src/HOL/Number_Theory/Euclidean_Algorithm.thy
 changeset 64164 38c407446400 parent 64163 62c9e5c05928 child 64177 006f303fb173
1.1 --- a/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Tue Oct 11 16:44:13 2016 +0200
1.2 +++ b/src/HOL/Number_Theory/Euclidean_Algorithm.thy	Wed Oct 12 20:38:47 2016 +0200
1.3 @@ -6,39 +6,6 @@
1.4  imports "~~/src/HOL/GCD" Factorial_Ring
1.5  begin
1.7 -class divide_modulo = semidom_divide + modulo +
1.8 -  assumes div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
1.9 -begin
1.10 -
1.11 -lemma zero_mod_left [simp]: "0 mod a = 0"
1.12 -  using div_mod_equality[of 0 a 0] by simp
1.13 -
1.14 -lemma dvd_mod_iff [simp]:
1.15 -  assumes "k dvd n"
1.16 -  shows   "(k dvd m mod n) = (k dvd m)"
1.17 -proof -
1.18 -  thm div_mod_equality
1.19 -  from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
1.21 -  also have "(m div n) * n + m mod n = m"
1.22 -    using div_mod_equality[of m n 0] by simp
1.23 -  finally show ?thesis .
1.24 -qed
1.25 -
1.26 -lemma mod_0_imp_dvd:
1.27 -  assumes "a mod b = 0"
1.28 -  shows   "b dvd a"
1.29 -proof -
1.30 -  have "b dvd ((a div b) * b)" by simp
1.31 -  also have "(a div b) * b = a"
1.32 -    using div_mod_equality[of a b 0] by (simp add: assms)
1.33 -  finally show ?thesis .
1.34 -qed
1.35 -
1.36 -end
1.37 -
1.38 -
1.39 -
1.40  text \<open>
1.41    A Euclidean semiring is a semiring upon which the Euclidean algorithm can be
1.42    implemented. It must provide:
1.43 @@ -50,7 +17,7 @@
1.44    The existence of these functions makes it possible to derive gcd and lcm functions
1.45    for any Euclidean semiring.
1.46  \<close>
1.47 -class euclidean_semiring = divide_modulo + normalization_semidom +
1.48 +class euclidean_semiring = semiring_modulo + normalization_semidom +
1.49    fixes euclidean_size :: "'a \<Rightarrow> nat"
1.50    assumes size_0 [simp]: "euclidean_size 0 = 0"
1.51    assumes mod_size_less:
1.52 @@ -59,6 +26,30 @@
1.53      "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
1.54  begin
1.56 +lemma zero_mod_left [simp]: "0 mod a = 0"
1.57 +  using mod_div_equality [of 0 a] by simp
1.58 +
1.59 +lemma dvd_mod_iff:
1.60 +  assumes "k dvd n"
1.61 +  shows   "(k dvd m mod n) = (k dvd m)"
1.62 +proof -
1.63 +  from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
1.65 +  also have "(m div n) * n + m mod n = m"
1.66 +    using mod_div_equality [of m n] by simp
1.67 +  finally show ?thesis .
1.68 +qed
1.69 +
1.70 +lemma mod_0_imp_dvd:
1.71 +  assumes "a mod b = 0"
1.72 +  shows   "b dvd a"
1.73 +proof -
1.74 +  have "b dvd ((a div b) * b)" by simp
1.75 +  also have "(a div b) * b = a"
1.76 +    using mod_div_equality [of a b] by (simp add: assms)
1.77 +  finally show ?thesis .
1.78 +qed
1.79 +
1.80  lemma euclidean_size_normalize [simp]:
1.81    "euclidean_size (normalize a) = euclidean_size a"
1.82  proof (cases "a = 0")
1.83 @@ -81,36 +72,11 @@
1.84    obtains s and t where "a = s * b + t"
1.85      and "euclidean_size t < euclidean_size b"
1.86  proof -
1.87 -  from div_mod_equality [of a b 0]
1.88 +  from mod_div_equality [of a b]
1.89       have "a = a div b * b + a mod b" by simp
1.90    with that and assms show ?thesis by (auto simp add: mod_size_less)
1.91  qed
1.93 -lemma zero_mod_left [simp]: "0 mod a = 0"
1.94 -  using div_mod_equality[of 0 a 0] by simp
1.95 -
1.96 -lemma dvd_mod_iff [simp]:
1.97 -  assumes "k dvd n"
1.98 -  shows   "(k dvd m mod n) = (k dvd m)"
1.99 -proof -
1.100 -  thm div_mod_equality
1.101 -  from assms have "(k dvd m mod n) \<longleftrightarrow> (k dvd ((m div n) * n + m mod n))"
1.103 -  also have "(m div n) * n + m mod n = m"
1.104 -    using div_mod_equality[of m n 0] by simp
1.105 -  finally show ?thesis .
1.106 -qed
1.108 -lemma mod_0_imp_dvd:
1.109 -  assumes "a mod b = 0"
1.110 -  shows   "b dvd a"
1.111 -proof -
1.112 -  have "b dvd ((a div b) * b)" by simp
1.113 -  also have "(a div b) * b = a"
1.114 -    using div_mod_equality[of a b 0] by (simp add: assms)
1.115 -  finally show ?thesis .
1.116 -qed
1.118  lemma dvd_euclidean_size_eq_imp_dvd:
1.119    assumes "a \<noteq> 0" and b_dvd_a: "b dvd a" and size_eq: "euclidean_size a = euclidean_size b"
1.120    shows "a dvd b"
1.121 @@ -118,7 +84,7 @@
1.122    assume "\<not> a dvd b"
1.123    hence "b mod a \<noteq> 0" using mod_0_imp_dvd[of b a] by blast
1.124    then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
1.125 -  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by simp
1.126 +  from b_dvd_a have b_dvd_mod: "b dvd b mod a" by (simp add: dvd_mod_iff)
1.127    from b_dvd_mod obtain c where "b mod a = b * c" unfolding dvd_def by blast
1.128      with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
1.129    with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
1.130 @@ -541,7 +507,7 @@
1.131                (s' * x + t' * y) - r' div r * (s * x + t * y)" by (simp add: algebra_simps)
1.132        also have "s' * x + t' * y = r'" by fact
1.133        also have "s * x + t * y = r" by fact
1.134 -      also have "r' - r' div r * r = r' mod r" using div_mod_equality[of r' r]
1.135 +      also have "r' - r' div r * r = r' mod r" using mod_div_equality [of r' r]
1.136          by (simp add: algebra_simps)
1.137        finally show "(s' - r' div r * s) * x + (t' - r' div r * t) * y = r' mod r" .
1.138      qed (auto simp: gcd_eucl_non_0 algebra_simps div_mod_equality')