1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| A.Asperti, C.Sacerdoti Coen, *)
8 (* ||A|| E.Tassi, S.Zacchiroli *)
10 (* \ / Matita is distributed under the terms of the *)
11 (* v GNU Lesser General Public License Version 2.1 *)
13 (**************************************************************************)
15 set "baseuri" "cic:/matita/nat/nth_prime".
17 include "nat/primes.ma".
18 include "nat/lt_arith.ma".
20 (* upper bound by Bertrand's conjecture. *)
21 (* Too difficult to prove.
22 let rec nth_prime n \def
24 [ O \Rightarrow (S(S O))
26 let previous_prime \def S (nth_prime p) in
27 min_aux previous_prime ((S(S O))*previous_prime) primeb].
29 theorem example8 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
30 normalize.reflexivity.
33 theorem example9 : nth_prime (S(S(S O))) = (S(S(S(S(S(S(S O))))))).
34 normalize.reflexivity.
37 theorem example10 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
38 normalize.reflexivity.
41 theorem smallest_factor_fact: \forall n:nat.
42 n < smallest_factor (S n!).
44 apply not_le_to_lt.unfold Not.
46 apply (not_divides_S_fact n (smallest_factor(S n!))).
47 apply lt_SO_smallest_factor.
48 unfold lt.apply le_S_S.apply le_SO_fact.
50 apply divides_smallest_factor_n.
51 unfold lt.apply le_S_S.apply le_O_n.
54 theorem ex_prime: \forall n. (S O) \le n \to \exists m.
55 n < m \land m \le S n! \land (prime m).
58 apply (ex_intro nat ? (S(S O))).
59 split.split.apply (le_n (S(S O))).
60 apply (le_n (S(S O))).apply (primeb_to_Prop (S(S O))).
61 apply (ex_intro nat ? (smallest_factor (S (S n1)!))).
63 apply smallest_factor_fact.
64 apply le_smallest_factor_n.
65 (* Andrea: ancora hint non lo trova *)
66 apply prime_smallest_factor_n.unfold lt.
67 apply le_S.apply le_SSO_fact.
68 unfold lt.apply le_S_S.assumption.
71 let rec nth_prime n \def
73 [ O \Rightarrow (S(S O))
75 let previous_prime \def (nth_prime p) in
76 let upper_bound \def S previous_prime! in
77 min_aux upper_bound (S previous_prime) primeb].
80 theorem example11 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
81 normalize.reflexivity.
84 theorem example12: nth_prime (S(S(S O))) = (S(S(S(S(S(S(S O))))))).
85 normalize.reflexivity.
88 theorem example13 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
89 normalize.reflexivity.
92 alias num (instance 0) = "natural number".
93 theorem example14 : nth_prime 18 = 67.
94 normalize.reflexivity.
98 theorem prime_nth_prime : \forall n:nat.prime (nth_prime n).
100 apply (nat_case n).simplify.
101 apply (primeb_to_Prop (S(S O))).
104 (let previous_prime \def (nth_prime m) in
105 let upper_bound \def S previous_prime! in
106 prime (min_aux upper_bound (S previous_prime) primeb)).
107 apply primeb_true_to_prime.
108 apply f_min_aux_true.
109 apply (ex_intro nat ? (smallest_factor (S (nth_prime m)!))).
111 apply smallest_factor_fact.
113 [2: apply le_smallest_factor_n
115 | apply (le_plus_n_r (S (nth_prime m)) (S (fact (nth_prime m))))
117 apply prime_to_primeb_true.
118 apply prime_smallest_factor_n.unfold lt.
119 apply le_S_S.apply le_SO_fact.
122 (* properties of nth_prime *)
123 theorem increasing_nth_prime: increasing nth_prime.
127 (let previous_prime \def (nth_prime n) in
128 let upper_bound \def S previous_prime! in
129 (S previous_prime) \le min_aux upper_bound (S previous_prime) primeb).
134 variant lt_nth_prime_n_nth_prime_Sn :\forall n:nat.
135 (nth_prime n) < (nth_prime (S n)) \def increasing_nth_prime.
137 theorem injective_nth_prime: injective nat nat nth_prime.
138 apply increasing_to_injective.
139 apply increasing_nth_prime.
142 theorem lt_SO_nth_prime_n : \forall n:nat. (S O) \lt nth_prime n.
143 intros. elim n.unfold lt.apply le_n.
144 apply (trans_lt ? (nth_prime n1)).
145 assumption.apply lt_nth_prime_n_nth_prime_Sn.
148 theorem lt_O_nth_prime_n : \forall n:nat. O \lt nth_prime n.
149 intros.apply (trans_lt O (S O)).
150 unfold lt. apply le_n.apply lt_SO_nth_prime_n.
153 theorem lt_n_nth_prime_n : \forall n:nat. n \lt nth_prime n.
156 [apply lt_O_nth_prime_n
157 |apply (lt_to_le_to_lt ? (S (nth_prime n1)))
158 [unfold.apply le_S_S.assumption
159 |apply lt_nth_prime_n_nth_prime_Sn
164 theorem ex_m_le_n_nth_prime_m:
165 \forall n: nat. nth_prime O \le n \to
166 \exists m. nth_prime m \le n \land n < nth_prime (S m).
168 apply increasing_to_le2.
169 exact lt_nth_prime_n_nth_prime_Sn.assumption.
172 theorem lt_nth_prime_to_not_prime: \forall n,m. nth_prime n < m \to m < nth_prime (S n)
175 apply primeb_false_to_not_prime.
176 letin previous_prime \def (nth_prime n).
177 letin upper_bound \def (S previous_prime!).
178 apply (lt_min_aux_to_false primeb (S previous_prime) upper_bound m).
181 apply (transitive_le (S m) (nth_prime (S n)) (min_aux (S (fact (nth_prime n))) (S (nth_prime n)) primeb) ? ?);
183 |apply (le_n (min_aux (S (fact (nth_prime n))) (S (nth_prime n)) primeb)).
187 (* nth_prime enumerates all primes *)
188 theorem prime_to_nth_prime : \forall p:nat. prime p \to
189 \exists i. nth_prime i = p.
191 cut (\exists m. nth_prime m \le p \land p < nth_prime (S m)).
193 cut (nth_prime a < p \lor nth_prime a = p).
197 apply (lt_nth_prime_to_not_prime a).assumption.assumption.
198 apply (ex_intro nat ? a).assumption.
199 apply le_to_or_lt_eq.assumption.
200 apply ex_m_le_n_nth_prime_m.
201 simplify.unfold prime in H.elim H.assumption.