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".
let rec list_divides l n \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) ].
-
-definition list_n : nat \to list nat \def
- \lambda n.let rec aux n k \def
+let rec list_n_aux n k \def
match n with
[ O => nil nat
- | S n1 => k::aux n1 (S k) ]
- in aux (pred n) 2.
+ | 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
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.
-
-axiom sieve_monotonic : \forall n.sieve (S n) = sieve n \lor sieve (S n) = (S n)::sieve n.
-
-axiom daemon : False.
-
-axiom in_list_cons_case : \forall A,x,a,l.in_list A x (a::l) \to
- x = a \lor in_list A x l.
-
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
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 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.
+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 x \leq k \land \forall p.in_list ? p l1 \to \lnot p \divides x) \land
- ((x \leq k \land \forall p.in_list ? p l1 \to \lnot p \divides x) \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 H4;elim (not_le_Sn_O ? H4)]]
- simplify;split;intros;
- [elim (H p);elim (H4 H3);assumption
- |elim (H p);apply (H6 H3 H4);rewrite > Hcut;intros;unfold in H7;
- elim H7;elim H8;clear H7 H8;generalize in match H9;elim a
- [simplify in H7;destruct H7
- |simplify in H8;destruct H8]]
+ |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;intros
- [elim (H1 p);elim (H5 H4);assumption
- |elim (H1 p);apply (H7 H4 H5);intros;unfold in H8;
- elim H8;elim H9;clear H8 H9;generalize in match H10;elim a
- [simplify in H8;destruct H8
- |simplify in H9;destruct H9]]
- |simplify;elim (H k (filter ? l (\lambda x.notb (divides_b t1 x))) (t1::l1) ? ? ? p)
- [split;intros
- [apply H5;assumption
- |apply H6;assumption]
- |intro;split;intros
- [elim (in_list_cons_case ? ? ? ? H5);
- [rewrite > H6;split
+ [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 daemon (* aggiungere hp: ogni elemento di l2 è >= 2 *)
- |intros;elim (le_to_or_lt_eq ? ? (divides_to_le ? ? ? H7))
- [elim (divides_to_prime_divides ? ? H8 H9 H7);elim H10;
- elim H11;clear H10 H11;elim (H3 t1);elim H10
- [clear H10 H11;elim (H16 ? ? H12);elim (H2 a);clear H10;
- apply H11
+ [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
- |apply (trans_le ? ? ? ? H15);
- apply (trans_le ? ? ? H13);
+ |elim H17;apply (trans_le ? ? ? ? H20);
+ apply (trans_le ? ? ? H15);
apply lt_to_le;assumption
- |intros;
- (* sfruttare il fatto che a < t1
- e t1 è il minimo di t1::l *)
- elim daemon]
- |unfold;apply (ex_intro ? ? []);
- apply (ex_intro ? ? l);
- reflexivity]
- |elim daemon (* aggiungere hp: ogni elemento di l2 è >= 2 *)
+ |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 t1);elim H11
+ [elim H13;apply lt_to_le;assumption
+ |apply in_list_head]
|assumption]]
- |elim (H3 t1);elim H7
- [assumption
- |apply (ex_intro ? ? []);apply (ex_intro ? ? l);reflexivity]]
+ |elim (H3 t1);elim H9
+ [elim H11;assumption
+ |apply in_list_head]]
|intros;elim (le_to_or_lt_eq t1 x)
[assumption
- |rewrite > H8 in H7;lapply (in_list_filter_to_p_true ? ? ? H7);
+ |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
- |elim daemon (* aggiungere hp: ogni elemento di l2 è >= 2 *)]
- |(* sfruttare il fatto che t1 è il minimo di t1::l *)
- elim daemon]]
- |elim (H2 p1);elim (H7 H6);split
+ |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 H10;apply in_list_cons;apply (in_list_filter ? ? ? H11);]]
- |elim (decidable_eq_nat p1 t1)
- [rewrite > H8;apply (ex_intro ? ? []);apply (ex_intro ? ? l1);
- reflexivity
- |apply in_list_cons;elim (H2 p1);apply (H10 H5 H6);intros;
+ |intros;apply H12;apply in_list_cons;apply (in_list_filter ? ? ? H13)]]
+ |elim (decidable_eq_nat p t1)
+ [rewrite > H10;apply in_list_head
+ |apply in_list_cons;elim (H2 p);apply (H12 H7 H8);intros;
apply (trans_le ? t1)
- [elim (decidable_lt p1 t1)
+ [elim (decidable_lt p t1)
[assumption
- |lapply (not_lt_to_le ? ? H12);
- lapply (decidable_divides t1 p1)
+ |lapply (not_lt_to_le ? ? H14);
+ lapply (decidable_divides t1 p)
[elim Hletin1
- [elim H5;lapply (H15 ? H13)
- [elim H8;symmetry;assumption
- |(* per il solito discorso l2 >= 2 *)
- elim daemon]
- |elim (Not_lt_n_n p1);apply H7;apply in_list_filter_r
- [elim (H3 p1);apply (in_list_tail ? ? t1)
- [apply H15;split
- [assumption
- |intros;elim H5;intro;lapply (H18 ? H19)
- [rewrite > Hletin2 in H16;elim (H9 H16);
- lapply (H21 t1)
+ [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 p2);elim (H20 H16);
- elim H22;assumption]]
- |unfold;intro;apply H13;rewrite > H16;apply divides_n_n;]
- |rewrite > (not_divides_to_divides_b_false ? ? ? H13);
+ |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 daemon (* usare il solito >= 2 *)]]]
- |elim daemon (* come sopra *)]]
- |elim daemon (* t1::l è ordinata *)]]]
- |intro;elim (H3 x);split;intros;
+ |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 H5
+ [elim H7
[assumption
- |apply in_list_cons;apply (in_list_filter ? ? ? H7)]
- |intros;elim (in_list_cons_case ? ? ? ? H8)
- [rewrite > H9;intro;lapply (in_list_filter_to_p_true ? ? ? H7);
- rewrite > (divides_to_divides_b_true ? ? ? H10) in Hletin
+ |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 daemon (* dal fatto che ogni elemento di t1::l è >= 2 *)]
- |elim H5
- [apply H11;assumption
- |apply in_list_cons;apply (in_list_filter ? ? ? H7)]]]
- |elim H7;elim (in_list_cons_case ? ? ? ? (H6 ?))
- [elim (H9 x)
- [rewrite > H10;unfold;
- apply (ex_intro ? ? []);apply (ex_intro ? ? l1);
- reflexivity
- |apply divides_n_n;]
- |split
- [assumption
- |intros;apply H9;apply in_list_cons;assumption]
+ |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 (H9 t1)
+ |lapply (H11 t1)
[rewrite > (not_divides_to_divides_b_false ? ? ? Hletin);
[reflexivity
- |(* solito >= 2 *)
- elim daemon]
+ |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 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 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 sieve_soundness : \forall n.\forall p.
- in_list ? p (sieve n) \to
- p \leq n \land prime p.
-intros;unfold sieve in H;lapply (sieve_prime n (S n) (list_n n) [] ? ? ? p)
- [intros;split;intros;
- [elim daemon (* H1 is absurd *)
- |elim daemon (* da sistemare, bisognerebbe raggiungere l'assurdo sapendo che p1 è primo e < 2 *)]
- |intro;split;intros
+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
- [elim daemon (* vero, dalla H1 *)
- |intros;elim daemon (* H2 è assurda *)]
- |elim H1;
- (* vera supponendo x >= 2, il che in effetti è doveroso sistemare nel teoremone di prima *)
- elim daemon]
- |elim daemon (* sempre vero, da dimostrare *)
- |elim Hletin;elim (H1 H);split
- [elim daemon (* si può ottenere da H *)
- |assumption]]
+ [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 h1 t1 => match t1 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
+ [reflexivity
+ |change in H2 with (andb (checker (t1::l1)) (leb t (t1+(t1+O))) = true);
+ apply (andb_true_true ? ? H2)]
+qed.
+
+theorem checker_sound : \forall l1,l2,l,x,y.l = l1@(x::y::l2) \to
+ checker l = true \to x \leq 2*y.
+intro;elim l1 0
+ [simplify;intros 5;rewrite > H;simplify;intro;
+ apply leb_true_to_le;apply (andb_true_true_r ? ? H1);
+ |simplify;intros;rewrite > H1 in H2;lapply (checker_cons ? ? H2);
+ apply (H l2 ? ? ? ? Hletin);reflexivity]
qed.
definition bertrand \def \lambda n.
definition not_bertrand \def \lambda n.
\forall p.n < p \to p \le 2*n \to \not (prime p).
+(*
+lemma list_of_primes_SO: \forall l.list_of_primes 1 l \to
+l = [].
+intro.cases l;intros
+ [reflexivity
+ |apply False_ind.unfold in H.
+ absurd ((prime n) \land n \le 1)
+ [apply H.
+ apply in_list_head
+ |intro.elim H1.
+ elim H2.
+ apply (lt_to_not_le ? ? H4 H3)
+ ]
+ ]
+qed.
+*)
+
+lemma min_prim : \forall n.\exists p. n < p \land prime p \land
+ \forall q.prime q \to q < p \to q \leq n.
+intro;elim (le_to_or_lt_eq ? ? (le_O_n n))
+ [apply (ex_intro ? ? (min_aux (S (n!)) (S n) primeb));
+ split
+ [split
+ [apply le_min_aux;
+ |apply primeb_true_to_prime;apply f_min_aux_true;elim (ex_prime n);
+ [apply (ex_intro ? ? a);elim H1;elim H2;split
+ [split
+ [assumption
+ |rewrite > plus_n_O;apply le_plus
+ [assumption
+ |apply le_O_n]]
+ |apply prime_to_primeb_true;assumption]
+ |assumption]]
+ |intros;apply not_lt_to_le;intro;lapply (lt_min_aux_to_false ? ? ? ? H3 H2);
+ rewrite > (prime_to_primeb_true ? H1) in Hletin;destruct Hletin]
+ |apply (ex_intro ? ? 2);split
+ [split
+ [rewrite < H;apply lt_O_S
+ |apply primeb_true_to_prime;reflexivity]
+ |intros;elim (lt_to_not_le ? ? H2);apply prime_to_lt_SO;assumption]]
+qed.
+
+theorem list_of_primes_to_bertrand: \forall n,pn,l.0 < n \to prime pn \to n <pn \to
+list_of_primes pn l \to
+(\forall p. prime p \to p \le pn \to in_list nat p l) \to
+(\forall p. in_list nat p l \to 2 < p \to
+\exists pp. in_list nat pp l \land pp < p \land p \le 2*pp) \to bertrand n.
+intros.
+elim (min_prim n).
+apply (ex_intro ? ? a).
+elim H6.clear H6.elim H7.clear H7.
+split
+ [split
+ [assumption
+ |elim (le_to_or_lt_eq ? ? (prime_to_lt_SO ? H9))
+ [elim (H5 a)
+ [elim H10.clear H10.elim H11.clear H11.
+ apply (trans_le ? ? ? H12).
+ apply le_times_r.
+ apply H8
+ [unfold in H3.
+ elim (H3 a1 H10).
+ assumption
+ |assumption
+ ]
+ |apply H4
+ [assumption
+ |apply not_lt_to_le.intro.
+ apply (lt_to_not_le ? ? H2).
+ apply H8;assumption
+ ]
+ |assumption
+ ]
+ |rewrite < H7.
+ apply O_lt_const_to_le_times_const.
+ assumption
+ ]
+ ]
+ |assumption
+ ]
+qed.
+
+let rec check_list l \def
+ match l with
+ [ nil \Rightarrow true
+ | cons (hd:nat) tl \Rightarrow
+ match tl with
+ [ nil \Rightarrow eqb hd 2
+ | cons hd1 tl1 \Rightarrow
+ (leb (S hd1) hd \land leb hd (2*hd1) \land check_list tl)
+ ]
+ ]
+.
+
+lemma check_list1: \forall n,m,l.(check_list (n::m::l)) = true \to
+m < n \land n \le 2*m \land (check_list (m::l)) = true \land ((check_list l) = true).
+intros 3.
+change in ⊢ (? ? % ?→?) with (leb (S m) n \land leb n (2*m) \land check_list (m::l)).
+intro.
+lapply (andb_true_true ? ? H) as H1.
+lapply (andb_true_true_r ? ? H) as H2.clear H.
+lapply (andb_true_true ? ? H1) as H3.
+lapply (andb_true_true_r ? ? H1) as H4.clear H1.
+split
+ [split
+ [split
+ [apply leb_true_to_le.assumption
+ |apply leb_true_to_le.assumption
+ ]
+ |assumption
+ ]
+ |generalize in match H2.
+ cases l
+ [intro.reflexivity
+ |change in ⊢ (? ? % ?→?) with (leb (S n1) m \land leb m (2*n1) \land check_list (n1::l1)).
+ intro.
+ lapply (andb_true_true_r ? ? H) as H2.
+ assumption
+ ]
+ ]
+qed.
+
+theorem check_list2: \forall l. check_list l = true \to
+\forall p. in_list nat p l \to 2 < p \to
+\exists pp. in_list nat pp l \land pp < p \land p \le 2*pp.
+intro.elim l 2
+ [intros.apply False_ind.apply (not_in_list_nil ? ? H1)
+ |cases l1;intros
+ [lapply (in_list_singleton_to_eq ? ? ? H2) as H4.
+ apply False_ind.
+ apply (lt_to_not_eq ? ? H3).
+ apply sym_eq.apply eqb_true_to_eq.
+ rewrite > H4.apply H1
+ |elim (check_list1 ? ? ? H1).clear H1.
+ elim H4.clear H4.
+ elim H1.clear H1.
+ elim (in_list_cons_case ? ? ? ? H2)
+ [apply (ex_intro ? ? n).
+ split
+ [split
+ [apply in_list_cons.apply in_list_head
+ |rewrite > H1.assumption
+ ]
+ |rewrite > H1.assumption
+ ]
+ |elim (H H6 p H1 H3).clear H.
+ apply (ex_intro ? ? a).
+ elim H8.clear H8.
+ elim H.clear H.
+ split
+ [split
+ [apply in_list_cons.assumption
+ |assumption
+ ]
+ |assumption
+ ]
+ ]
+ ]
+ ]
+qed.
+
+(* qualcosa che non va con gli S *)
+lemma le_to_bertrand : \forall n.O < n \to n \leq exp 2 8 \to bertrand n.
+intros.
+apply (list_of_primes_to_bertrand ? (S(exp 2 8)) (sieve (S(exp 2 8))))
+ [assumption
+ |apply primeb_true_to_prime.reflexivity
+ |apply (le_to_lt_to_lt ? ? ? H1).
+ apply le_n
+ |lapply (sieve_sound1 (S(exp 2 8))) as H
+ [elim H.assumption
+ |apply leb_true_to_le.reflexivity
+ ]
+ |intros.apply (sieve_sound2 ? ? H3 H2)
+ |apply check_list2.
+ reflexivity
+ ]
+qed.
+
+(*lemma pippo : \forall k,n.in_list ? (nth_prime (S k)) (sieve n) \to
+ \exists l.sieve n = l@((nth_prime (S k))::(sieve (nth_prime k))).
+intros;elim H;elim H1;clear H H1;apply (ex_intro ? ? a);
+cut (a1 = sieve (nth_prime k))
+ [rewrite < Hcut;assumption
+ |lapply (sieve_sorted n);generalize in match H2*)
+
+(* old proof by Wilmer
+lemma le_to_bertrand : \forall n.O < n \to n \leq exp 2 8 \to bertrand n.
+intros;
+elim (min_prim n);apply (ex_intro ? ? a);elim H2;elim H3;clear H2 H3;
+cut (a \leq 257)
+ [|apply not_lt_to_le;intro;apply (le_to_not_lt ? ? H1);apply (H4 ? ? H2);
+ apply primeb_true_to_prime;reflexivity]
+split
+ [split
+ [assumption
+ |elim (prime_to_nth_prime a H6);generalize in match H2;cases a1
+ [simplify;intro;rewrite < H3;rewrite < plus_n_O;
+ change in \vdash (? % ?) with (1+1);apply le_plus;assumption
+ |intro;lapply (H4 (nth_prime n1))
+ [apply (trans_le ? (2*(nth_prime n1)))
+ [rewrite < H3;
+ cut (\exists l1,l2.sieve 257 = l1@((nth_prime (S n1))::((nth_prime n1)::l2)))
+ [elim Hcut1;elim H7;clear Hcut1 H7;
+ apply (checker_sound a2 a3 (sieve 257))
+ [apply H8
+ |reflexivity]
+ |elim (sieve_sound2 257 (nth_prime (S n1)) ? ?)
+ [elim (sieve_sound2 257 (nth_prime n1) ? ?)
+ [elim H8;
+ cut (\forall p.in_list ? p (a3@(nth_prime n1::a4)) \to prime p)
+ [|rewrite < H9;intros;apply (in_list_sieve_to_prime 257 p ? H10);
+ apply leb_true_to_le;reflexivity]
+ apply (ex_intro ? ? a2);apply (ex_intro ? ? a4);
+ elim H7;clear H7 H8;
+ cut ((nth_prime n1)::a4 = a5)
+ [|generalize in match H10;
+ lapply (sieve_sorted 257);
+ generalize in match Hletin1;
+ rewrite > H9 in ⊢ (? %→? ? % ?→?);
+ generalize in match Hcut1;
+ generalize in match a2;
+ elim a3 0
+ [intro;elim l
+ [change in H11 with (nth_prime n1::a4 = nth_prime (S n1)::a5);
+ destruct H11;elim (eq_to_not_lt ? ? Hcut2);
+ apply increasing_nth_prime
+ |change in H12 with (nth_prime n1::a4 = t::(l1@(nth_prime (S n1)::a5)));
+ destruct H12;
+ change in H11 with (sorted_gt (nth_prime n1::l1@(nth_prime (S n1)::a5)));
+ lapply (sorted_to_minimum ? ? ? H11 (nth_prime (S n1)))
+ [unfold in Hletin2;elim (le_to_not_lt ? ? (lt_to_le ? ? Hletin2));
+ apply increasing_nth_prime
+ |apply (ex_intro ? ? l1);apply (ex_intro ? ? a5);reflexivity]]
+ |intros 5;elim l1
+ [change in H12 with (t::(l@(nth_prime n1::a4)) = nth_prime (S n1)::a5);
+ destruct H12;cut (l = [])
+ [rewrite > Hcut2;reflexivity
+ |change in H11 with (sorted_gt (nth_prime (S n1)::(l@(nth_prime n1::a4))));
+ generalize in match H11;generalize in match H8;cases l;intros
+ [reflexivity
+ |lapply (sorted_cons_to_sorted ? ? ? H13);
+ lapply (sorted_to_minimum ? ? ? H13 n2)
+ [simplify in Hletin2;lapply (sorted_to_minimum ? ? ? Hletin2 (nth_prime n1))
+ [unfold in Hletin3;unfold in Hletin4;
+ elim (lt_nth_prime_to_not_prime ? ? Hletin4 Hletin3);
+ apply H12;
+ apply (ex_intro ? ? [nth_prime (S n1)]);
+ apply (ex_intro ? ? (l2@(nth_prime n1::a4)));
+ reflexivity
+ |apply (ex_intro ? ? l2);apply (ex_intro ? ? a4);reflexivity]
+ |simplify;apply in_list_head]]]
+ |change in H13 with (t::(l@(nth_prime n1::a4)) = t1::(l2@(nth_prime (S n1)::a5)));
+ destruct H13;apply (H7 l2 ? ? Hcut3)
+ [intros;apply H8;simplify;apply in_list_cons;
+ assumption
+ |simplify in H12;
+ apply (sorted_cons_to_sorted ? ? ? H12)]]]]
+ rewrite > Hcut2 in ⊢ (? ? ? (? ? ? (? ? ? %)));
+ apply H10
+ |apply (trans_le ? ? ? Hletin);apply lt_to_le;
+ apply (trans_le ? ? ? H5 Hcut)
+ |apply prime_nth_prime]
+ |rewrite > H3;assumption
+ |apply prime_nth_prime]]
+ |apply le_times_r;assumption]
+ |apply prime_nth_prime
+ |rewrite < H3;apply increasing_nth_prime]]]
+ |assumption]
+qed. *)
+
lemma not_not_bertrand_to_bertrand1: \forall n.
\lnot (not_bertrand n) \to \forall x. n \le x \to x \le 2*n \to
(\forall p.x < p \to p \le 2*n \to \not (prime p))
]
qed.
-theorem le_to_bertrand:
+theorem le_to_bertrand2:
\forall n. (exp 2 8) \le n \to bertrand n.
intros.
apply not_not_bertrand_to_bertrand.unfold.intro.
]
qed.
+theorem bertrand_n :
+\forall n. O < n \to bertrand n.
+intros;elim (decidable_le n 256)
+ [apply le_to_bertrand;assumption
+ |apply le_to_bertrand2;apply lt_to_le;apply not_le_to_lt;apply H1]
+qed.
+
(* test
theorem mod_exp: eqb (mod (exp 2 8) 13) O = false.
reflexivity.
projects / helm.git / blobdiff