--- a/src/HOL/NumberTheory/IntPrimes.thy Tue Sep 29 22:15:54 2009 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,423 +0,0 @@
-(* Title: HOL/NumberTheory/IntPrimes.thy
- ID: $Id$
- Author: Thomas M. Rasmussen
- Copyright 2000 University of Cambridge
-*)
-
-header {* Divisibility and prime numbers (on integers) *}
-
-theory IntPrimes
-imports Main Primes
-begin
-
-text {*
- The @{text dvd} relation, GCD, Euclid's extended algorithm, primes,
- congruences (all on the Integers). Comparable to theory @{text
- Primes}, but @{text dvd} is included here as it is not present in
- main HOL. Also includes extended GCD and congruences not present in
- @{text Primes}.
-*}
-
-
-subsection {* Definitions *}
-
-consts
- xzgcda :: "int * int * int * int * int * int * int * int => int * int * int"
-
-recdef xzgcda
- "measure ((\<lambda>(m, n, r', r, s', s, t', t). nat r)
- :: int * int * int * int *int * int * int * int => nat)"
- "xzgcda (m, n, r', r, s', s, t', t) =
- (if r \<le> 0 then (r', s', t')
- else xzgcda (m, n, r, r' mod r,
- s, s' - (r' div r) * s,
- t, t' - (r' div r) * t))"
-
-definition
- zprime :: "int \<Rightarrow> bool" where
- "zprime p = (1 < p \<and> (\<forall>m. 0 <= m & m dvd p --> m = 1 \<or> m = p))"
-
-definition
- xzgcd :: "int => int => int * int * int" where
- "xzgcd m n = xzgcda (m, n, m, n, 1, 0, 0, 1)"
-
-definition
- zcong :: "int => int => int => bool" ("(1[_ = _] '(mod _'))") where
- "[a = b] (mod m) = (m dvd (a - b))"
-
-subsection {* Euclid's Algorithm and GCD *}
-
-
-lemma zrelprime_zdvd_zmult_aux:
- "zgcd n k = 1 ==> k dvd m * n ==> 0 \<le> m ==> k dvd m"
- by (metis abs_of_nonneg dvd_triv_right zgcd_greatest_iff zgcd_zmult_distrib2_abs zmult_1_right)
-
-lemma zrelprime_zdvd_zmult: "zgcd n k = 1 ==> k dvd m * n ==> k dvd m"
- apply (case_tac "0 \<le> m")
- apply (blast intro: zrelprime_zdvd_zmult_aux)
- apply (subgoal_tac "k dvd -m")
- apply (rule_tac [2] zrelprime_zdvd_zmult_aux, auto)
- done
-
-lemma zgcd_geq_zero: "0 <= zgcd x y"
- by (auto simp add: zgcd_def)
-
-text{*This is merely a sanity check on zprime, since the previous version
- denoted the empty set.*}
-lemma "zprime 2"
- apply (auto simp add: zprime_def)
- apply (frule zdvd_imp_le, simp)
- apply (auto simp add: order_le_less dvd_def)
- done
-
-lemma zprime_imp_zrelprime:
- "zprime p ==> \<not> p dvd n ==> zgcd n p = 1"
- apply (auto simp add: zprime_def)
- apply (metis zgcd_geq_zero zgcd_zdvd1 zgcd_zdvd2)
- done
-
-lemma zless_zprime_imp_zrelprime:
- "zprime p ==> 0 < n ==> n < p ==> zgcd n p = 1"
- apply (erule zprime_imp_zrelprime)
- apply (erule zdvd_not_zless, assumption)
- done
-
-lemma zprime_zdvd_zmult:
- "0 \<le> (m::int) ==> zprime p ==> p dvd m * n ==> p dvd m \<or> p dvd n"
- by (metis zgcd_zdvd1 zgcd_zdvd2 zgcd_pos zprime_def zrelprime_dvd_mult)
-
-lemma zgcd_zadd_zmult [simp]: "zgcd (m + n * k) n = zgcd m n"
- apply (rule zgcd_eq [THEN trans])
- apply (simp add: mod_add_eq)
- apply (rule zgcd_eq [symmetric])
- done
-
-lemma zgcd_zdvd_zgcd_zmult: "zgcd m n dvd zgcd (k * m) n"
-by (simp add: zgcd_greatest_iff)
-
-lemma zgcd_zmult_zdvd_zgcd:
- "zgcd k n = 1 ==> zgcd (k * m) n dvd zgcd m n"
- apply (simp add: zgcd_greatest_iff)
- apply (rule_tac n = k in zrelprime_zdvd_zmult)
- prefer 2
- apply (simp add: zmult_commute)
- apply (metis zgcd_1 zgcd_commute zgcd_left_commute)
- done
-
-lemma zgcd_zmult_cancel: "zgcd k n = 1 ==> zgcd (k * m) n = zgcd m n"
- by (simp add: zgcd_def nat_abs_mult_distrib gcd_mult_cancel)
-
-lemma zgcd_zgcd_zmult:
- "zgcd k m = 1 ==> zgcd n m = 1 ==> zgcd (k * n) m = 1"
- by (simp add: zgcd_zmult_cancel)
-
-lemma zdvd_iff_zgcd: "0 < m ==> m dvd n \<longleftrightarrow> zgcd n m = m"
- by (metis abs_of_pos zdvd_mult_div_cancel zgcd_0 zgcd_commute zgcd_geq_zero zgcd_zdvd2 zgcd_zmult_eq_self)
-
-
-
-subsection {* Congruences *}
-
-lemma zcong_1 [simp]: "[a = b] (mod 1)"
- by (unfold zcong_def, auto)
-
-lemma zcong_refl [simp]: "[k = k] (mod m)"
- by (unfold zcong_def, auto)
-
-lemma zcong_sym: "[a = b] (mod m) = [b = a] (mod m)"
- unfolding zcong_def minus_diff_eq [of a, symmetric] dvd_minus_iff ..
-
-lemma zcong_zadd:
- "[a = b] (mod m) ==> [c = d] (mod m) ==> [a + c = b + d] (mod m)"
- apply (unfold zcong_def)
- apply (rule_tac s = "(a - b) + (c - d)" in subst)
- apply (rule_tac [2] dvd_add, auto)
- done
-
-lemma zcong_zdiff:
- "[a = b] (mod m) ==> [c = d] (mod m) ==> [a - c = b - d] (mod m)"
- apply (unfold zcong_def)
- apply (rule_tac s = "(a - b) - (c - d)" in subst)
- apply (rule_tac [2] dvd_diff, auto)
- done
-
-lemma zcong_trans:
- "[a = b] (mod m) ==> [b = c] (mod m) ==> [a = c] (mod m)"
-unfolding zcong_def by (auto elim!: dvdE simp add: algebra_simps)
-
-lemma zcong_zmult:
- "[a = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"
- apply (rule_tac b = "b * c" in zcong_trans)
- apply (unfold zcong_def)
- apply (metis zdiff_zmult_distrib2 dvd_mult zmult_commute)
- apply (metis zdiff_zmult_distrib2 dvd_mult)
- done
-
-lemma zcong_scalar: "[a = b] (mod m) ==> [a * k = b * k] (mod m)"
- by (rule zcong_zmult, simp_all)
-
-lemma zcong_scalar2: "[a = b] (mod m) ==> [k * a = k * b] (mod m)"
- by (rule zcong_zmult, simp_all)
-
-lemma zcong_zmult_self: "[a * m = b * m] (mod m)"
- apply (unfold zcong_def)
- apply (rule dvd_diff, simp_all)
- done
-
-lemma zcong_square:
- "[| zprime p; 0 < a; [a * a = 1] (mod p)|]
- ==> [a = 1] (mod p) \<or> [a = p - 1] (mod p)"
- apply (unfold zcong_def)
- apply (rule zprime_zdvd_zmult)
- apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)
- prefer 4
- apply (simp add: zdvd_reduce)
- apply (simp_all add: zdiff_zmult_distrib zmult_commute zdiff_zmult_distrib2)
- done
-
-lemma zcong_cancel:
- "0 \<le> m ==>
- zgcd k m = 1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
- apply safe
- prefer 2
- apply (blast intro: zcong_scalar)
- apply (case_tac "b < a")
- prefer 2
- apply (subst zcong_sym)
- apply (unfold zcong_def)
- apply (rule_tac [!] zrelprime_zdvd_zmult)
- apply (simp_all add: zdiff_zmult_distrib)
- apply (subgoal_tac "m dvd (-(a * k - b * k))")
- apply simp
- apply (subst dvd_minus_iff, assumption)
- done
-
-lemma zcong_cancel2:
- "0 \<le> m ==>
- zgcd k m = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
- by (simp add: zmult_commute zcong_cancel)
-
-lemma zcong_zgcd_zmult_zmod:
- "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd m n = 1
- ==> [a = b] (mod m * n)"
- apply (auto simp add: zcong_def dvd_def)
- apply (subgoal_tac "m dvd n * ka")
- apply (subgoal_tac "m dvd ka")
- apply (case_tac [2] "0 \<le> ka")
- apply (metis zdvd_mult_div_cancel dvd_refl dvd_mult_left zmult_commute zrelprime_zdvd_zmult)
- apply (metis abs_dvd_iff abs_of_nonneg zadd_0 zgcd_0_left zgcd_commute zgcd_zadd_zmult zgcd_zdvd_zgcd_zmult zgcd_zmult_distrib2_abs zmult_1_right zmult_commute)
- apply (metis mult_le_0_iff zdvd_mono zdvd_mult_cancel dvd_triv_left zero_le_mult_iff zle_anti_sym zle_linear zle_refl zmult_commute zrelprime_zdvd_zmult)
- apply (metis dvd_triv_left)
- done
-
-lemma zcong_zless_imp_eq:
- "0 \<le> a ==>
- a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
- apply (unfold zcong_def dvd_def, auto)
- apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
- apply (metis diff_add_cancel mod_pos_pos_trivial zadd_0 zadd_commute zmod_eq_0_iff mod_add_right_eq)
- done
-
-lemma zcong_square_zless:
- "zprime p ==> 0 < a ==> a < p ==>
- [a * a = 1] (mod p) ==> a = 1 \<or> a = p - 1"
- apply (cut_tac p = p and a = a in zcong_square)
- apply (simp add: zprime_def)
- apply (auto intro: zcong_zless_imp_eq)
- done
-
-lemma zcong_not:
- "0 < a ==> a < m ==> 0 < b ==> b < a ==> \<not> [a = b] (mod m)"
- apply (unfold zcong_def)
- apply (rule zdvd_not_zless, auto)
- done
-
-lemma zcong_zless_0:
- "0 \<le> a ==> a < m ==> [a = 0] (mod m) ==> a = 0"
- apply (unfold zcong_def dvd_def, auto)
- apply (metis div_pos_pos_trivial linorder_not_less div_mult_self1_is_id)
- done
-
-lemma zcong_zless_unique:
- "0 < m ==> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
- apply auto
- prefer 2 apply (metis zcong_sym zcong_trans zcong_zless_imp_eq)
- apply (unfold zcong_def dvd_def)
- apply (rule_tac x = "a mod m" in exI, auto)
- apply (metis zmult_div_cancel)
- done
-
-lemma zcong_iff_lin: "([a = b] (mod m)) = (\<exists>k. b = a + m * k)"
- unfolding zcong_def
- apply (auto elim!: dvdE simp add: algebra_simps)
- apply (rule_tac x = "-k" in exI) apply simp
- done
-
-lemma zgcd_zcong_zgcd:
- "0 < m ==>
- zgcd a m = 1 ==> [a = b] (mod m) ==> zgcd b m = 1"
- by (auto simp add: zcong_iff_lin)
-
-lemma zcong_zmod_aux:
- "a - b = (m::int) * (a div m - b div m) + (a mod m - b mod m)"
- by(simp add: zdiff_zmult_distrib2 add_diff_eq eq_diff_eq add_ac)
-
-lemma zcong_zmod: "[a = b] (mod m) = [a mod m = b mod m] (mod m)"
- apply (unfold zcong_def)
- apply (rule_tac t = "a - b" in ssubst)
- apply (rule_tac m = m in zcong_zmod_aux)
- apply (rule trans)
- apply (rule_tac [2] k = m and m = "a div m - b div m" in zdvd_reduce)
- apply (simp add: zadd_commute)
- done
-
-lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
- apply auto
- apply (metis pos_mod_conj zcong_zless_imp_eq zcong_zmod)
- apply (metis zcong_refl zcong_zmod)
- done
-
-lemma zcong_zminus [iff]: "[a = b] (mod -m) = [a = b] (mod m)"
- by (auto simp add: zcong_def)
-
-lemma zcong_zero [iff]: "[a = b] (mod 0) = (a = b)"
- by (auto simp add: zcong_def)
-
-lemma "[a = b] (mod m) = (a mod m = b mod m)"
- apply (case_tac "m = 0", simp add: DIVISION_BY_ZERO)
- apply (simp add: linorder_neq_iff)
- apply (erule disjE)
- prefer 2 apply (simp add: zcong_zmod_eq)
- txt{*Remainding case: @{term "m<0"}*}
- apply (rule_tac t = m in zminus_zminus [THEN subst])
- apply (subst zcong_zminus)
- apply (subst zcong_zmod_eq, arith)
- apply (frule neg_mod_bound [of _ a], frule neg_mod_bound [of _ b])
- apply (simp add: zmod_zminus2_eq_if del: neg_mod_bound)
- done
-
-subsection {* Modulo *}
-
-lemma zmod_zdvd_zmod:
- "0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"
- by (rule mod_mod_cancel)
-
-
-subsection {* Extended GCD *}
-
-declare xzgcda.simps [simp del]
-
-lemma xzgcd_correct_aux1:
- "zgcd r' r = k --> 0 < r -->
- (\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn))"
- apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
- z = s and aa = t' and ab = t in xzgcda.induct)
- apply (subst zgcd_eq)
- apply (subst xzgcda.simps, auto)
- apply (case_tac "r' mod r = 0")
- prefer 2
- apply (frule_tac a = "r'" in pos_mod_sign, auto)
- apply (rule exI)
- apply (rule exI)
- apply (subst xzgcda.simps, auto)
- done
-
-lemma xzgcd_correct_aux2:
- "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> 0 < r -->
- zgcd r' r = k"
- apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
- z = s and aa = t' and ab = t in xzgcda.induct)
- apply (subst zgcd_eq)
- apply (subst xzgcda.simps)
- apply (auto simp add: linorder_not_le)
- apply (case_tac "r' mod r = 0")
- prefer 2
- apply (frule_tac a = "r'" in pos_mod_sign, auto)
- apply (metis Pair_eq simps zle_refl)
- done
-
-lemma xzgcd_correct:
- "0 < n ==> (zgcd m n = k) = (\<exists>s t. xzgcd m n = (k, s, t))"
- apply (unfold xzgcd_def)
- apply (rule iffI)
- apply (rule_tac [2] xzgcd_correct_aux2 [THEN mp, THEN mp])
- apply (rule xzgcd_correct_aux1 [THEN mp, THEN mp], auto)
- done
-
-
-text {* \medskip @{term xzgcd} linear *}
-
-lemma xzgcda_linear_aux1:
- "(a - r * b) * m + (c - r * d) * (n::int) =
- (a * m + c * n) - r * (b * m + d * n)"
- by (simp add: zdiff_zmult_distrib zadd_zmult_distrib2 zmult_assoc)
-
-lemma xzgcda_linear_aux2:
- "r' = s' * m + t' * n ==> r = s * m + t * n
- ==> (r' mod r) = (s' - (r' div r) * s) * m + (t' - (r' div r) * t) * (n::int)"
- apply (rule trans)
- apply (rule_tac [2] xzgcda_linear_aux1 [symmetric])
- apply (simp add: eq_diff_eq mult_commute)
- done
-
-lemma order_le_neq_implies_less: "(x::'a::order) \<le> y ==> x \<noteq> y ==> x < y"
- by (rule iffD2 [OF order_less_le conjI])
-
-lemma xzgcda_linear [rule_format]:
- "0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->
- r' = s' * m + t' * n --> r = s * m + t * n --> rn = sn * m + tn * n"
- apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
- z = s and aa = t' and ab = t in xzgcda.induct)
- apply (subst xzgcda.simps)
- apply (simp (no_asm))
- apply (rule impI)+
- apply (case_tac "r' mod r = 0")
- apply (simp add: xzgcda.simps, clarify)
- apply (subgoal_tac "0 < r' mod r")
- apply (rule_tac [2] order_le_neq_implies_less)
- apply (rule_tac [2] pos_mod_sign)
- apply (cut_tac m = m and n = n and r' = r' and r = r and s' = s' and
- s = s and t' = t' and t = t in xzgcda_linear_aux2, auto)
- done
-
-lemma xzgcd_linear:
- "0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"
- apply (unfold xzgcd_def)
- apply (erule xzgcda_linear, assumption, auto)
- done
-
-lemma zgcd_ex_linear:
- "0 < n ==> zgcd m n = k ==> (\<exists>s t. k = s * m + t * n)"
- apply (simp add: xzgcd_correct, safe)
- apply (rule exI)+
- apply (erule xzgcd_linear, auto)
- done
-
-lemma zcong_lineq_ex:
- "0 < n ==> zgcd a n = 1 ==> \<exists>x. [a * x = 1] (mod n)"
- apply (cut_tac m = a and n = n and k = 1 in zgcd_ex_linear, safe)
- apply (rule_tac x = s in exI)
- apply (rule_tac b = "s * a + t * n" in zcong_trans)
- prefer 2
- apply simp
- apply (unfold zcong_def)
- apply (simp (no_asm) add: zmult_commute)
- done
-
-lemma zcong_lineq_unique:
- "0 < n ==>
- zgcd a n = 1 ==> \<exists>!x. 0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"
- apply auto
- apply (rule_tac [2] zcong_zless_imp_eq)
- apply (tactic {* stac (thm "zcong_cancel2" RS sym) 6 *})
- apply (rule_tac [8] zcong_trans)
- apply (simp_all (no_asm_simp))
- prefer 2
- apply (simp add: zcong_sym)
- apply (cut_tac a = a and n = n in zcong_lineq_ex, auto)
- apply (rule_tac x = "x * b mod n" in exI, safe)
- apply (simp_all (no_asm_simp))
- apply (metis zcong_scalar zcong_zmod zmod_zmult1_eq zmult_1 zmult_assoc)
- done
-
-end