--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| A.Asperti, C.Sacerdoti Coen, *)
+(* ||A|| E.Tassi, S.Zacchiroli *)
+(* \ / *)
+(* \ / Matita is distributed under the terms of the *)
+(* v GNU Lesser General Public License Version 2.1 *)
+(* *)
+(**************************************************************************)
+
+set "baseuri" "cic:/matita/library_autobatch/nat/nth_prime".
+
+include "auto/nat/primes.ma".
+include "auto/nat/lt_arith.ma".
+
+(* upper bound by Bertrand's conjecture. *)
+(* Too difficult to prove.
+let rec nth_prime n \def
+match n with
+ [ O \Rightarrow (S(S O))
+ | (S p) \Rightarrow
+ let previous_prime \def S (nth_prime p) in
+ min_aux previous_prime ((S(S O))*previous_prime) primeb].
+
+theorem example8 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
+normalize.reflexivity.
+qed.
+
+theorem example9 : nth_prime (S(S(S O))) = (S(S(S(S(S(S(S O))))))).
+normalize.reflexivity.
+qed.
+
+theorem example10 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
+normalize.reflexivity.
+qed. *)
+
+theorem smallest_factor_fact: \forall n:nat.
+n < smallest_factor (S n!).
+intros.
+apply not_le_to_lt.
+unfold Not.
+intro.
+apply (not_divides_S_fact n (smallest_factor(S n!)))
+[ apply lt_SO_smallest_factor.
+ unfold lt.autobatch
+ (*apply le_S_S.
+ apply le_SO_fact*)
+| assumption
+| autobatch
+ (*apply divides_smallest_factor_n.
+ unfold lt.
+ apply le_S_S.
+ apply le_O_n*)
+]
+qed.
+
+theorem ex_prime: \forall n. (S O) \le n \to \exists m.
+n < m \land m \le S n! \land (prime m).
+intros.
+elim H
+[ apply (ex_intro nat ? (S(S O))).
+ split;autobatch
+ (*[ split
+ [ apply (le_n (S(S O)))
+ | apply (le_n (S(S O)))
+ ]
+ | apply (primeb_to_Prop (S(S O)))
+ ]*)
+| apply (ex_intro nat ? (smallest_factor (S (S n1)!))).
+ split
+ [ autobatch
+ (*split
+ [ apply smallest_factor_fact
+ | apply le_smallest_factor_n
+ ]*)
+ | (* Andrea: ancora hint non lo trova *)
+ apply prime_smallest_factor_n.
+ unfold lt.autobatch
+ (*apply le_S.
+ apply le_SSO_fact.
+ unfold lt.
+ apply le_S_S.
+ assumption*)
+ ]
+]
+qed.
+
+let rec nth_prime n \def
+match n with
+ [ O \Rightarrow (S(S O))
+ | (S p) \Rightarrow
+ let previous_prime \def (nth_prime p) in
+ let upper_bound \def S previous_prime! in
+ min_aux (upper_bound - (S previous_prime)) upper_bound primeb].
+
+(* it works, but nth_prime 4 takes already a few minutes -
+it must compute factorial of 7 ...*)
+(*
+theorem example11 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
+autobatch.
+(*normalize.reflexivity.*)
+qed.
+
+theorem example12: nth_prime (S(S(S O))) = (S(S(S(S(S(S(S O))))))).
+autobatch.
+(*normalize.reflexivity.*)
+qed.
+
+theorem example13 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
+autobatch.
+(*normalize.reflexivity.*)
+qed.
+*)
+(*
+theorem example14 : nth_prime (S(S(S(S(S O))))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
+normalize.reflexivity.
+*)
+
+theorem prime_nth_prime : \forall n:nat.prime (nth_prime n).
+intro.
+apply (nat_case n)
+[ autobatch
+ (*simplify.
+ apply (primeb_to_Prop (S(S O)))*)
+| intro.
+ change with
+ (let previous_prime \def (nth_prime m) in
+ let upper_bound \def S previous_prime! in
+ prime (min_aux (upper_bound - (S previous_prime)) upper_bound primeb)).
+ apply primeb_true_to_prime.
+ apply f_min_aux_true.
+ apply (ex_intro nat ? (smallest_factor (S (nth_prime m)!))).
+ split
+ [ split
+ [ cut (S (nth_prime m)!-(S (nth_prime m)! - (S (nth_prime m))) = (S (nth_prime m)))
+ [ rewrite > Hcut.
+ exact (smallest_factor_fact (nth_prime m))
+ | (* maybe we could factorize this proof *)
+ apply plus_to_minus.
+ autobatch
+ (*apply plus_minus_m_m.
+ apply le_S_S.
+ apply le_n_fact_n*)
+ ]
+ | apply le_smallest_factor_n
+ ]
+ | apply prime_to_primeb_true.
+ apply prime_smallest_factor_n.
+ unfold lt.autobatch
+ (*apply le_S_S.
+ apply le_SO_fact*)
+ ]
+]
+qed.
+
+(* properties of nth_prime *)
+theorem increasing_nth_prime: increasing nth_prime.
+unfold increasing.
+intros.
+change with
+(let previous_prime \def (nth_prime n) in
+let upper_bound \def S previous_prime! in
+(S previous_prime) \le min_aux (upper_bound - (S previous_prime)) upper_bound primeb).
+intros.
+cut (upper_bound - (upper_bound -(S previous_prime)) = (S previous_prime))
+[ rewrite < Hcut in \vdash (? % ?).
+ apply le_min_aux
+| apply plus_to_minus.
+ autobatch
+ (*apply plus_minus_m_m.
+ apply le_S_S.
+ apply le_n_fact_n*)
+]
+qed.
+
+variant lt_nth_prime_n_nth_prime_Sn :\forall n:nat.
+(nth_prime n) < (nth_prime (S n)) \def increasing_nth_prime.
+
+theorem injective_nth_prime: injective nat nat nth_prime.
+autobatch.
+(*apply increasing_to_injective.
+apply increasing_nth_prime.*)
+qed.
+
+theorem lt_SO_nth_prime_n : \forall n:nat. (S O) \lt nth_prime n.
+intros.
+(*usando la tattica autobatch qui, dopo svariati minuti la computazione non era
+ * ancora terminata
+ *)
+elim n
+[ unfold lt.autobatch
+ (*apply le_n*)
+| autobatch
+ (*apply (trans_lt ? (nth_prime n1))
+ [ assumption
+ | apply lt_nth_prime_n_nth_prime_Sn
+ ]*)
+]
+qed.
+
+theorem lt_O_nth_prime_n : \forall n:nat. O \lt nth_prime n.
+intros.
+autobatch.
+(*apply (trans_lt O (S O))
+[ unfold lt.
+ apply le_n
+| apply lt_SO_nth_prime_n
+]*)
+qed.
+
+theorem ex_m_le_n_nth_prime_m:
+\forall n: nat. nth_prime O \le n \to
+\exists m. nth_prime m \le n \land n < nth_prime (S m).
+autobatch.
+(*intros.
+apply increasing_to_le2
+[ exact lt_nth_prime_n_nth_prime_Sn
+| assumption
+]*)
+qed.
+
+theorem lt_nth_prime_to_not_prime: \forall n,m. nth_prime n < m \to m < nth_prime (S n)
+\to \lnot (prime m).
+intros.
+apply primeb_false_to_not_prime.
+letin previous_prime \def (nth_prime n).
+letin upper_bound \def (S previous_prime!).
+apply (lt_min_aux_to_false primeb upper_bound (upper_bound - (S previous_prime)) m)
+[ cut (S (nth_prime n)!-(S (nth_prime n)! - (S (nth_prime n))) = (S (nth_prime n)))
+ [ rewrite > Hcut.
+ assumption
+ | apply plus_to_minus.
+ autobatch
+ (*apply plus_minus_m_m.
+ apply le_S_S.
+ apply le_n_fact_n*)
+ ]
+| assumption
+]
+qed.
+
+(* nth_prime enumerates all primes *)
+theorem prime_to_nth_prime : \forall p:nat. prime p \to
+\exists i. nth_prime i = p.
+intros.
+cut (\exists m. nth_prime m \le p \land p < nth_prime (S m))
+[ elim Hcut.
+ elim H1.
+ cut (nth_prime a < p \lor nth_prime a = p)
+ [ elim Hcut1
+ [ absurd (prime p)
+ [ assumption
+ | autobatch
+ (*apply (lt_nth_prime_to_not_prime a);assumption*)
+ ]
+ | autobatch
+ (*apply (ex_intro nat ? a).
+ assumption*)
+ ]
+ | autobatch
+ (*apply le_to_or_lt_eq.
+ assumption*)
+ ]
+| apply ex_m_le_n_nth_prime_m.
+ simplify.
+ unfold prime in H.
+ elim H.
+ assumption
+]
+qed.
+
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