src/HOL/NumberTheory/IntPrimes.thy
author ballarin
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Methods rule_tac etc support static (Isar) contexts.
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(*  Title:      HOL/NumberTheory/IntPrimes.thy
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    ID:         $Id$
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    Author:     Thomas M. Rasmussen
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    Copyright   2000  University of Cambridge
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Changes by Jeremy Avigad, 2003/02/21:
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   Repaired definition of zprime_def, added "0 <= m &"
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   Added lemma zgcd_geq_zero
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   Repaired proof of zprime_imp_zrelprime
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*)
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header {* Divisibility and prime numbers (on integers) *}
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theory IntPrimes = Primes:
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text {*
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  The @{text dvd} relation, GCD, Euclid's extended algorithm, primes,
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  congruences (all on the Integers).  Comparable to theory @{text
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  Primes}, but @{text dvd} is included here as it is not present in
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  main HOL.  Also includes extended GCD and congruences not present in
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  @{text Primes}.
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*}
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subsection {* Definitions *}
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consts
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  xzgcda :: "int * int * int * int * int * int * int * int => int * int * int"
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recdef xzgcda
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  "measure ((\<lambda>(m, n, r', r, s', s, t', t). nat r)
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    :: int * int * int * int *int * int * int * int => nat)"
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  "xzgcda (m, n, r', r, s', s, t', t) =
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	(if r \<le> 0 then (r', s', t')
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	 else xzgcda (m, n, r, r' mod r, 
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		      s, s' - (r' div r) * s, 
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		      t, t' - (r' div r) * t))"
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constdefs
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  zgcd :: "int * int => int"
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  "zgcd == \<lambda>(x,y). int (gcd (nat (abs x), nat (abs y)))"
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  zprime :: "int set"
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  "zprime == {p. 1 < p \<and> (\<forall>m. 0 <= m & m dvd p --> m = 1 \<or> m = p)}"
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  xzgcd :: "int => int => int * int * int"
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  "xzgcd m n == xzgcda (m, n, m, n, 1, 0, 0, 1)"
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  zcong :: "int => int => int => bool"    ("(1[_ = _] '(mod _'))")
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  "[a = b] (mod m) == m dvd (a - b)"
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text {* \medskip @{term gcd} lemmas *}
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lemma gcd_add1_eq: "gcd (m + k, k) = gcd (m + k, m)"
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  by (simp add: gcd_commute)
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lemma gcd_diff2: "m \<le> n ==> gcd (n, n - m) = gcd (n, m)"
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  apply (subgoal_tac "n = m + (n - m)")
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   apply (erule ssubst, rule gcd_add1_eq, simp)
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  done
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subsection {* Euclid's Algorithm and GCD *}
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lemma zgcd_0 [simp]: "zgcd (m, 0) = abs m"
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  by (simp add: zgcd_def zabs_def)
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lemma zgcd_0_left [simp]: "zgcd (0, m) = abs m"
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  by (simp add: zgcd_def zabs_def)
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lemma zgcd_zminus [simp]: "zgcd (-m, n) = zgcd (m, n)"
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  by (simp add: zgcd_def)
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lemma zgcd_zminus2 [simp]: "zgcd (m, -n) = zgcd (m, n)"
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  by (simp add: zgcd_def)
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lemma zgcd_non_0: "0 < n ==> zgcd (m, n) = zgcd (n, m mod n)"
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  apply (frule_tac b = n and a = m in pos_mod_sign)
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  apply (simp del: pos_mod_sign add: zgcd_def zabs_def nat_mod_distrib)
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  apply (auto simp add: gcd_non_0 nat_mod_distrib [symmetric] zmod_zminus1_eq_if)
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  apply (frule_tac a = m in pos_mod_bound)
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  apply (simp del: pos_mod_bound add: nat_diff_distrib gcd_diff2 nat_le_eq_zle)
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  done
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lemma zgcd_eq: "zgcd (m, n) = zgcd (n, m mod n)"
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  apply (case_tac "n = 0", simp add: DIVISION_BY_ZERO)
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  apply (auto simp add: linorder_neq_iff zgcd_non_0)
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  apply (cut_tac m = "-m" and n = "-n" in zgcd_non_0, auto)
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  done
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lemma zgcd_1 [simp]: "zgcd (m, 1) = 1"
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  by (simp add: zgcd_def zabs_def)
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lemma zgcd_0_1_iff [simp]: "(zgcd (0, m) = 1) = (abs m = 1)"
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  by (simp add: zgcd_def zabs_def)
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lemma zgcd_zdvd1 [iff]: "zgcd (m, n) dvd m"
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  by (simp add: zgcd_def zabs_def int_dvd_iff)
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lemma zgcd_zdvd2 [iff]: "zgcd (m, n) dvd n"
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  by (simp add: zgcd_def zabs_def int_dvd_iff)
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lemma zgcd_greatest_iff: "k dvd zgcd (m, n) = (k dvd m \<and> k dvd n)"
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  by (simp add: zgcd_def zabs_def int_dvd_iff dvd_int_iff nat_dvd_iff)
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lemma zgcd_commute: "zgcd (m, n) = zgcd (n, m)"
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  by (simp add: zgcd_def gcd_commute)
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lemma zgcd_1_left [simp]: "zgcd (1, m) = 1"
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  by (simp add: zgcd_def gcd_1_left)
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lemma zgcd_assoc: "zgcd (zgcd (k, m), n) = zgcd (k, zgcd (m, n))"
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  by (simp add: zgcd_def gcd_assoc)
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lemma zgcd_left_commute: "zgcd (k, zgcd (m, n)) = zgcd (m, zgcd (k, n))"
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  apply (rule zgcd_commute [THEN trans])
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  apply (rule zgcd_assoc [THEN trans])
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  apply (rule zgcd_commute [THEN arg_cong])
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  done
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lemmas zgcd_ac = zgcd_assoc zgcd_commute zgcd_left_commute
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  -- {* addition is an AC-operator *}
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lemma zgcd_zmult_distrib2: "0 \<le> k ==> k * zgcd (m, n) = zgcd (k * m, k * n)"
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  by (simp del: zmult_zminus_right
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      add: zmult_zminus_right [symmetric] nat_mult_distrib zgcd_def zabs_def
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          zmult_less_0_iff gcd_mult_distrib2 [symmetric] zmult_int [symmetric])
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lemma zgcd_zmult_distrib2_abs: "zgcd (k * m, k * n) = abs k * zgcd (m, n)"
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  by (simp add: zabs_def zgcd_zmult_distrib2)
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lemma zgcd_self [simp]: "0 \<le> m ==> zgcd (m, m) = m"
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  by (cut_tac k = m and m = 1 and n = 1 in zgcd_zmult_distrib2, simp_all)
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lemma zgcd_zmult_eq_self [simp]: "0 \<le> k ==> zgcd (k, k * n) = k"
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  by (cut_tac k = k and m = 1 and n = n in zgcd_zmult_distrib2, simp_all)
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lemma zgcd_zmult_eq_self2 [simp]: "0 \<le> k ==> zgcd (k * n, k) = k"
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  by (cut_tac k = k and m = n and n = 1 in zgcd_zmult_distrib2, simp_all)
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lemma zrelprime_zdvd_zmult_aux:
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     "zgcd (n, k) = 1 ==> k dvd m * n ==> 0 \<le> m ==> k dvd m"
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  apply (subgoal_tac "m = zgcd (m * n, m * k)")
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   apply (erule ssubst, rule zgcd_greatest_iff [THEN iffD2])
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   apply (simp_all add: zgcd_zmult_distrib2 [symmetric] int_0_le_mult_iff)
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  done
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lemma zrelprime_zdvd_zmult: "zgcd (n, k) = 1 ==> k dvd m * n ==> k dvd m"
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  apply (case_tac "0 \<le> m")
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   apply (blast intro: zrelprime_zdvd_zmult_aux)
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  apply (subgoal_tac "k dvd -m")
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   apply (rule_tac [2] zrelprime_zdvd_zmult_aux, auto)
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  done
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   156
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lemma zgcd_geq_zero: "0 <= zgcd(x,y)"
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  by (auto simp add: zgcd_def)
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text{*This is merely a sanity check on zprime, since the previous version
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      denoted the empty set.*}
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lemma "2 \<in> zprime"
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  apply (auto simp add: zprime_def) 
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  apply (frule zdvd_imp_le, simp) 
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  apply (auto simp add: order_le_less dvd_def) 
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  done
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lemma zprime_imp_zrelprime:
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    "p \<in> zprime ==> \<not> p dvd n ==> zgcd (n, p) = 1"
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  apply (auto simp add: zprime_def)
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  apply (drule_tac x = "zgcd(n, p)" in allE)
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  apply (auto simp add: zgcd_zdvd2 [of n p] zgcd_geq_zero)
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  apply (insert zgcd_zdvd1 [of n p], auto)
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  done
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lemma zless_zprime_imp_zrelprime:
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    "p \<in> zprime ==> 0 < n ==> n < p ==> zgcd (n, p) = 1"
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  apply (erule zprime_imp_zrelprime)
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  apply (erule zdvd_not_zless, assumption)
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  done
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lemma zprime_zdvd_zmult:
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    "0 \<le> (m::int) ==> p \<in> zprime ==> p dvd m * n ==> p dvd m \<or> p dvd n"
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  apply safe
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  apply (rule zrelprime_zdvd_zmult)
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   apply (rule zprime_imp_zrelprime, auto)
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  done
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lemma zgcd_zadd_zmult [simp]: "zgcd (m + n * k, n) = zgcd (m, n)"
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  apply (rule zgcd_eq [THEN trans])
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  apply (simp add: zmod_zadd1_eq)
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  apply (rule zgcd_eq [symmetric])
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  done
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lemma zgcd_zdvd_zgcd_zmult: "zgcd (m, n) dvd zgcd (k * m, n)"
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  apply (simp add: zgcd_greatest_iff)
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  apply (blast intro: zdvd_trans)
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  done
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lemma zgcd_zmult_zdvd_zgcd:
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    "zgcd (k, n) = 1 ==> zgcd (k * m, n) dvd zgcd (m, n)"
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  apply (simp add: zgcd_greatest_iff)
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  apply (rule_tac n = k in zrelprime_zdvd_zmult)
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   prefer 2
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   apply (simp add: zmult_commute)
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  apply (subgoal_tac "zgcd (k, zgcd (k * m, n)) = zgcd (k * m, zgcd (k, n))")
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   apply simp
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  apply (simp (no_asm) add: zgcd_ac)
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  done
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lemma zgcd_zmult_cancel: "zgcd (k, n) = 1 ==> zgcd (k * m, n) = zgcd (m, n)"
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  by (simp add: zgcd_def nat_abs_mult_distrib gcd_mult_cancel)
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lemma zgcd_zgcd_zmult:
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    "zgcd (k, m) = 1 ==> zgcd (n, m) = 1 ==> zgcd (k * n, m) = 1"
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  by (simp add: zgcd_zmult_cancel)
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   217
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lemma zdvd_iff_zgcd: "0 < m ==> (m dvd n) = (zgcd (n, m) = m)"
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  apply safe
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   apply (rule_tac [2] n = "zgcd (n, m)" in zdvd_trans)
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    apply (rule_tac [3] zgcd_zdvd1, simp_all)
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  apply (unfold dvd_def, auto)
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   223
  done
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subsection {* Congruences *}
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lemma zcong_1 [simp]: "[a = b] (mod 1)"
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  by (unfold zcong_def, auto)
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lemma zcong_refl [simp]: "[k = k] (mod m)"
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  by (unfold zcong_def, auto)
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4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
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   233
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lemma zcong_sym: "[a = b] (mod m) = [b = a] (mod m)"
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  apply (unfold zcong_def dvd_def, auto)
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   236
   apply (rule_tac [!] x = "-k" in exI, auto)
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  done
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   238
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   239
lemma zcong_zadd:
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    "[a = b] (mod m) ==> [c = d] (mod m) ==> [a + c = b + d] (mod m)"
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   241
  apply (unfold zcong_def)
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   242
  apply (rule_tac s = "(a - b) + (c - d)" in subst)
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   243
   apply (rule_tac [2] zdvd_zadd, auto)
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  done
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   245
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lemma zcong_zdiff:
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    "[a = b] (mod m) ==> [c = d] (mod m) ==> [a - c = b - d] (mod m)"
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   248
  apply (unfold zcong_def)
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   249
  apply (rule_tac s = "(a - b) - (c - d)" in subst)
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   250
   apply (rule_tac [2] zdvd_zdiff, auto)
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   251
  done
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   252
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lemma zcong_trans:
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    "[a = b] (mod m) ==> [b = c] (mod m) ==> [a = c] (mod m)"
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   255
  apply (unfold zcong_def dvd_def, auto)
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   256
  apply (rule_tac x = "k + ka" in exI)
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diff changeset
   257
  apply (simp add: zadd_ac zadd_zmult_distrib2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   258
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   259
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   260
lemma zcong_zmult:
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   261
    "[a = b] (mod m) ==> [c = d] (mod m) ==> [a * c = b * d] (mod m)"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   262
  apply (rule_tac b = "b * c" in zcong_trans)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   263
   apply (unfold zcong_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   264
   apply (rule_tac s = "c * (a - b)" in subst)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   265
    apply (rule_tac [3] s = "b * (c - d)" in subst)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   266
     prefer 4
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   267
     apply (blast intro: zdvd_zmult)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   268
    prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   269
    apply (blast intro: zdvd_zmult)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   270
   apply (simp_all add: zdiff_zmult_distrib2 zmult_commute)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   271
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   272
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   273
lemma zcong_scalar: "[a = b] (mod m) ==> [a * k = b * k] (mod m)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   274
  by (rule zcong_zmult, simp_all)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   275
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   276
lemma zcong_scalar2: "[a = b] (mod m) ==> [k * a = k * b] (mod m)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   277
  by (rule zcong_zmult, simp_all)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   278
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   279
lemma zcong_zmult_self: "[a * m = b * m] (mod m)"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   280
  apply (unfold zcong_def)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   281
  apply (rule zdvd_zdiff, simp_all)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   282
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   283
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   284
lemma zcong_square:
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   285
   "[|p \<in> zprime;  0 < a;  [a * a = 1] (mod p)|]
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   286
    ==> [a = 1] (mod p) \<or> [a = p - 1] (mod p)"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   287
  apply (unfold zcong_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   288
  apply (rule zprime_zdvd_zmult)
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   289
    apply (rule_tac [3] s = "a * a - 1 + p * (1 - a)" in subst)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   290
     prefer 4
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   291
     apply (simp add: zdvd_reduce)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   292
    apply (simp_all add: zdiff_zmult_distrib zmult_commute zdiff_zmult_distrib2)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   293
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   294
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   295
lemma zcong_cancel:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   296
  "0 \<le> m ==>
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   297
    zgcd (k, m) = 1 ==> [a * k = b * k] (mod m) = [a = b] (mod m)"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   298
  apply safe
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   299
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   300
   apply (blast intro: zcong_scalar)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   301
  apply (case_tac "b < a")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   302
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   303
   apply (subst zcong_sym)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   304
   apply (unfold zcong_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   305
   apply (rule_tac [!] zrelprime_zdvd_zmult)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   306
     apply (simp_all add: zdiff_zmult_distrib)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   307
  apply (subgoal_tac "m dvd (-(a * k - b * k))")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   308
   apply (simp add: zminus_zdiff_eq)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   309
  apply (subst zdvd_zminus_iff, assumption)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   310
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   311
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   312
lemma zcong_cancel2:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   313
  "0 \<le> m ==>
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   314
    zgcd (k, m) = 1 ==> [k * a = k * b] (mod m) = [a = b] (mod m)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   315
  by (simp add: zmult_commute zcong_cancel)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   316
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   317
lemma zcong_zgcd_zmult_zmod:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   318
  "[a = b] (mod m) ==> [a = b] (mod n) ==> zgcd (m, n) = 1
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   319
    ==> [a = b] (mod m * n)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   320
  apply (unfold zcong_def dvd_def, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   321
  apply (subgoal_tac "m dvd n * ka")
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   322
   apply (subgoal_tac "m dvd ka")
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   323
    apply (case_tac [2] "0 \<le> ka")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   324
     prefer 3
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   325
     apply (subst zdvd_zminus_iff [symmetric])
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   326
     apply (rule_tac n = n in zrelprime_zdvd_zmult)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   327
      apply (simp add: zgcd_commute)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   328
     apply (simp add: zmult_commute zdvd_zminus_iff)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   329
    prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   330
    apply (rule_tac n = n in zrelprime_zdvd_zmult)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   331
     apply (simp add: zgcd_commute)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   332
    apply (simp add: zmult_commute)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   333
   apply (auto simp add: dvd_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   334
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   335
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   336
lemma zcong_zless_imp_eq:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   337
  "0 \<le> a ==>
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   338
    a < m ==> 0 \<le> b ==> b < m ==> [a = b] (mod m) ==> a = b"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   339
  apply (unfold zcong_def dvd_def, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   340
  apply (drule_tac f = "\<lambda>z. z mod m" in arg_cong)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   341
  apply (cut_tac z = a and w = b in zless_linear, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   342
   apply (subgoal_tac [2] "(a - b) mod m = a - b")
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   343
    apply (rule_tac [3] mod_pos_pos_trivial, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   344
  apply (subgoal_tac "(m + (a - b)) mod m = m + (a - b)")
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   345
   apply (rule_tac [2] mod_pos_pos_trivial, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   346
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   347
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   348
lemma zcong_square_zless:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   349
  "p \<in> zprime ==> 0 < a ==> a < p ==>
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   350
    [a * a = 1] (mod p) ==> a = 1 \<or> a = p - 1"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   351
  apply (cut_tac p = p and a = a in zcong_square)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   352
     apply (simp add: zprime_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   353
    apply (auto intro: zcong_zless_imp_eq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   354
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   355
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   356
lemma zcong_not:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   357
    "0 < a ==> a < m ==> 0 < b ==> b < a ==> \<not> [a = b] (mod m)"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   358
  apply (unfold zcong_def)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   359
  apply (rule zdvd_not_zless, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   360
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   361
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   362
lemma zcong_zless_0:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   363
    "0 \<le> a ==> a < m ==> [a = 0] (mod m) ==> a = 0"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   364
  apply (unfold zcong_def dvd_def, auto)
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   365
  apply (subgoal_tac "0 < m")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   366
   apply (simp add: int_0_le_mult_iff)
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   367
   apply (subgoal_tac "m * k < m * 1")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   368
    apply (drule zmult_zless_cancel1 [THEN iffD1])
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   369
    apply (auto simp add: linorder_neq_iff)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   370
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   371
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   372
lemma zcong_zless_unique:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   373
    "0 < m ==> (\<exists>!b. 0 \<le> b \<and> b < m \<and> [a = b] (mod m))"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   374
  apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   375
   apply (subgoal_tac [2] "[b = y] (mod m)")
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   376
    apply (case_tac [2] "b = 0")
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   377
     apply (case_tac [3] "y = 0")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   378
      apply (auto intro: zcong_trans zcong_zless_0 zcong_zless_imp_eq order_less_le
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   379
        simp add: zcong_sym)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   380
  apply (unfold zcong_def dvd_def)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   381
  apply (rule_tac x = "a mod m" in exI, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   382
  apply (rule_tac x = "-(a div m)" in exI)
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13193
diff changeset
   383
  apply (simp add:zdiff_eq_eq eq_zdiff_eq zadd_commute)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   384
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   385
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   386
lemma zcong_iff_lin: "([a = b] (mod m)) = (\<exists>k. b = a + m * k)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   387
  apply (unfold zcong_def dvd_def, auto)
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   388
   apply (rule_tac [!] x = "-k" in exI, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   389
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   390
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   391
lemma zgcd_zcong_zgcd:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   392
  "0 < m ==>
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   393
    zgcd (a, m) = 1 ==> [a = b] (mod m) ==> zgcd (b, m) = 1"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   394
  by (auto simp add: zcong_iff_lin)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   395
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   396
lemma zcong_zmod_aux:
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   397
     "a - b = (m::int) * (a div m - b div m) + (a mod m - b mod m)"
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   398
  by(simp add: zdiff_zmult_distrib2 zadd_zdiff_eq eq_zdiff_eq zadd_ac)
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13193
diff changeset
   399
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   400
lemma zcong_zmod: "[a = b] (mod m) = [a mod m = b mod m] (mod m)"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   401
  apply (unfold zcong_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   402
  apply (rule_tac t = "a - b" in ssubst)
14174
f3cafd2929d5 Methods rule_tac etc support static (Isar) contexts.
ballarin
parents: 13837
diff changeset
   403
  apply (rule_tac m = m in zcong_zmod_aux)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   404
  apply (rule trans)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   405
   apply (rule_tac [2] k = m and m = "a div m - b div m" in zdvd_reduce)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   406
  apply (simp add: zadd_commute)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   407
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   408
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   409
lemma zcong_zmod_eq: "0 < m ==> [a = b] (mod m) = (a mod m = b mod m)"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   410
  apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   411
   apply (rule_tac m = m in zcong_zless_imp_eq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   412
       prefer 5
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   413
       apply (subst zcong_zmod [symmetric], simp_all)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   414
  apply (unfold zcong_def dvd_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   415
  apply (rule_tac x = "a div m - b div m" in exI)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   416
  apply (rule_tac m1 = m in zcong_zmod_aux [THEN trans], auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   417
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   418
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   419
lemma zcong_zminus [iff]: "[a = b] (mod -m) = [a = b] (mod m)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   420
  by (auto simp add: zcong_def)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   421
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   422
lemma zcong_zero [iff]: "[a = b] (mod 0) = (a = b)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   423
  by (auto simp add: zcong_def)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   424
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   425
lemma "[a = b] (mod m) = (a mod m = b mod m)"
13183
c7290200b3f4 conversion of IntDiv.thy to Isar format
paulson
parents: 11868
diff changeset
   426
  apply (case_tac "m = 0", simp add: DIVISION_BY_ZERO)
13193
d5234c261813 finished an incomplete proof
paulson
parents: 13187
diff changeset
   427
  apply (simp add: linorder_neq_iff)
d5234c261813 finished an incomplete proof
paulson
parents: 13187
diff changeset
   428
  apply (erule disjE)  
d5234c261813 finished an incomplete proof
paulson
parents: 13187
diff changeset
   429
   prefer 2 apply (simp add: zcong_zmod_eq)
d5234c261813 finished an incomplete proof
paulson
parents: 13187
diff changeset
   430
  txt{*Remainding case: @{term "m<0"}*}
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   431
  apply (rule_tac t = m in zminus_zminus [THEN subst])
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   432
  apply (subst zcong_zminus)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   433
  apply (subst zcong_zmod_eq, arith)
13193
d5234c261813 finished an incomplete proof
paulson
parents: 13187
diff changeset
   434
  apply (frule neg_mod_bound [of _ a], frule neg_mod_bound [of _ b]) 
13788
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13630
diff changeset
   435
  apply (simp add: zmod_zminus2_eq_if del: neg_mod_bound)
13193
d5234c261813 finished an incomplete proof
paulson
parents: 13187
diff changeset
   436
  done
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   437
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   438
subsection {* Modulo *}
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   439
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   440
lemma zmod_zdvd_zmod:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   441
    "0 < (m::int) ==> m dvd b ==> (a mod b mod m) = (a mod m)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   442
  apply (unfold dvd_def, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   443
  apply (subst zcong_zmod_eq [symmetric])
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   444
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   445
   apply (subst zcong_iff_lin)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   446
   apply (rule_tac x = "k * (a div (m * k))" in exI)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   447
   apply (simp add:zmult_assoc [symmetric], assumption)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   448
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   449
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   450
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   451
subsection {* Extended GCD *}
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   452
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   453
declare xzgcda.simps [simp del]
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   454
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   455
lemma xzgcd_correct_aux1:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   456
  "zgcd (r', r) = k --> 0 < r -->
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   457
    (\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn))"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   458
  apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   459
    z = s and aa = t' and ab = t in xzgcda.induct)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   460
  apply (subst zgcd_eq)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   461
  apply (subst xzgcda.simps, auto)
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   462
  apply (case_tac "r' mod r = 0")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   463
   prefer 2
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   464
   apply (frule_tac a = "r'" in pos_mod_sign, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   465
  apply (rule exI)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   466
  apply (rule exI)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   467
  apply (subst xzgcda.simps, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   468
  apply (simp add: zabs_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   469
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   470
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   471
lemma xzgcd_correct_aux2:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   472
  "(\<exists>sn tn. xzgcda (m, n, r', r, s', s, t', t) = (k, sn, tn)) --> 0 < r -->
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   473
    zgcd (r', r) = k"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   474
  apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   475
    z = s and aa = t' and ab = t in xzgcda.induct)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   476
  apply (subst zgcd_eq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   477
  apply (subst xzgcda.simps)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   478
  apply (auto simp add: linorder_not_le)
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   479
  apply (case_tac "r' mod r = 0")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   480
   prefer 2
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   481
   apply (frule_tac a = "r'" in pos_mod_sign, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   482
  apply (erule_tac P = "xzgcda ?u = ?v" in rev_mp)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   483
  apply (subst xzgcda.simps, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   484
  apply (simp add: zabs_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   485
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   486
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   487
lemma xzgcd_correct:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   488
    "0 < n ==> (zgcd (m, n) = k) = (\<exists>s t. xzgcd m n = (k, s, t))"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   489
  apply (unfold xzgcd_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   490
  apply (rule iffI)
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   491
   apply (rule_tac [2] xzgcd_correct_aux2 [THEN mp, THEN mp])
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   492
    apply (rule xzgcd_correct_aux1 [THEN mp, THEN mp], auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   493
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   494
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   495
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   496
text {* \medskip @{term xzgcd} linear *}
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   497
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   498
lemma xzgcda_linear_aux1:
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   499
  "(a - r * b) * m + (c - r * d) * (n::int) =
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   500
   (a * m + c * n) - r * (b * m + d * n)"
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   501
  by (simp add: zdiff_zmult_distrib zadd_zmult_distrib2 zmult_assoc)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   502
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   503
lemma xzgcda_linear_aux2:
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   504
  "r' = s' * m + t' * n ==> r = s * m + t * n
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   505
    ==> (r' mod r) = (s' - (r' div r) * s) * m + (t' - (r' div r) * t) * (n::int)"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   506
  apply (rule trans)
13524
604d0f3622d6 *** empty log message ***
wenzelm
parents: 13517
diff changeset
   507
   apply (rule_tac [2] xzgcda_linear_aux1 [symmetric])
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13193
diff changeset
   508
  apply (simp add: eq_zdiff_eq zmult_commute)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   509
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   510
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   511
lemma order_le_neq_implies_less: "(x::'a::order) \<le> y ==> x \<noteq> y ==> x < y"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   512
  by (rule iffD2 [OF order_less_le conjI])
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   513
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   514
lemma xzgcda_linear [rule_format]:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   515
  "0 < r --> xzgcda (m, n, r', r, s', s, t', t) = (rn, sn, tn) -->
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   516
    r' = s' * m + t' * n -->  r = s * m + t * n --> rn = sn * m + tn * n"
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   517
  apply (rule_tac u = m and v = n and w = r' and x = r and y = s' and
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   518
    z = s and aa = t' and ab = t in xzgcda.induct)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   519
  apply (subst xzgcda.simps)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   520
  apply (simp (no_asm))
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   521
  apply (rule impI)+
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   522
  apply (case_tac "r' mod r = 0")
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   523
   apply (simp add: xzgcda.simps, clarify)
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   524
  apply (subgoal_tac "0 < r' mod r")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   525
   apply (rule_tac [2] order_le_neq_implies_less)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   526
   apply (rule_tac [2] pos_mod_sign)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   527
    apply (cut_tac m = m and n = n and r' = r' and r = r and s' = s' and
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   528
      s = s and t' = t' and t = t in xzgcda_linear_aux2, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   529
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   530
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   531
lemma xzgcd_linear:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   532
    "0 < n ==> xzgcd m n = (r, s, t) ==> r = s * m + t * n"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   533
  apply (unfold xzgcd_def)
13837
8dd150d36c65 Reorganized, moving many results about the integer dvd relation from IntPrimes
paulson
parents: 13833
diff changeset
   534
  apply (erule xzgcda_linear, assumption, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   535
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   536
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   537
lemma zgcd_ex_linear:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   538
    "0 < n ==> zgcd (m, n) = k ==> (\<exists>s t. k = s * m + t * n)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   539
  apply (simp add: xzgcd_correct, safe)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   540
  apply (rule exI)+
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   541
  apply (erule xzgcd_linear, auto)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   542
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   543
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   544
lemma zcong_lineq_ex:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   545
    "0 < n ==> zgcd (a, n) = 1 ==> \<exists>x. [a * x = 1] (mod n)"
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   546
  apply (cut_tac m = a and n = n and k = 1 in zgcd_ex_linear, safe)
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   547
  apply (rule_tac x = s in exI)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   548
  apply (rule_tac b = "s * a + t * n" in zcong_trans)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   549
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   550
   apply simp
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   551
  apply (unfold zcong_def)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   552
  apply (simp (no_asm) add: zmult_commute zdvd_zminus_iff)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   553
  done
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   554
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   555
lemma zcong_lineq_unique:
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   556
  "0 < n ==>
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   557
    zgcd (a, n) = 1 ==> \<exists>!x. 0 \<le> x \<and> x < n \<and> [a * x = b] (mod n)"
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   558
  apply auto
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   559
   apply (rule_tac [2] zcong_zless_imp_eq)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   560
       apply (tactic {* stac (thm "zcong_cancel2" RS sym) 6 *})
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   561
         apply (rule_tac [8] zcong_trans)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   562
          apply (simp_all (no_asm_simp))
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   563
   prefer 2
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   564
   apply (simp add: zcong_sym)
13833
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   565
  apply (cut_tac a = a and n = n in zcong_lineq_ex, auto)
f8dcb1d9ea68 zprime_def fixes by Jeremy Avigad
paulson
parents: 13788
diff changeset
   566
  apply (rule_tac x = "x * b mod n" in exI, safe)
13788
fd03c4ab89d4 pos/neg_mod_sign/bound are now simp rules.
nipkow
parents: 13630
diff changeset
   567
    apply (simp_all (no_asm_simp))
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   568
  apply (subst zcong_zmod)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   569
  apply (subst zmod_zmult1_eq [symmetric])
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   570
  apply (subst zcong_zmod [symmetric])
11868
56db9f3a6b3e Numerals now work for the integers: the binary numerals for 0 and 1 rewrite
paulson
parents: 11701
diff changeset
   571
  apply (subgoal_tac "[a * x * b = 1 * b] (mod n)")
11049
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   572
   apply (rule_tac [2] zcong_zmult)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   573
    apply (simp_all add: zmult_assoc)
7eef34adb852 HOL-NumberTheory: converted to new-style format and proper document setup;
wenzelm
parents: 10147
diff changeset
   574
  done
9508
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   575
4d01dbf6ded7 Chinese Remainder Theorem, Wilsons Theorem, etc., by T M Masmussen
paulson
parents:
diff changeset
   576
end