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 (fact n)).
45 change with smallest_factor (S (fact n)) \le n \to False.intro.
46 apply not_divides_S_fact n (smallest_factor(S (fact n))).
47 apply lt_SO_smallest_factor.
48 simplify.apply le_S_S.apply le_SO_fact.
50 apply divides_smallest_factor_n.
51 simplify.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 (fact 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 (fact (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.
67 change with (S(S O)) \le S (fact (S n1)).
68 apply le_S.apply le_SSO_fact.
69 simplify.apply le_S_S.assumption.
72 let rec nth_prime n \def
74 [ O \Rightarrow (S(S O))
76 let previous_prime \def (nth_prime p) in
77 let upper_bound \def S (fact previous_prime) in
78 min_aux (upper_bound - (S previous_prime)) upper_bound primeb].
80 (* it works, but nth_prime 4 takes already a few minutes -
81 it must compute factorial of 7 ...
83 theorem example11 : nth_prime (S(S O)) = (S(S(S(S(S O))))).
84 normalize.reflexivity.
87 theorem example12: nth_prime (S(S(S O))) = (S(S(S(S(S(S(S O))))))).
88 normalize.reflexivity.
91 theorem example13 : nth_prime (S(S(S(S O)))) = (S(S(S(S(S(S(S(S(S(S(S O))))))))))).
92 normalize.reflexivity.
95 theorem prime_nth_prime : \forall n:nat.prime (nth_prime n).
98 change with prime (S(S O)).
99 apply primeb_to_Prop (S(S O)).
102 let previous_prime \def (nth_prime m) in
103 let upper_bound \def S (fact previous_prime) in
104 prime (min_aux (upper_bound - (S previous_prime)) upper_bound primeb).
105 apply primeb_true_to_prime.
106 apply f_min_aux_true.
107 apply ex_intro nat ? (smallest_factor (S (fact (nth_prime m)))).
109 cut S (fact (nth_prime m))-(S (fact (nth_prime m)) - (S (nth_prime m))) = (S (nth_prime m)).
110 rewrite > Hcut.exact smallest_factor_fact (nth_prime m).
111 (* maybe we could factorize this proof *)
114 apply plus_minus_m_m.
117 apply le_smallest_factor_n.
118 apply prime_to_primeb_true.
119 apply prime_smallest_factor_n.
120 change with (S(S O)) \le S (fact (nth_prime m)).
121 apply le_S_S.apply le_SO_fact.
124 (* properties of nth_prime *)
125 theorem increasing_nth_prime: increasing nth_prime.
126 change with \forall n:nat. (nth_prime n) < (nth_prime (S n)).
129 let previous_prime \def (nth_prime n) in
130 let upper_bound \def S (fact previous_prime) in
131 (S previous_prime) \le min_aux (upper_bound - (S previous_prime)) upper_bound primeb.
133 cut upper_bound - (upper_bound -(S previous_prime)) = (S previous_prime).
134 rewrite < Hcut in \vdash (? % ?).
138 apply plus_minus_m_m.
143 variant lt_nth_prime_n_nth_prime_Sn :\forall n:nat.
144 (nth_prime n) < (nth_prime (S n)) \def increasing_nth_prime.
146 theorem injective_nth_prime: injective nat nat nth_prime.
147 apply increasing_to_injective.
148 apply increasing_nth_prime.
151 theorem lt_SO_nth_prime_n : \forall n:nat. (S O) \lt nth_prime n.
152 intros. elim n.simplify.apply le_n.
153 apply trans_lt ? (nth_prime n1).
154 assumption.apply lt_nth_prime_n_nth_prime_Sn.
157 theorem lt_O_nth_prime_n : \forall n:nat. O \lt nth_prime n.
158 intros.apply trans_lt O (S O).
159 simplify. apply le_n.apply lt_SO_nth_prime_n.
162 theorem ex_m_le_n_nth_prime_m:
163 \forall n: nat. nth_prime O \le n \to
164 \exists m. nth_prime m \le n \land n < nth_prime (S m).
166 apply increasing_to_le2.
167 exact lt_nth_prime_n_nth_prime_Sn.assumption.
170 theorem lt_nth_prime_to_not_prime: \forall n,m. nth_prime n < m \to m < nth_prime (S n)
173 apply primeb_false_to_not_prime.
174 letin previous_prime \def nth_prime n.
175 letin upper_bound \def S (fact previous_prime).
176 apply lt_min_aux_to_false primeb upper_bound (upper_bound - (S previous_prime)) m.
177 cut S (fact (nth_prime n))-(S (fact (nth_prime n)) - (S (nth_prime n))) = (S (nth_prime n)).
178 rewrite > Hcut.assumption.
181 apply plus_minus_m_m.
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.simplify in H.elim H.assumption.