X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fnat%2Fbertrand.ma;h=ce92ee4edc9d7015079831ff14314ae9c1f90aeb;hb=68dbcd02022874a025a9444aa1125b0458816fbb;hp=3f613c8d762f81089a3c13a94c819eb46a61ec71;hpb=1ff3965d308be074f3ed5181b3c38921f289b6a9;p=helm.git diff --git a/helm/software/matita/library/nat/bertrand.ma b/helm/software/matita/library/nat/bertrand.ma index 3f613c8d7..ce92ee4ed 100644 --- a/helm/software/matita/library/nat/bertrand.ma +++ b/helm/software/matita/library/nat/bertrand.ma @@ -15,8 +15,10 @@ include "nat/sqrt.ma". include "nat/chebyshev_teta.ma". include "nat/chebyshev.ma". -include "list/list.ma". +include "list/in.ma". +include "list/sort.ma". include "nat/o.ma". +include "nat/sieve.ma". let rec list_divides l n \def match l with @@ -32,493 +34,6 @@ definition lprim : nat \to list nat \def | false => aux m1 (n-m1::acc)]] in aux (pred n) []. -let rec filter A l p on l \def - match l with - [ nil => nil A - | cons (a:A) (tl:list A) => match (p a) with - [ true => a::(filter A tl p) - | false => filter A tl p ]]. - -let rec length A (l:list A) on l \def - match l with - [ nil => O - | cons (a:A) (tl:list A) => S (length A tl) ]. - -let rec list_n_aux n k \def - match n with - [ O => nil nat - | S n1 => k::list_n_aux n1 (S k) ]. - -definition list_n : nat \to list nat \def - \lambda n.list_n_aux (pred n) 2. - -let rec sieve_aux l1 l2 t on t \def - match t with - [ O => l1 - | S t1 => match l2 with - [ nil => l1 - | cons n tl => sieve_aux (n::l1) (filter nat tl (\lambda x.notb (divides_b n x))) t1]]. - -definition sieve : nat \to list nat \def - \lambda m.sieve_aux [] (list_n m) m. - -definition ord_list \def - \lambda l. - \forall a,b,l1,l2.l = l1@(a::b::l2) \to b \leq a. - -definition in_list \def - \lambda A.\lambda a:A.\lambda l:list A. - \exists l1,l2.l = l1@(a::l2). - -lemma in_list_filter_to_p_true : \forall l,x,p. -in_list nat x (filter nat l p) \to p x = true. -intros;elim H;elim H1;clear H H1;generalize in match H2;generalize in match a;elim l 0 - [simplify;intro;elim l1 - [simplify in H;destruct H - |simplify in H1;destruct H1] - |intros;simplify in H1;apply (bool_elim ? (p t));intro; - rewrite > H3 in H1;simplify in H1 - [generalize in match H1;elim l2 - [simplify in H4;destruct H4;assumption - |simplify in H5;destruct H5;apply (H l3);assumption] - |apply (H l2);assumption]] -qed. - -lemma in_list_cons : \forall l,x,y.in_list nat x l \to in_list nat x (y::l). -intros;unfold in H;unfold;elim H;elim H1;apply (ex_intro ? ? (y::a)); -apply (ex_intro ? ? a1);simplify;rewrite < H2;reflexivity. -qed. - -lemma in_list_tail : \forall l,x,y.in_list nat x (y::l) \to x \neq y \to in_list nat x l. -intros;elim H;elim H2;generalize in match H3;elim a - [simplify in H4;destruct H4;elim H1;reflexivity - |simplify in H5;destruct H5;apply (ex_intro ? ? l1);apply (ex_intro ? ? a1); - reflexivity] -qed. - -lemma in_list_filter : \forall l,p,x.in_list nat x (filter nat l p) \to in_list nat x l. -intros;elim H;elim H1;generalize in match H2;generalize in match a;elim l 0 - [simplify;intro;elim l1 - [simplify in H3;destruct H3 - |simplify in H4;destruct H4] - |intros;simplify in H4;apply (bool_elim ? (p t));intro - [rewrite > H5 in H4;simplify in H4;generalize in match H4;elim l2 - [simplify in H6;destruct H6;apply (ex_intro ? ? []);apply (ex_intro ? ? l1); - simplify;reflexivity - |simplify in H7;destruct H7;apply in_list_cons;apply (H3 ? Hcut1);] - |rewrite > H5 in H4;simplify in H4;apply in_list_cons;apply (H3 ? H4);]] -qed. - -lemma in_list_filter_r : \forall l,p,x.in_list nat x l \to p x = true \to in_list nat x (filter nat l p). -intros;elim H;elim H2;rewrite > H3;elim a - [simplify;rewrite > H1;simplify;apply (ex_intro ? ? []);apply (ex_intro ? ? (filter nat a1 p)); - reflexivity - |simplify;elim (p t);simplify - [apply in_list_cons;assumption - |assumption]] -qed. - -lemma in_list_head : \forall x,l.in_list nat x (x::l). -intros;apply (ex_intro ? ? []);apply (ex_intro ? ? l);reflexivity; -qed. - -lemma in_list_cons_case : \forall A,x,a,l.in_list A x (a::l) \to - x = a \lor in_list A x l. -intros;elim H;elim H1;clear H H1;generalize in match H2;elim a1 - [simplify in H;destruct H;left;reflexivity - |simplify in H1;destruct H1;right; - apply (ex_intro ? ? l1); - apply (ex_intro ? ? a2); - reflexivity] -qed. - -lemma divides_to_prime_divides : \forall n,m.1 < m \to m < n \to m \divides n \to - \exists p.p \leq m \land prime p \land p \divides n. -intros;apply (ex_intro ? ? (nth_prime (max_prime_factor m)));split - [split - [apply divides_to_le - [apply lt_to_le;assumption - |apply divides_max_prime_factor_n;assumption] - |apply prime_nth_prime;] - |apply (transitive_divides ? ? ? ? H2);apply divides_max_prime_factor_n; - assumption] -qed. - - -lemma le_length_filter : \forall A,l,p.length A (filter A l p) \leq length A l. -intros;elim l - [simplify;apply le_n - |simplify;apply (bool_elim ? (p t));intro - [simplify;apply le_S_S;assumption - |simplify;apply le_S;assumption]] -qed. - -inductive sorted (P:nat \to nat \to Prop): list nat \to Prop \def -| sort_nil : sorted P [] -| sort_cons : \forall x,l.sorted P l \to (\forall y.in_list ? y l \to P x y) - \to sorted P (x::l). - -definition sorted_lt : list nat \to Prop \def \lambda l.sorted lt l. - -definition sorted_gt : list nat \to Prop \def \lambda l.sorted gt l. - -lemma sorted_cons_to_sorted : \forall P,x,l.sorted P (x::l) \to sorted P l. -intros;inversion H;intros - [destruct H1 - |destruct H4;assumption] -qed. - -lemma sorted_to_minimum : \forall P,x,l.sorted P (x::l) \to - \forall y.in_list ? y l \to P x y. -intros;inversion H;intros; - [destruct H2 - |destruct H5;apply H4;assumption] -qed. - -lemma not_in_list_nil : \forall A,a.\lnot in_list A a []. -intros;intro;elim H;elim H1;generalize in match H2;elim a1 - [simplify in H3;destruct H3 - |simplify in H4;destruct H4] -qed. - -lemma sieve_prime : \forall t,k,l2,l1. - (\forall p.(in_list ? p l1 \to prime p \land p \leq k \land \forall x.in_list ? x l2 \to p < x) \land - (prime p \to p \leq k \to (\forall x.in_list ? x l2 \to p < x) \to in_list ? p l1)) \to - (\forall x.(in_list ? x l2 \to 2 \leq x \land x \leq k \land \forall p.in_list ? p l1 \to \lnot p \divides x) \land - (2 \leq x \to x \leq k \to (\forall p.in_list ? p l1 \to \lnot p \divides x) \to - in_list ? x l2)) \to - length ? l2 \leq t \to - sorted_gt l1 \to - sorted_lt l2 \to - sorted_gt (sieve_aux l1 l2 t) \land - \forall p.(in_list ? p (sieve_aux l1 l2 t) \to prime p \land p \leq k) \land - (prime p \to p \leq k \to in_list ? p (sieve_aux l1 l2 t)). -intro.elim t 0 - [intros;cut (l2 = []) - [|generalize in match H2;elim l2 - [reflexivity - |simplify in H6;elim (not_le_Sn_O ? H6)]] - simplify;split - [assumption - |intro;elim (H p);split;intros - [elim (H5 H7);assumption - |apply (H6 H7 H8);rewrite > Hcut;intros;elim (not_in_list_nil ? ? H9)]] - |intros 4;elim l2 - [simplify;split; - [assumption - |intro;elim (H1 p);split;intros - [elim (H6 H8);assumption - |apply (H7 H8 H9);intros;elim (not_in_list_nil ? ? H10)]] - |simplify;elim (H k (filter ? l (\lambda x.notb (divides_b t1 x))) (t1::l1)) - [split; - [assumption - |intro;apply H8;] - |split;intros - [elim (in_list_cons_case ? ? ? ? H7); - [rewrite > H8;split - [split - [unfold;intros;split - [elim (H3 t1);elim H9 - [elim H11;assumption - |apply in_list_head] - |intros;elim (le_to_or_lt_eq ? ? (divides_to_le ? ? ? H9)) - [elim (divides_to_prime_divides ? ? H10 H11 H9);elim H12; - elim H13;clear H13 H12;elim (H3 t1);elim H12 - [clear H13 H12;elim (H18 ? ? H14);elim (H2 a); - apply H13 - [assumption - |elim H17;apply (trans_le ? ? ? ? H20); - apply (trans_le ? ? ? H15); - apply lt_to_le;assumption - |intros;apply (trans_le ? (S m)) - [apply le_S_S;assumption - |apply (trans_le ? ? ? H11); - elim (in_list_cons_case ? ? ? ? H19) - [rewrite > H20;apply le_n - |apply lt_to_le;apply (sorted_to_minimum ? ? ? H6);assumption]]] - |unfold;apply (ex_intro ? ? []); - apply (ex_intro ? ? l); - reflexivity] - |elim (H3 t1);elim H11 - [elim H13;apply lt_to_le;assumption - |apply in_list_head] - |assumption]] - |elim (H3 t1);elim H9 - [elim H11;assumption - |apply (ex_intro ? ? []);apply (ex_intro ? ? l);reflexivity]] - |intros;elim (le_to_or_lt_eq t1 x) - [assumption - |rewrite > H10 in H9;lapply (in_list_filter_to_p_true ? ? ? H9); - lapply (divides_n_n x); - rewrite > (divides_to_divides_b_true ? ? ? Hletin1) in Hletin - [simplify in Hletin;destruct Hletin - |rewrite < H10;elim (H3 t1);elim H11 - [elim H13;apply lt_to_le;assumption - |apply in_list_head]] - |apply lt_to_le;apply (sorted_to_minimum ? ? ? H6);apply (in_list_filter ? ? ? H9)]] - |elim (H2 p);elim (H9 H8);split - [assumption - |intros;apply H12;apply in_list_cons;apply (in_list_filter ? ? ? H13)]] - |elim (decidable_eq_nat p t1) - [rewrite > H10;apply (ex_intro ? ? []);apply (ex_intro ? ? l1); - reflexivity - |apply in_list_cons;elim (H2 p);apply (H12 H7 H8);intros; - apply (trans_le ? t1) - [elim (decidable_lt p t1) - [assumption - |lapply (not_lt_to_le ? ? H14); - lapply (decidable_divides t1 p) - [elim Hletin1 - [elim H7;lapply (H17 ? H15) - [elim H10;symmetry;assumption - |elim (H3 t1);elim H18 - [elim H20;assumption - |apply in_list_head]] - |elim (Not_lt_n_n p);apply H9;apply in_list_filter_r - [elim (H3 p);apply (in_list_tail ? ? t1) - [apply H17 - [apply prime_to_lt_SO;assumption - |assumption - |intros;elim H7;intro;lapply (H20 ? H21) - [rewrite > Hletin2 in H18;elim (H11 H18); - lapply (H23 t1) - [elim (lt_to_not_le ? ? Hletin3 Hletin) - |apply (ex_intro ? ? []);apply (ex_intro ? ? l); - reflexivity] - |apply prime_to_lt_SO;elim (H2 p1);elim (H22 H18); - elim H24;assumption]] - |unfold;intro;apply H15;rewrite > H18;apply divides_n_n] - |rewrite > (not_divides_to_divides_b_false ? ? ? H15); - [reflexivity - |elim (H3 t1);elim H16 - [elim H18;apply lt_to_le;assumption - |apply in_list_head]]]] - |elim (H3 t1);elim H15 - [elim H17;apply lt_to_le;assumption - |apply in_list_head]]] - |elim (in_list_cons_case ? ? ? ? H13) - [rewrite > H14;apply le_n - |apply lt_to_le;apply (sorted_to_minimum ? ? ? H6);assumption]]]] - |elim (H3 x);split;intros; - [split - [elim H7 - [assumption - |apply in_list_cons;apply (in_list_filter ? ? ? H9)] - |intros;elim (in_list_cons_case ? ? ? ? H10) - [rewrite > H11;intro;lapply (in_list_filter_to_p_true ? ? ? H9); - rewrite > (divides_to_divides_b_true ? ? ? H12) in Hletin - [simplify in Hletin;destruct Hletin - |elim (H3 t1);elim H13 - [elim H15;apply lt_to_le;assumption - |apply in_list_head]] - |elim H7 - [apply H13;assumption - |apply in_list_cons;apply (in_list_filter ? ? ? H9)]]] - |elim (in_list_cons_case ? ? ? ? (H8 ? ? ?)) - [elim (H11 x) - [rewrite > H12;apply in_list_head - |apply divides_n_n] - |assumption - |assumption - |intros;apply H11;apply in_list_cons;assumption - |apply in_list_filter_r; - [assumption - |lapply (H11 t1) - [rewrite > (not_divides_to_divides_b_false ? ? ? Hletin); - [reflexivity - |elim (H3 t1);elim H13 - [elim H15;apply lt_to_le;assumption - |apply in_list_head]] - |apply in_list_head]]]] - |apply (trans_le ? ? ? (le_length_filter ? ? ?));apply le_S_S_to_le; - apply H4 - |apply sort_cons - [assumption - |intros;unfold;elim (H2 y);elim (H8 H7); - apply H11;apply in_list_head] - |generalize in match (sorted_cons_to_sorted ? ? ? H6);elim l - [simplify;assumption - |simplify;elim (notb (divides_b t1 t2));simplify - [lapply (sorted_cons_to_sorted ? ? ? H8);lapply (H7 Hletin); - apply (sort_cons ? ? ? Hletin1);intros; - apply (sorted_to_minimum ? ? ? H8);apply (in_list_filter ? ? ? H9); - |apply H7;apply (sorted_cons_to_sorted ? ? ? H8)]]]]] -qed. - -lemma in_list_singleton_to_eq : \forall A,x,y.in_list A x [y] \to x = y. -intros;elim H;elim H1;generalize in match H2;elim a - [simplify in H3;destruct H3;reflexivity - |simplify in H4;destruct H4;generalize in match Hcut1;elim l - [simplify in H4;destruct H4 - |simplify in H5;destruct H5]] -qed. - -lemma le_list_n_aux_k_k : \forall n,m,k.in_list ? n (list_n_aux m k) \to - k \leq n. -intros 2;elim m - [simplify in H;elim (not_in_list_nil ? ? H) - |simplify in H1;elim H1;elim H2;generalize in match H3;elim a - [simplify in H4;destruct H4;apply le_n - |simplify in H5;destruct H5;apply lt_to_le;apply (H (S k)); - apply (ex_intro ? ? l);apply (ex_intro ? ? a1);assumption]] -qed. - -lemma in_list_SSO_list_n : \forall n.2 \leq n \to in_list ? 2 (list_n n). -intros;elim H - [simplify;apply (ex_intro ? ? []);apply (ex_intro ? ? []); - simplify;reflexivity - |generalize in match H2;elim H1 - [simplify;apply (ex_intro ? ? []);apply (ex_intro ? ? [3]);simplify;reflexivity - |simplify;apply (ex_intro ? ? []);apply (ex_intro ? ? (list_n_aux n2 3)); - simplify;reflexivity]] -qed. - -lemma le_SSO_list_n : \forall m,n.in_list nat n (list_n m) \to 2 \leq n. -intros;unfold list_n in H;apply (le_list_n_aux_k_k ? ? ? H); -qed. - -lemma le_list_n_aux : \forall n,m,k.in_list ? n (list_n_aux m k) \to n \leq k+m-1. -intros 2;elim m - [simplify in H;elim (not_in_list_nil ? ? H) - |simplify in H1;elim H1;elim H2;generalize in match H3;elim a - [simplify in H4;destruct H4;rewrite < plus_n_Sm;simplify;rewrite < minus_n_O; - rewrite > plus_n_O in \vdash (? % ?);apply le_plus_r;apply le_O_n - |simplify in H5;destruct H5;rewrite < plus_n_Sm;apply (H (S k)); - apply (ex_intro ? ? l);apply (ex_intro ? ? a1);assumption]] -qed. - -lemma le_list_n : \forall n,m.in_list ? n (list_n m) \to n \leq m. -intros;unfold list_n in H;lapply (le_list_n_aux ? ? ? H); -simplify in Hletin;generalize in match H;generalize in match Hletin;elim m - [simplify in H2;elim (not_in_list_nil ? ? H2) - |simplify in H2;assumption] -qed. - - -lemma le_list_n_aux_r : \forall n,m.O < m \to \forall k.k \leq n \to n \leq k+m-1 \to in_list ? n (list_n_aux m k). -intros 3;elim H 0 - [intros;simplify;rewrite < plus_n_Sm in H2;simplify in H2; - rewrite < plus_n_O in H2;rewrite < minus_n_O in H2; - rewrite > (antisymmetric_le k n H1 H2);apply in_list_head - |intros 5;simplify;generalize in match H2;elim H3 - [apply in_list_head - |apply in_list_cons;apply H6 - [apply le_S_S;assumption - |rewrite < plus_n_Sm in H7;apply H7]]] -qed. - -lemma le_list_n_r : \forall n,m.S O < m \to 2 \leq n \to n \leq m \to in_list ? n (list_n m). -intros;unfold list_n;apply le_list_n_aux_r - [elim H;simplify - [apply lt_O_S - |generalize in match H4;elim H3; - [apply lt_O_S - |simplify in H7;apply le_S;assumption]] - |assumption - |simplify;generalize in match H2;elim H;simplify;assumption] -qed. - -lemma le_length_list_n : \forall n. length ? (list_n n) \leq n. -intro;cut (\forall n,k.length ? (list_n_aux n k) \leq (S n)) - [elim n;simplify - [apply le_n - |apply Hcut] - |intro;elim n1;simplify - [apply le_O_n - |apply le_S_S;apply H]] -qed. - -lemma sorted_list_n_aux : \forall n,k.sorted_lt (list_n_aux n k). -intro.elim n 0 - [simplify;intro;apply sort_nil - |intro;simplify;intros 2;apply sort_cons - [apply H - |intros;lapply (le_list_n_aux_k_k ? ? ? H1);assumption]] -qed. - -definition list_of_primes \def \lambda n.\lambda l. -\forall p.in_list nat p l \to prime p \land p \leq n. - -lemma sieve_sound1 : \forall n.2 \leq n \to -sorted_gt (sieve n) \land list_of_primes n (sieve n). -intros;elim (sieve_prime n n (list_n n) []) - [split - [assumption - |intro;unfold sieve in H3;elim (H2 p);elim (H3 H5);split;assumption] - |split;intros - [elim (not_in_list_nil ? ? H1) - |lapply (lt_to_not_le ? ? (H3 2 ?)) - [apply in_list_SSO_list_n;assumption - |elim Hletin;apply prime_to_lt_SO;assumption]] - |split;intros - [split - [split - [apply (le_SSO_list_n ? ? H1) - |apply (le_list_n ? ? H1)] - |intros;elim (not_in_list_nil ? ? H2)] - |apply le_list_n_r;assumption] - |apply le_length_list_n - |apply sort_nil - |elim n;simplify - [apply sort_nil - |elim n1;simplify - [apply sort_nil - |simplify;apply sort_cons - [apply sorted_list_n_aux - |intros;lapply (le_list_n_aux_k_k ? ? ? H3); - assumption]]]] -qed. - -lemma sieve_sorted : \forall n.sorted_gt (sieve n). -intros;elim (decidable_le 2 n) - [elim (sieve_sound1 ? H);assumption - |generalize in match (le_S_S_to_le ? ? (not_le_to_lt ? ? H));cases n - [intro;apply sort_nil - |intros;lapply (le_S_S_to_le ? ? H1);rewrite < (le_n_O_to_eq ? Hletin); - apply sort_nil]] -qed. - -lemma in_list_sieve_to_prime : \forall n,p.2 \leq n \to in_list ? p (sieve n) \to - prime p. -intros;elim (sieve_sound1 ? H);elim (H3 ? H1);assumption; -qed. - -lemma in_list_sieve_to_leq : \forall n,p.2 \leq n \to in_list ? p (sieve n) \to - p \leq n. -intros;elim (sieve_sound1 ? H);elim (H3 ? H1);assumption; -qed. - -lemma sieve_sound2 : \forall n,p.p \leq n \to prime p \to in_list ? p (sieve n). -intros;elim (sieve_prime n n (list_n n) []) - [elim (H3 p);apply H5;assumption - |split - [intro;elim (not_in_list_nil ? ? H2) - |intros;lapply (lt_to_not_le ? ? (H4 2 ?)) - [apply in_list_SSO_list_n;apply (trans_le ? ? ? ? H); - apply prime_to_lt_SO;assumption - |elim Hletin;apply prime_to_lt_SO;assumption]] - |split;intros - [split;intros - [split - [apply (le_SSO_list_n ? ? H2) - |apply (le_list_n ? ? H2)] - |elim (not_in_list_nil ? ? H3)] - |apply le_list_n_r - [apply (trans_le ? ? ? H2 H3) - |assumption - |assumption]] - |apply le_length_list_n - |apply sort_nil - |elim n;simplify - [apply sort_nil - |elim n1;simplify - [apply sort_nil - |simplify;apply sort_cons - [apply sorted_list_n_aux - |intros;lapply (le_list_n_aux_k_k ? ? ? H4); - assumption]]]] -qed. - let rec checker l \def match l with [ nil => true @@ -527,10 +42,10 @@ let rec checker l \def | cons h2 t2 => (andb (checker t1) (leb h1 (2*h2))) ]]. lemma checker_cons : \forall t,l.checker (t::l) = true \to checker l = true. -intros 2;simplify;intro;generalize in match H;elim l +intros 2;simplify;intro;elim l in H ⊢ % [reflexivity - |change in H2 with (andb (checker (t1::l1)) (leb t (t1+(t1+O))) = true); - apply (andb_true_true ? ? H2)] + |change in H1 with (andb (checker (a::l1)) (leb t (a+(a+O))) = true); + apply (andb_true_true ? ? H1)] qed. theorem checker_sound : \forall l1,l2,l,x,y.l = l1@(x::y::l2) \to @@ -694,7 +209,7 @@ intro.elim l 2 |rewrite > H1.assumption ] |elim (H H6 p H1 H3).clear H. - apply (ex_intro ? ? a). + apply (ex_intro ? ? a1). elim H8.clear H8. elim H.clear H. split @@ -722,8 +237,7 @@ apply (list_of_primes_to_bertrand ? (S(exp 2 8)) (sieve (S(exp 2 8)))) |apply leb_true_to_le.reflexivity ] |intros.apply (sieve_sound2 ? ? H3 H2) - |(* se tolgo l'argomento l'apply diventa lenta *) - apply (check_list2 (sieve (S(exp 2 8)))). + |apply check_list2. reflexivity ] qed. @@ -1096,7 +610,7 @@ qed. theorem le_B_split1_teta:\forall n.18 \le n \to not_bertrand n \to B_split1 (2*n) \le teta (2 * n / 3). -intros.unfold B_split1.unfold teta. +intros. unfold B_split1.unfold teta. apply (trans_le ? (pi_p (S (2*n)) primeb (λp:nat.(p)\sup(bool_to_nat (eqb (k (2*n) p) 1))))) [apply le_pi_p.intros. apply le_exp