src/HOL/Rings.thy
changeset 66807 c3631f32dfeb
parent 66793 deabce3ccf1f
child 66808 1907167b6038
     1.1 --- a/src/HOL/Rings.thy	Sun Oct 08 22:28:21 2017 +0200
     1.2 +++ b/src/HOL/Rings.thy	Sun Oct 08 22:28:21 2017 +0200
     1.3 @@ -1558,6 +1558,82 @@
     1.4  end
     1.5  
     1.6  
     1.7 +text \<open>Quotient and remainder in integral domains\<close>
     1.8 +
     1.9 +class semidom_modulo = algebraic_semidom + semiring_modulo
    1.10 +begin
    1.11 +
    1.12 +lemma mod_0 [simp]: "0 mod a = 0"
    1.13 +  using div_mult_mod_eq [of 0 a] by simp
    1.14 +
    1.15 +lemma mod_by_0 [simp]: "a mod 0 = a"
    1.16 +  using div_mult_mod_eq [of a 0] by simp
    1.17 +
    1.18 +lemma mod_by_1 [simp]:
    1.19 +  "a mod 1 = 0"
    1.20 +proof -
    1.21 +  from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp
    1.22 +  then have "a + a mod 1 = a + 0" by simp
    1.23 +  then show ?thesis by (rule add_left_imp_eq)
    1.24 +qed
    1.25 +
    1.26 +lemma mod_self [simp]:
    1.27 +  "a mod a = 0"
    1.28 +  using div_mult_mod_eq [of a a] by simp
    1.29 +
    1.30 +lemma dvd_imp_mod_0 [simp]:
    1.31 +  assumes "a dvd b"
    1.32 +  shows "b mod a = 0"
    1.33 +  using assms minus_div_mult_eq_mod [of b a] by simp
    1.34 +
    1.35 +lemma mod_0_imp_dvd: 
    1.36 +  assumes "a mod b = 0"
    1.37 +  shows   "b dvd a"
    1.38 +proof -
    1.39 +  have "b dvd ((a div b) * b)" by simp
    1.40 +  also have "(a div b) * b = a"
    1.41 +    using div_mult_mod_eq [of a b] by (simp add: assms)
    1.42 +  finally show ?thesis .
    1.43 +qed
    1.44 +
    1.45 +lemma mod_eq_0_iff_dvd:
    1.46 +  "a mod b = 0 \<longleftrightarrow> b dvd a"
    1.47 +  by (auto intro: mod_0_imp_dvd)
    1.48 +
    1.49 +lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
    1.50 +  "a dvd b \<longleftrightarrow> b mod a = 0"
    1.51 +  by (simp add: mod_eq_0_iff_dvd)
    1.52 +
    1.53 +lemma dvd_mod_iff: 
    1.54 +  assumes "c dvd b"
    1.55 +  shows "c dvd a mod b \<longleftrightarrow> c dvd a"
    1.56 +proof -
    1.57 +  from assms have "(c dvd a mod b) \<longleftrightarrow> (c dvd ((a div b) * b + a mod b))" 
    1.58 +    by (simp add: dvd_add_right_iff)
    1.59 +  also have "(a div b) * b + a mod b = a"
    1.60 +    using div_mult_mod_eq [of a b] by simp
    1.61 +  finally show ?thesis .
    1.62 +qed
    1.63 +
    1.64 +lemma dvd_mod_imp_dvd:
    1.65 +  assumes "c dvd a mod b" and "c dvd b"
    1.66 +  shows "c dvd a"
    1.67 +  using assms dvd_mod_iff [of c b a] by simp
    1.68 +
    1.69 +end
    1.70 +
    1.71 +class idom_modulo = idom + semidom_modulo
    1.72 +begin
    1.73 +
    1.74 +subclass idom_divide ..
    1.75 +
    1.76 +lemma div_diff [simp]:
    1.77 +  "c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> (a - b) div c = a div c - b div c"
    1.78 +  using div_add [of _  _ "- b"] by (simp add: dvd_neg_div)
    1.79 +
    1.80 +end
    1.81 +
    1.82 +
    1.83  class ordered_semiring = semiring + ordered_comm_monoid_add +
    1.84    assumes mult_left_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c * a \<le> c * b"
    1.85    assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"