2*n)*A n))
- [@le_A_BA |@le_times [@le_B_exp |//]]
-qed.
-
-theorem lt_4_to_le_A_exp: ∀n. 4 < n →
- A(2*n) ≤ exp 2 ((2*n)-2)*A n.
-#n #lt4n @(transitive_le ? (B(2*n)*A n))
- [@le_A_BA|@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_A_exp1: ∀n.
- A(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_A_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_A: monotonic nat le A.
-#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 (A n1)) >commutative_times @le_times
- [@lt_O_exp @lt_O_S |@Hcut]
- |>bigop_Sfalse in ⊢ (??%); //
- ]
- ]
-qed.
-
-(*
-theorem le_A_exp2: \forall n. O < n \to
-A(n) \le (exp (S(S O)) ((S(S O)) * (S(S O)) * n)).
-intros.
-apply (trans_le ? (A (exp (S(S O)) (S(log (S(S O)) n)))))
- [apply monotonic_A.
- apply lt_to_le.
- apply lt_exp_log.
- apply le_n
- |apply (trans_le ? ((exp (S(S O)) ((S(S O))*(exp (S(S O)) (S(log (S(S O)) n)))))))
- [apply le_A_exp1
- |apply le_exp
- [apply lt_O_S
- |rewrite > assoc_times.apply le_times_r.
- change with ((S(S O))*((S(S O))\sup(log (S(S O)) n))≤(S(S O))*n).
- apply le_times_r.
- apply le_exp_log.
- assumption
- ]
- ]
- ]
-qed.
-*)
-
-(* example *)
-example A_1: A 1 = 1. // qed.
-
-example A_2: A 2 = 2. // qed.
-
-example A_3: A 3 = 6. // qed.
-
-example A_4: A 4 = 12. // qed.
-
-(*
-(* a better result *)
-theorem le_A_exp3: \forall n. S O < n \to
-A(n) \le exp (pred n) (2*(exp 2 (2 * n)).
-intro.
-apply (nat_elim1 n).
-intros.
-elim (or_eq_eq_S m).
-elim H2
- [elim (le_to_or_lt_eq (S O) a)
- [rewrite > H3 in ⊢ (? % ?).
- apply (trans_le ? ((exp (S(S O)) ((S(S O)*a)))*A a))
- [apply le_A_exp
- |apply (trans_le ? (((S(S O)))\sup((S(S O))*a)*
- ((pred a)\sup((S(S O)))*((S(S O)))\sup((S(S O))*a))))
- [apply le_times_r.
- apply H
- [rewrite > H3.
- rewrite > times_n_SO in ⊢ (? % ?).
- rewrite > sym_times.
- apply lt_times_l1
- [apply lt_to_le.assumption
- |apply le_n
- ]
- |assumption
- ]
- |rewrite > sym_times.
- rewrite > assoc_times.
- rewrite < exp_plus_times.
- apply (trans_le ?
- (pred a\sup((S(S O)))*(S(S O))\sup(S(S O))*(S(S O))\sup((S(S O))*m)))
- [rewrite > assoc_times.
- apply le_times_r.
- rewrite < exp_plus_times.
- apply le_exp
- [apply lt_O_S
- |rewrite < H3.
- simplify.
- rewrite < plus_n_O.
- apply le_S.apply le_S.
- apply le_n
- ]
- |apply le_times_l.
- rewrite > times_exp.
- apply monotonic_exp1.
- rewrite > H3.
- rewrite > sym_times.
- cases a
- [apply le_n
- |simplify.
- rewrite < plus_n_Sm.
- apply le_S.
- apply le_n
- ]
- ]
- ]
- ]
- |rewrite < H4 in H3.
- simplify in H3.
- rewrite > H3.
- simplify.
- apply le_S_S.apply le_S_S.apply le_O_n
- |apply not_lt_to_le.intro.
- apply (lt_to_not_le ? ? H1).
- rewrite > H3.
- apply (le_n_O_elim a)
- [apply le_S_S_to_le.assumption
- |apply le_O_n
- ]
- ]
- |elim (le_to_or_lt_eq O a (le_O_n ?))
- [apply (trans_le ? (A ((S(S O))*(S a))))
- [apply monotonic_A.
- rewrite > H3.
- rewrite > times_SSO.
- apply le_n_Sn
- |apply (trans_le ? ((exp (S(S O)) ((S(S O)*(S a))))*A (S a)))
- [apply le_A_exp
- |apply (trans_le ? (((S(S O)))\sup((S(S O))*S a)*
- (a\sup((S(S O)))*((S(S O)))\sup((S(S O))*(S a)))))
- [apply le_times_r.
- apply H
- [rewrite > H3.
- apply le_S_S.
- simplify.
- rewrite > plus_n_SO.
- apply le_plus_r.
- rewrite < plus_n_O.
- assumption
- |apply le_S_S.assumption
- ]
- |rewrite > sym_times.
- rewrite > assoc_times.
- rewrite < exp_plus_times.
- apply (trans_le ?
- (a\sup((S(S O)))*(S(S O))\sup(S(S O))*(S(S O))\sup((S(S O))*m)))
- [rewrite > assoc_times.
- apply le_times_r.
- rewrite < exp_plus_times.
- apply le_exp
- [apply lt_O_S
- |rewrite > times_SSO.
- rewrite < H3.
- simplify.
- rewrite < plus_n_Sm.
- rewrite < plus_n_O.
- apply le_n
- ]
- |apply le_times_l.
- rewrite > times_exp.
- apply monotonic_exp1.
- rewrite > H3.
- rewrite > sym_times.
- apply le_n
- ]
- ]
- ]
- ]
- |rewrite < H4 in H3.simplify in H3.
- apply False_ind.
- apply (lt_to_not_le ? ? H1).
- rewrite > H3.
- apply le_
- ]
- ]
-qed.
-*)
-
-theorem le_A_exp4: ∀n. 1 < n →
- A(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))*A a))
- [@le_A_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 ? (A (2*(S a))))
- [@monotonic_A >Hm normalize <plus_n_Sm @le_n_Sn
- |@(transitive_le … (le_A_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_A_exp: ∀n. n ≤ 8 →
- A(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_A_exp5: ∀n. A(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_A_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 ? (A (2*(S a))))
- [@monotonic_A >Hm normalize <plus_n_Sm @le_n_Sn
- |@(transitive_le ? ((exp 2 ((2*(S a))-2))*A (S a)))
- [@lt_4_to_le_A_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_A_exp @le_S_S_to_le @not_le_to_lt //
- ]
-qed.
-
-theorem le_exp_Al:∀n. O < n → exp 2 n ≤ A (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_A|@le_n]
- ]
- ]
- ]
-qed.
-
-theorem le_exp_A2:∀n. 1 < n → exp 2 (n / 2) \le A n.
-#n #lt1n @(transitive_le ? (A(n/2*2)))
- [>commutative_times @le_exp_Al
- 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_A >(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 leA_prim: ∀n.
- exp n (prim n) \le A 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 (A 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 ? (A (2*n)))
- [@le_B_A |@le_Al]
- ]
-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 ? (A (4*n)))
- [@le_B_A4 // |@le_Al]
- ]
-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 (A n) 2))
- [>exp_Sn >exp_Sn whd in match (exp ? 0); <times_n_1 @leA_r2
- |>commutative_times <exp_exp_times @daemon (* monotonic_exp1
- apply le_A_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.
-
-(*
-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 / commitdiff