\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.
+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
|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
- (\forall p.in_list ? p (sieve n) \to
- prime p \land p \leq n).
+sorted_gt (sieve n) \land list_of_primes n (sieve n).
intros;elim (sieve_prime n n (list_n n) [])
[split
[assumption
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))
|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)
+ |(* se tolgo l'argomento l'apply diventa lenta *)
+ apply (check_list2 (sieve (S(exp 2 8)))).
+ 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);
[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;
|apply prime_nth_prime
|rewrite < H3;apply increasing_nth_prime]]]
|assumption]
-qed.
+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
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