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
| false => aux m1 (n-m1::acc)]]
in aux (pred n) [].
-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.
-
-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.
-
-definition sorted_lt \def sorted ? lt.
-definition sorted_gt \def sorted ? gt.
-
-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 a x))) (a::l1))
- [split;
- [assumption
- |intro;apply H8;]
- |split;intros
- [elim (in_list_cons_case ? ? ? ? H7);
- [rewrite > H8;split
- [split
- [unfold;intros;split
- [elim (H3 a);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 a);elim H12
- [clear H13 H12;elim (H18 ? ? H14);elim (H2 a1);
- 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]]]
- |apply in_list_head]
- |elim (H3 a);elim H11
- [elim H13;apply lt_to_le;assumption
- |apply in_list_head]
- |assumption]]
- |elim (H3 a);elim H9
- [elim H11;assumption
- |apply in_list_head]]
- |intros;elim (le_to_or_lt_eq a 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 a);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 a)
- [rewrite > H10;apply in_list_head
- |apply in_list_cons;elim (H2 p);apply (H12 H7 H8);intros;
- apply (trans_le ? a)
- [elim (decidable_lt p a)
- [assumption
- |lapply (not_lt_to_le ? ? H14);
- lapply (decidable_divides a p)
- [elim Hletin1
- [elim H7;lapply (H17 ? H15)
- [elim H10;symmetry;assumption
- |elim (H3 a);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 ? ? a)
- [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 a)
- [elim (lt_to_not_le ? ? Hletin3 Hletin)
- |apply in_list_head]
- |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 a);elim H16
- [elim H18;apply lt_to_le;assumption
- |apply in_list_head]]]]
- |elim (H3 a);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 a);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 a)
- [rewrite > (not_divides_to_divides_b_false ? ? ? Hletin);
- [reflexivity
- |elim (H3 a);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 a a1));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 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 (in_list_cons_case ? ? ? ? H1)
- [rewrite > H2;apply le_n
- |apply lt_to_le;apply H;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 in_list_head
- |generalize in match H2;elim H1;simplify;apply in_list_head]
-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 (in_list_cons_case ? ? ? ? H1)
- [rewrite > H2;rewrite < plus_n_Sm;simplify;rewrite < minus_n_O;
- rewrite > plus_n_O in \vdash (? % ?);apply le_plus_r;apply le_O_n
- |rewrite < plus_n_Sm;apply (H (S k));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;simplify;apply sort_nil
- |intros;lapply (le_S_S_to_le ? ? H1);rewrite < (le_n_O_to_eq ? Hletin);
- simplify;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
| 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 (a::l1)) (leb t (a+(a+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
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