(* Title: HOL/Number_Theory/Eratosthenes.thy
Author: Florian Haftmann, TU Muenchen
*)
section \<open>The sieve of Eratosthenes\<close>
theory Eratosthenes
imports Main "HOL-Computational_Algebra.Primes"
begin
subsection \<open>Preliminary: strict divisibility\<close>
context dvd
begin
abbreviation dvd_strict :: "'a \ 'a \ bool" (infixl "dvd'_strict" 50)
where
"b dvd_strict a \ b dvd a \ \ a dvd b"
end
subsection \<open>Main corpus\<close>
text \<open>The sieve is modelled as a list of booleans, where \<^const>\<open>False\<close> means \emph{marked out}.\<close>
type_synonym marks = "bool list"
definition numbers_of_marks :: "nat \ marks \ nat set"
where
"numbers_of_marks n bs = fst ` {x \ set (enumerate n bs). snd x}"
lemma numbers_of_marks_simps [simp, code]:
"numbers_of_marks n [] = {}"
"numbers_of_marks n (True # bs) = insert n (numbers_of_marks (Suc n) bs)"
"numbers_of_marks n (False # bs) = numbers_of_marks (Suc n) bs"
by (auto simp add: numbers_of_marks_def intro!: image_eqI)
lemma numbers_of_marks_Suc:
"numbers_of_marks (Suc n) bs = Suc ` numbers_of_marks n bs"
by (auto simp add: numbers_of_marks_def enumerate_Suc_eq image_iff Bex_def)
lemma numbers_of_marks_replicate_False [simp]:
"numbers_of_marks n (replicate m False) = {}"
by (auto simp add: numbers_of_marks_def enumerate_replicate_eq)
lemma numbers_of_marks_replicate_True [simp]:
"numbers_of_marks n (replicate m True) = {n..
by (auto simp add: numbers_of_marks_def enumerate_replicate_eq image_def)
lemma in_numbers_of_marks_eq:
"m \ numbers_of_marks n bs \ m \ {n.. bs ! (m - n)"
by (simp add: numbers_of_marks_def in_set_enumerate_eq image_iff add.commute)
lemma sorted_list_of_set_numbers_of_marks:
"sorted_list_of_set (numbers_of_marks n bs) = map fst (filter snd (enumerate n bs))"
by (auto simp add: numbers_of_marks_def distinct_map
intro!: sorted_filter distinct_filter inj_onI sorted_distinct_set_unique)
text \<open>Marking out multiples in a sieve\<close>
definition mark_out :: "nat \ marks \ marks"
where
"mark_out n bs = map (\(q, b). b \ \ Suc n dvd Suc (Suc q)) (enumerate n bs)"
lemma mark_out_Nil [simp]: "mark_out n [] = []"
by (simp add: mark_out_def)
lemma length_mark_out [simp]: "length (mark_out n bs) = length bs"
by (simp add: mark_out_def)
lemma numbers_of_marks_mark_out:
"numbers_of_marks n (mark_out m bs) = {q \ numbers_of_marks n bs. \ Suc m dvd Suc q - n}"
by (auto simp add: numbers_of_marks_def mark_out_def in_set_enumerate_eq image_iff
nth_enumerate_eq less_eq_dvd_minus)
text \<open>Auxiliary operation for efficient implementation\<close>
definition mark_out_aux :: "nat \ nat \ marks \ marks"
where
"mark_out_aux n m bs =
map (\<lambda>(q, b). b \<and> (q < m + n \<or> \<not> Suc n dvd Suc (Suc q) + (n - m mod Suc n))) (enumerate n bs)"
lemma mark_out_code [code]: "mark_out n bs = mark_out_aux n n bs"
proof -
have aux: False
if A: "Suc n dvd Suc (Suc a)"
and B: "a < n + n"
and C: "n \ a"
for a
proof (cases "n = 0")
case True
with A B C show ?thesis by simp
next
case False
define m where "m = Suc n"
then have "m > 0" by simp
from False have "n > 0" by simp
from A obtain q where q: "Suc (Suc a) = Suc n * q" by (rule dvdE)
have "q > 0"
proof (rule ccontr)
assume "\ q > 0"
with q show False by simp
qed
with \<open>n > 0\<close> have "Suc n * q \<ge> 2" by (auto simp add: gr0_conv_Suc)
with q have a: "a = Suc n * q - 2" by simp
with B have "q + n * q < n + n + 2" by auto
then have "m * q < m * 2" by (simp add: m_def)
with \<open>m > 0\<close> have "q < 2" by simp
with \<open>q > 0\<close> have "q = 1" by simp
with a have "a = n - 1" by simp
with \<open>n > 0\<close> C show False by simp
qed
show ?thesis
by (auto simp add: mark_out_def mark_out_aux_def in_set_enumerate_eq intro: aux)
qed
lemma mark_out_aux_simps [simp, code]:
"mark_out_aux n m [] = []"
"mark_out_aux n 0 (b # bs) = False # mark_out_aux n n bs"
"mark_out_aux n (Suc m) (b # bs) = b # mark_out_aux n m bs"
proof goal_cases
case 1
show ?case
by (simp add: mark_out_aux_def)
next
case 2
show ?case
by (auto simp add: mark_out_code [symmetric] mark_out_aux_def mark_out_def
enumerate_Suc_eq in_set_enumerate_eq less_eq_dvd_minus)
next
case 3
{ define v where "v = Suc m"
define w where "w = Suc n"
fix q
assume "m + n \ q"
then obtain r where q: "q = m + n + r" by (auto simp add: le_iff_add)
{ fix u
from w_def have "u mod w < w" by simp
then have "u + (w - u mod w) = w + (u - u mod w)"
by simp
then have "u + (w - u mod w) = w + u div w * w"
by (simp add: minus_mod_eq_div_mult)
}
then have "w dvd v + w + r + (w - v mod w) \ w dvd m + w + r + (w - m mod w)"
by (simp add: add.assoc add.left_commute [of m] add.left_commute [of v]
dvd_add_left_iff dvd_add_right_iff)
moreover from q have "Suc q = m + w + r" by (simp add: w_def)
moreover from q have "Suc (Suc q) = v + w + r" by (simp add: v_def w_def)
ultimately have "w dvd Suc (Suc (q + (w - v mod w))) \ w dvd Suc (q + (w - m mod w))"
by (simp only: add_Suc [symmetric])
then have "Suc n dvd Suc (Suc (Suc (q + n) - Suc m mod Suc n)) \
Suc n dvd Suc (Suc (q + n - m mod Suc n))"
by (simp add: v_def w_def Suc_diff_le trans_le_add2)
}
then show ?case
by (auto simp add: mark_out_aux_def
enumerate_Suc_eq in_set_enumerate_eq not_less)
qed
text \<open>Main entry point to sieve\<close>
fun sieve :: "nat \ marks \ marks"
where
"sieve n [] = []"
| "sieve n (False # bs) = False # sieve (Suc n) bs"
| "sieve n (True # bs) = True # sieve (Suc n) (mark_out n bs)"
text \<open>
There are the following possible optimisations here:
\begin{itemize}
\item \<^const>\<open>sieve\<close> can abort as soon as \<^term>\<open>n\<close> is too big to let
\<^const>\<open>mark_out\<close> have any effect.
\item Search for further primes can be given up as soon as the search
position exceeds the square root of the maximum candidate.
\end{itemize}
This is left as an constructive exercise to the reader.
\<close>
lemma numbers_of_marks_sieve:
"numbers_of_marks (Suc n) (sieve n bs) =
{q \<in> numbers_of_marks (Suc n) bs. \<forall>m \<in> numbers_of_marks (Suc n) bs. \<not> m dvd_strict q}"
proof (induct n bs rule: sieve.induct)
case 1
show ?case by simp
next
case 2
then show ?case by simp
next
case (3 n bs)
have aux: "n \ Suc ` M \ n > 0 \ n - 1 \ M" (is "?lhs \ ?rhs") for M n
proof
show ?rhs if ?lhs using that by auto
show ?lhs if ?rhs
proof -
from that have "n > 0" and "n - 1 \ M" by auto
then have "Suc (n - 1) \ Suc ` M" by blast
with \<open>n > 0\<close> show "n \<in> Suc ` M" by simp
qed
qed
have aux1: False if "Suc (Suc n) \ m" and "m dvd Suc n" for m :: nat
proof -
from \<open>m dvd Suc n\<close> obtain q where "Suc n = m * q" ..
with \<open>Suc (Suc n) \<le> m\<close> have "Suc (m * q) \<le> m" by simp
then have "m * q < m" by arith
then have "q = 0" by simp
with \<open>Suc n = m * q\<close> show ?thesis by simp
qed
have aux2: "m dvd q"
if 1: "\q>0. 1 < q \ Suc n < q \ q \ Suc (n + length bs) \
bs ! (q - Suc (Suc n)) \<longrightarrow> \<not> Suc n dvd q \<longrightarrow> q dvd m \<longrightarrow> m dvd q"
and 2: "\ Suc n dvd m" "q dvd m"
and 3: "Suc n < q" "q \ Suc (n + length bs)" "bs ! (q - Suc (Suc n))"
for m q :: nat
proof -
from 1 have *: "\q. Suc n < q \ q \ Suc (n + length bs) \
bs ! (q - Suc (Suc n)) \<Longrightarrow> \<not> Suc n dvd q \<Longrightarrow> q dvd m \<Longrightarrow> m dvd q"
by auto
from 2 have "\ Suc n dvd q" by (auto elim: dvdE)
moreover note 3
moreover note \<open>q dvd m\<close>
ultimately show ?thesis by (auto intro: *)
qed
from 3 show ?case
apply (simp_all add: numbers_of_marks_mark_out numbers_of_marks_Suc Compr_image_eq
inj_image_eq_iff in_numbers_of_marks_eq Ball_def imp_conjL aux)
apply safe
apply (simp_all add: less_diff_conv2 le_diff_conv2 dvd_minus_self not_less)
apply (clarsimp dest!: aux1)
apply (simp add: Suc_le_eq less_Suc_eq_le)
apply (rule aux2)
apply (clarsimp dest!: aux1)+
done
qed
text \<open>Relation of the sieve algorithm to actual primes\<close>
definition primes_upto :: "nat \ nat list"
where
"primes_upto n = sorted_list_of_set {m. m \ n \ prime m}"
lemma set_primes_upto: "set (primes_upto n) = {m. m \ n \ prime m}"
by (simp add: primes_upto_def)
lemma sorted_primes_upto [iff]: "sorted (primes_upto n)"
by (simp add: primes_upto_def)
lemma distinct_primes_upto [iff]: "distinct (primes_upto n)"
by (simp add: primes_upto_def)
lemma set_primes_upto_sieve:
"set (primes_upto n) = numbers_of_marks 2 (sieve 1 (replicate (n - 1) True))"
proof -
consider "n = 0 \ n = 1" | "n > 1" by arith
then show ?thesis
proof cases
case 1
then show ?thesis
by (auto simp add: numbers_of_marks_sieve numeral_2_eq_2 set_primes_upto
dest: prime_gt_Suc_0_nat)
next
case 2
{
fix m q
assume "Suc (Suc 0) \ q"
and "q < Suc n"
and "m dvd q"
then have "m < Suc n" by (auto dest: dvd_imp_le)
assume *: "\m\{Suc (Suc 0).. q dvd m"
and "m dvd q" and "m \ 1"
have "m = q"
proof (cases "m = 0")
case True with \<open>m dvd q\<close> show ?thesis by simp
next
case False with \<open>m \<noteq> 1\<close> have "Suc (Suc 0) \<le> m" by arith
with \<open>m < Suc n\<close> * \<open>m dvd q\<close> have "q dvd m" by simp
with \<open>m dvd q\<close> show ?thesis by (simp add: dvd_antisym)
qed
}
then have aux: "\m q. Suc (Suc 0) \ q \
q < Suc n \<Longrightarrow>
m dvd q \<Longrightarrow>
\<forall>m\<in>{Suc (Suc 0)..<Suc n}. m dvd q \<longrightarrow> q dvd m \<Longrightarrow>
m dvd q \<Longrightarrow> m \<noteq> q \<Longrightarrow> m = 1" by auto
from 2 show ?thesis
apply (auto simp add: numbers_of_marks_sieve numeral_2_eq_2 set_primes_upto
dest: prime_gt_Suc_0_nat)
apply (metis One_nat_def Suc_le_eq less_not_refl prime_nat_iff)
apply (metis One_nat_def Suc_le_eq aux prime_nat_iff)
done
qed
qed
lemma primes_upto_sieve [code]:
"primes_upto n = map fst (filter snd (enumerate 2 (sieve 1 (replicate (n - 1) True))))"
proof -
have "primes_upto n = sorted_list_of_set (numbers_of_marks 2 (sieve 1 (replicate (n - 1) True)))"
apply (rule sorted_distinct_set_unique)
apply (simp_all only: set_primes_upto_sieve numbers_of_marks_def)
apply auto
done
then show ?thesis
by (simp add: sorted_list_of_set_numbers_of_marks)
qed
lemma prime_in_primes_upto: "prime n \ n \ set (primes_upto n)"
by (simp add: set_primes_upto)
subsection \<open>Application: smallest prime beyond a certain number\<close>
definition smallest_prime_beyond :: "nat \ nat"
where
"smallest_prime_beyond n = (LEAST p. prime p \ p \ n)"
lemma prime_smallest_prime_beyond [iff]: "prime (smallest_prime_beyond n)" (is ?P)
and smallest_prime_beyond_le [iff]: "smallest_prime_beyond n \ n" (is ?Q)
proof -
let ?least = "LEAST p. prime p \ p \ n"
from primes_infinite obtain q where "prime q \ q \ n"
by (metis finite_nat_set_iff_bounded_le mem_Collect_eq nat_le_linear)
then have "prime ?least \ ?least \ n"
by (rule LeastI)
then show ?P and ?Q
by (simp_all add: smallest_prime_beyond_def)
qed
lemma smallest_prime_beyond_smallest: "prime p \ p \ n \ smallest_prime_beyond n \ p"
by (simp only: smallest_prime_beyond_def) (auto intro: Least_le)
lemma smallest_prime_beyond_eq:
"prime p \ p \ n \ (\q. prime q \ q \ n \ q \ p) \ smallest_prime_beyond n = p"
by (simp only: smallest_prime_beyond_def) (auto intro: Least_equality)
definition smallest_prime_between :: "nat \ nat \ nat option"
where
"smallest_prime_between m n =
(if (\<exists>p. prime p \<and> m \<le> p \<and> p \<le> n) then Some (smallest_prime_beyond m) else None)"
lemma smallest_prime_between_None:
"smallest_prime_between m n = None \ (\q. m \ q \ q \ n \ \ prime q)"
by (auto simp add: smallest_prime_between_def)
lemma smallest_prime_betwen_Some:
"smallest_prime_between m n = Some p \ smallest_prime_beyond m = p \ p \ n"
by (auto simp add: smallest_prime_between_def dest: smallest_prime_beyond_smallest [of _ m])
lemma [code]: "smallest_prime_between m n = List.find (\p. p \ m) (primes_upto n)"
proof -
have "List.find (\p. p \ m) (primes_upto n) = Some (smallest_prime_beyond m)"
if assms: "m \ p" "prime p" "p \ n" for p
proof -
define A where "A = {p. p \ n \ prime p \ m \ p}"
from assms have "smallest_prime_beyond m \ p"
by (auto intro: smallest_prime_beyond_smallest)
from this \<open>p \<le> n\<close> have *: "smallest_prime_beyond m \<le> n"
by (rule order_trans)
from assms have ex: "\p\n. prime p \ m \ p"
by auto
then have "finite A"
by (auto simp add: A_def)
with * have "Min A = smallest_prime_beyond m"
by (auto simp add: A_def intro: Min_eqI smallest_prime_beyond_smallest)
with ex sorted_primes_upto show ?thesis
by (auto simp add: set_primes_upto sorted_find_Min A_def)
qed
then show ?thesis
by (auto simp add: smallest_prime_between_def find_None_iff set_primes_upto
intro!: sym [of _ None])
qed
definition smallest_prime_beyond_aux :: "nat \ nat \ nat"
where
"smallest_prime_beyond_aux k n = smallest_prime_beyond n"
lemma [code]:
"smallest_prime_beyond_aux k n =
(case smallest_prime_between n (k * n) of
Some p \<Rightarrow> p
| None \<Rightarrow> smallest_prime_beyond_aux (Suc k) n)"
by (simp add: smallest_prime_beyond_aux_def smallest_prime_betwen_Some split: option.split)
lemma [code]: "smallest_prime_beyond n = smallest_prime_beyond_aux 2 n"
by (simp add: smallest_prime_beyond_aux_def)
end
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