src/HOL/Divides.thy
author haftmann
Fri Jul 11 09:02:22 2008 +0200 (2008-07-11)
changeset 27540 dc38e79f5a1c
parent 26748 4d51ddd6aa5c
child 27651 16a26996c30e
permissions -rw-r--r--
separate class dvd for divisibility predicate
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(*  Title:      HOL/Divides.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div, mod and the divides relation dvd *}
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theory Divides
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imports Nat Power Product_Type
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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begin
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subsection {* Syntactic division operations *}
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class dvd = times
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begin
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definition
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  dvd  :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50)
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where
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  [code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)"
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end
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class div = times +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract divisibility in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + dvd + div + 
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0: "a div 0 = 0"
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    and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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begin
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text {* @{const div} and @{const mod} *}
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lemma div_by_1: "a div 1 = a"
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  using mult_div [of 1 a] zero_neq_one by simp
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lemma mod_by_1: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_by_0: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mult_mod: "a * b mod b = 0"
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proof (cases "b = 0")
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  case True then show ?thesis by (simp add: mod_by_0)
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next
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  case False with mult_div have abb: "a * b div b = a" .
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  from mod_div_equality have "a * b div b * b + a * b mod b = a * b" .
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  with abb have "a * b + a * b mod b = a * b + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self: "a mod a = 0"
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  using mult_mod [of one] by simp
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lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using mult_div [of _ one] by simp
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lemma div_0: "0 div a = 0"
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proof (cases "a = 0")
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  case True then show ?thesis by (simp add: div_by_0)
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next
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  case False with mult_div have "0 * a div a = 0" .
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  then show ?thesis by simp
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qed
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lemma mod_0: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp 
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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text {* The @{const dvd} relation *}
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lemma dvdI [intro?]: "a = b * c \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>c. a = b * c \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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  with mod_div_equality [of b a] have "b div a * a = b" by simp
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  then have "b = a * (b div a)" unfolding mult_commute ..
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  then have "\<exists>c. b = a * c" ..
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  then show "a dvd b" unfolding dvd_def .
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next
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  assume "a dvd b"
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  then have "\<exists>c. b = a * c" unfolding dvd_def .
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  then obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then have "b mod a = c * a mod a" by (simp add: mult_commute)
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  then show "b mod a = 0" by (simp add: mult_mod)
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qed
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lemma dvd_refl: "a dvd a"
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  unfolding dvd_def_mod mod_self ..
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" unfolding dvd_def by auto
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  moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis unfolding dvd_def ..
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qed
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lemma zero_dvd_iff [noatp]: "0 dvd a \<longleftrightarrow> a = 0"
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  unfolding dvd_def by simp
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lemma dvd_0: "a dvd 0"
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unfolding dvd_def proof
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  show "0 = a * 0" by simp
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qed
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lemma one_dvd: "1 dvd a"
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  unfolding dvd_def by simp
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lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)"
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  unfolding dvd_def by (blast intro: mult_left_commute)
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lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult_commute)
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  apply (erule dvd_mult)
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  done
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lemma dvd_triv_right: "a dvd b * a"
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  by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left: "a dvd a * b"
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  by (rule dvd_mult2) (rule dvd_refl)
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lemma mult_dvd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> a * b dvd c * d"
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  apply (unfold dvd_def, clarify)
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  apply (rule_tac x = "k * ka" in exI)
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  apply (simp add: mult_ac)
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  done
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lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
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  by (simp add: dvd_def mult_assoc, blast)
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lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
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  unfolding mult_ac [of a] by (rule dvd_mult_left)
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end
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subsection {* Division on @{typ nat} *}
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text {*
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  We define @{const div} and @{const mod} on @{typ nat} by means
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  of a characteristic relation with two input arguments
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  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
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  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
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*}
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definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
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  "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
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text {* @{const divmod_rel} is total: *}
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lemma divmod_rel_ex:
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  obtains q r where "divmod_rel m n q r"
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proof (cases "n = 0")
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  case True with that show thesis
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    by (auto simp add: divmod_rel_def)
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next
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  case False
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  have "\<exists>q r. m = q * n + r \<and> r < n"
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  proof (induct m)
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    case 0 with `n \<noteq> 0`
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    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
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    then show ?case by blast
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  next
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    case (Suc m) then obtain q' r'
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      where m: "m = q' * n + r'" and n: "r' < n" by auto
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    then show ?case proof (cases "Suc r' < n")
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      case True
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      from m n have "Suc m = q' * n + Suc r'" by simp
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      with True show ?thesis by blast
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    next
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      case False then have "n \<le> Suc r'" by auto
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      moreover from n have "Suc r' \<le> n" by auto
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      ultimately have "n = Suc r'" by auto
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      with m have "Suc m = Suc q' * n + 0" by simp
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      with `n \<noteq> 0` show ?thesis by blast
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    qed
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  qed
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  with that show thesis
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    using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
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qed
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text {* @{const divmod_rel} is injective: *}
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lemma divmod_rel_unique_div:
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  assumes "divmod_rel m n q r"
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    and "divmod_rel m n q' r'"
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  shows "q = q'"
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proof (cases "n = 0")
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  case True with assms show ?thesis
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    by (simp add: divmod_rel_def)
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next
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  case False
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  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
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  apply (rule leI)
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  apply (subst less_iff_Suc_add)
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  apply (auto simp add: add_mult_distrib)
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  done
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  from `n \<noteq> 0` assms show ?thesis
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    by (auto simp add: divmod_rel_def
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      intro: order_antisym dest: aux sym)
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qed
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lemma divmod_rel_unique_mod:
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  assumes "divmod_rel m n q r"
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    and "divmod_rel m n q' r'"
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  shows "r = r'"
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proof -
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  from assms have "q = q'" by (rule divmod_rel_unique_div)
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  with assms show ?thesis by (simp add: divmod_rel_def)
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qed
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text {*
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  We instantiate divisibility on the natural numbers by
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  means of @{const divmod_rel}:
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*}
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instantiation nat :: semiring_div
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begin
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definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
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  [code func del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
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definition div_nat where
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  "m div n = fst (divmod m n)"
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definition mod_nat where
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  "m mod n = snd (divmod m n)"
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lemma divmod_div_mod:
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  "divmod m n = (m div n, m mod n)"
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  unfolding div_nat_def mod_nat_def by simp
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lemma divmod_eq:
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  assumes "divmod_rel m n q r" 
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  shows "divmod m n = (q, r)"
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  using assms by (auto simp add: divmod_def
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    dest: divmod_rel_unique_div divmod_rel_unique_mod)
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lemma div_eq:
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  assumes "divmod_rel m n q r" 
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  shows "m div n = q"
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  using assms by (auto dest: divmod_eq simp add: div_nat_def)
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lemma mod_eq:
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  assumes "divmod_rel m n q r" 
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  shows "m mod n = r"
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  using assms by (auto dest: divmod_eq simp add: mod_nat_def)
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lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
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proof -
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  from divmod_rel_ex
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    obtain q r where rel: "divmod_rel m n q r" .
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  moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
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    by simp_all
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  ultimately show ?thesis by simp
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qed
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lemma divmod_zero:
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  "divmod m 0 = (0, m)"
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proof -
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  from divmod_rel [of m 0] show ?thesis
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    unfolding divmod_div_mod divmod_rel_def by simp
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qed
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lemma divmod_base:
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  assumes "m < n"
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  shows "divmod m n = (0, m)"
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proof -
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  from divmod_rel [of m n] show ?thesis
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    unfolding divmod_div_mod divmod_rel_def
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    using assms by (cases "m div n = 0")
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      (auto simp add: gr0_conv_Suc [of "m div n"])
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qed
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lemma divmod_step:
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  assumes "0 < n" and "n \<le> m"
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  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
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proof -
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  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
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  with assms have m_div_n: "m div n \<ge> 1"
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    by (cases "m div n") (auto simp add: divmod_rel_def)
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  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
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    by (cases "m div n") (auto simp add: divmod_rel_def)
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  with divmod_eq have "divmod (m - n) n = (m div n - 1, m mod n)" by simp
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  moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
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  ultimately have "m div n = Suc ((m - n) div n)"
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    and "m mod n = (m - n) mod n" using m_div_n by simp_all
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  then show ?thesis using divmod_div_mod by simp
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qed
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text {* The ''recursion'' equations for @{const div} and @{const mod} *}
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lemma div_less [simp]:
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  fixes m n :: nat
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  assumes "m < n"
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  shows "m div n = 0"
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  using assms divmod_base divmod_div_mod by simp
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   329
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   330
lemma le_div_geq:
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   331
  fixes m n :: nat
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   332
  assumes "0 < n" and "n \<le> m"
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   333
  shows "m div n = Suc ((m - n) div n)"
haftmann@26100
   334
  using assms divmod_step divmod_div_mod by simp
paulson@14267
   335
haftmann@26100
   336
lemma mod_less [simp]:
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   337
  fixes m n :: nat
haftmann@26100
   338
  assumes "m < n"
haftmann@26100
   339
  shows "m mod n = m"
haftmann@26100
   340
  using assms divmod_base divmod_div_mod by simp
haftmann@26100
   341
haftmann@26100
   342
lemma le_mod_geq:
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   343
  fixes m n :: nat
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   344
  assumes "n \<le> m"
haftmann@26100
   345
  shows "m mod n = (m - n) mod n"
haftmann@26100
   346
  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
paulson@14267
   347
haftmann@25942
   348
instance proof
haftmann@26100
   349
  fix m n :: nat show "m div n * n + m mod n = m"
haftmann@26100
   350
    using divmod_rel [of m n] by (simp add: divmod_rel_def)
haftmann@25942
   351
next
haftmann@26100
   352
  fix n :: nat show "n div 0 = 0"
haftmann@26100
   353
    using divmod_zero divmod_div_mod [of n 0] by simp
haftmann@25942
   354
next
haftmann@26100
   355
  fix m n :: nat assume "n \<noteq> 0" then show "m * n div n = m"
haftmann@25942
   356
    by (induct m) (simp_all add: le_div_geq)
haftmann@25942
   357
qed
haftmann@26100
   358
haftmann@25942
   359
end
paulson@14267
   360
haftmann@26100
   361
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   362
haftmann@25942
   363
lemmas mod_div_equality = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
haftmann@26062
   364
lemmas mod_div_equality2 = mod_div_equality2 [of "n\<Colon>nat" m, standard]
haftmann@26062
   365
lemmas div_mod_equality = div_mod_equality [of "m\<Colon>nat" n k, standard]
haftmann@26062
   366
lemmas div_mod_equality2 = div_mod_equality2 [of "m\<Colon>nat" n k, standard]
haftmann@25942
   367
haftmann@25942
   368
ML {*
haftmann@25942
   369
structure CancelDivModData =
haftmann@25942
   370
struct
haftmann@25942
   371
haftmann@26100
   372
val div_name = @{const_name div};
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   373
val mod_name = @{const_name mod};
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   374
val mk_binop = HOLogic.mk_binop;
haftmann@26100
   375
val mk_sum = ArithData.mk_sum;
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   376
val dest_sum = ArithData.dest_sum;
haftmann@25942
   377
haftmann@25942
   378
(*logic*)
paulson@14267
   379
haftmann@25942
   380
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
haftmann@25942
   381
haftmann@25942
   382
val trans = trans
haftmann@25942
   383
haftmann@25942
   384
val prove_eq_sums =
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   385
  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
haftmann@26100
   386
  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
haftmann@25942
   387
haftmann@25942
   388
end;
haftmann@25942
   389
haftmann@25942
   390
structure CancelDivMod = CancelDivModFun(CancelDivModData);
haftmann@25942
   391
haftmann@26100
   392
val cancel_div_mod_proc = Simplifier.simproc @{theory}
haftmann@26100
   393
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
haftmann@25942
   394
haftmann@25942
   395
Addsimprocs[cancel_div_mod_proc];
haftmann@25942
   396
*}
haftmann@25942
   397
haftmann@26100
   398
text {* code generator setup *}
haftmann@26100
   399
haftmann@26100
   400
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@26100
   401
  let (q, r) = divmod (m - n) n in (Suc q, r))"
haftmann@26100
   402
  by (simp add: divmod_zero divmod_base divmod_step)
haftmann@26100
   403
    (simp add: divmod_div_mod)
haftmann@26100
   404
haftmann@26100
   405
code_modulename SML
haftmann@26100
   406
  Divides Nat
haftmann@26100
   407
haftmann@26100
   408
code_modulename OCaml
haftmann@26100
   409
  Divides Nat
haftmann@26100
   410
haftmann@26100
   411
code_modulename Haskell
haftmann@26100
   412
  Divides Nat
haftmann@26100
   413
haftmann@26100
   414
haftmann@26100
   415
subsubsection {* Quotient *}
haftmann@26100
   416
haftmann@26100
   417
lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard]
haftmann@26100
   418
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard]
haftmann@26100
   419
haftmann@26100
   420
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
haftmann@26100
   421
  by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   422
haftmann@26100
   423
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
haftmann@26100
   424
  by (simp add: div_geq)
haftmann@26100
   425
haftmann@26100
   426
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
haftmann@26100
   427
  by (rule mult_div) simp
haftmann@26100
   428
haftmann@26100
   429
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
haftmann@26100
   430
  by (simp add: mult_commute)
haftmann@26100
   431
haftmann@25942
   432
haftmann@25942
   433
subsubsection {* Remainder *}
haftmann@25942
   434
haftmann@25942
   435
lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard]
haftmann@26100
   436
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard]
haftmann@25942
   437
haftmann@26100
   438
lemma mod_less_divisor [simp]:
haftmann@26100
   439
  fixes m n :: nat
haftmann@26100
   440
  assumes "n > 0"
haftmann@26100
   441
  shows "m mod n < (n::nat)"
haftmann@26100
   442
  using assms divmod_rel unfolding divmod_rel_def by auto
paulson@14267
   443
haftmann@26100
   444
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   445
  fixes m n :: nat
haftmann@26100
   446
  shows "m mod n \<le> m"
haftmann@26100
   447
proof (rule add_leD2)
haftmann@26100
   448
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   449
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   450
qed
haftmann@26100
   451
haftmann@26100
   452
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
haftmann@25942
   453
  by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   454
haftmann@26100
   455
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
haftmann@26100
   456
  by (simp add: le_mod_geq)
haftmann@26100
   457
paulson@14267
   458
lemma mod_1 [simp]: "m mod Suc 0 = 0"
wenzelm@22718
   459
  by (induct m) (simp_all add: mod_geq)
paulson@14267
   460
haftmann@25942
   461
lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard]
paulson@14267
   462
paulson@14267
   463
lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
wenzelm@22718
   464
  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
wenzelm@22718
   465
   apply (simp add: add_commute)
haftmann@25942
   466
  apply (subst le_mod_geq [symmetric], simp_all)
wenzelm@22718
   467
  done
paulson@14267
   468
paulson@14267
   469
lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
wenzelm@22718
   470
  by (simp add: add_commute mod_add_self2)
paulson@14267
   471
paulson@14267
   472
lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
wenzelm@22718
   473
  by (induct k) (simp_all add: add_left_commute [of _ n])
paulson@14267
   474
paulson@14267
   475
lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
wenzelm@22718
   476
  by (simp add: mult_commute mod_mult_self1)
paulson@14267
   477
haftmann@26100
   478
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
wenzelm@22718
   479
  apply (cases "n = 0", simp)
wenzelm@22718
   480
  apply (cases "k = 0", simp)
wenzelm@22718
   481
  apply (induct m rule: nat_less_induct)
wenzelm@22718
   482
  apply (subst mod_if, simp)
wenzelm@22718
   483
  apply (simp add: mod_geq diff_mult_distrib)
wenzelm@22718
   484
  done
paulson@14267
   485
paulson@14267
   486
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
wenzelm@22718
   487
  by (simp add: mult_commute [of k] mod_mult_distrib)
paulson@14267
   488
paulson@14267
   489
lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
wenzelm@22718
   490
  apply (cases "n = 0", simp)
wenzelm@22718
   491
  apply (induct m, simp)
wenzelm@22718
   492
  apply (rename_tac k)
wenzelm@22718
   493
  apply (cut_tac m = "k * n" and n = n in mod_add_self2)
wenzelm@22718
   494
  apply (simp add: add_commute)
wenzelm@22718
   495
  done
paulson@14267
   496
paulson@14267
   497
lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
wenzelm@22718
   498
  by (simp add: mult_commute mod_mult_self_is_0)
paulson@14267
   499
paulson@14267
   500
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   501
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
wenzelm@22718
   502
  by (cut_tac m = m and n = n in mod_div_equality2, arith)
paulson@14267
   503
nipkow@15439
   504
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   505
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   506
  apply simp
wenzelm@22718
   507
  done
paulson@14267
   508
haftmann@26100
   509
subsubsection {* Quotient and Remainder *}
paulson@14267
   510
haftmann@26100
   511
lemma mod_div_decomp:
haftmann@26100
   512
  fixes n k :: nat
haftmann@26100
   513
  obtains m q where "m = n div k" and "q = n mod k"
haftmann@26100
   514
    and "n = m * k + q"
haftmann@26100
   515
proof -
haftmann@26100
   516
  from mod_div_equality have "n = n div k * k + n mod k" by auto
haftmann@26100
   517
  moreover have "n div k = n div k" ..
haftmann@26100
   518
  moreover have "n mod k = n mod k" ..
haftmann@26100
   519
  note that ultimately show thesis by blast
haftmann@26100
   520
qed
paulson@14267
   521
haftmann@26100
   522
lemma divmod_rel_mult1_eq:
haftmann@26100
   523
  "[| divmod_rel b c q r; c > 0 |]
haftmann@26100
   524
   ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
haftmann@26100
   525
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
paulson@14267
   526
paulson@14267
   527
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
nipkow@25134
   528
apply (cases "c = 0", simp)
haftmann@26100
   529
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
nipkow@25134
   530
done
paulson@14267
   531
paulson@14267
   532
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
nipkow@25134
   533
apply (cases "c = 0", simp)
haftmann@26100
   534
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN mod_eq])
nipkow@25134
   535
done
paulson@14267
   536
paulson@14267
   537
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
wenzelm@22718
   538
  apply (rule trans)
wenzelm@22718
   539
   apply (rule_tac s = "b*a mod c" in trans)
wenzelm@22718
   540
    apply (rule_tac [2] mod_mult1_eq)
wenzelm@22718
   541
   apply (simp_all add: mult_commute)
wenzelm@22718
   542
  done
paulson@14267
   543
nipkow@25162
   544
lemma mod_mult_distrib_mod:
nipkow@25162
   545
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
nipkow@25162
   546
apply (rule mod_mult1_eq' [THEN trans])
nipkow@25162
   547
apply (rule mod_mult1_eq)
nipkow@25162
   548
done
paulson@14267
   549
haftmann@26100
   550
lemma divmod_rel_add1_eq:
haftmann@26100
   551
  "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
haftmann@26100
   552
   ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
haftmann@26100
   553
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
paulson@14267
   554
paulson@14267
   555
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   556
lemma div_add1_eq:
nipkow@25134
   557
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
nipkow@25134
   558
apply (cases "c = 0", simp)
haftmann@26100
   559
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
nipkow@25134
   560
done
paulson@14267
   561
paulson@14267
   562
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
nipkow@25134
   563
apply (cases "c = 0", simp)
haftmann@26100
   564
apply (blast intro: divmod_rel_add1_eq [THEN mod_eq] divmod_rel)
nipkow@25134
   565
done
paulson@14267
   566
paulson@14267
   567
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   568
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   569
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   570
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   571
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   572
  done
paulson@10559
   573
haftmann@26100
   574
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
haftmann@26100
   575
      ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
haftmann@26100
   576
  by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   577
paulson@14267
   578
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
wenzelm@22718
   579
  apply (cases "b = 0", simp)
wenzelm@22718
   580
  apply (cases "c = 0", simp)
haftmann@26100
   581
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
wenzelm@22718
   582
  done
paulson@14267
   583
paulson@14267
   584
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
wenzelm@22718
   585
  apply (cases "b = 0", simp)
wenzelm@22718
   586
  apply (cases "c = 0", simp)
haftmann@26100
   587
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
wenzelm@22718
   588
  done
paulson@14267
   589
paulson@14267
   590
haftmann@25942
   591
subsubsection{*Cancellation of Common Factors in Division*}
paulson@14267
   592
paulson@14267
   593
lemma div_mult_mult_lemma:
wenzelm@22718
   594
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   595
  by (auto simp add: div_mult2_eq)
paulson@14267
   596
paulson@14267
   597
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   598
  apply (cases "b = 0")
wenzelm@22718
   599
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
wenzelm@22718
   600
  done
paulson@14267
   601
paulson@14267
   602
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
wenzelm@22718
   603
  apply (drule div_mult_mult1)
wenzelm@22718
   604
  apply (auto simp add: mult_commute)
wenzelm@22718
   605
  done
paulson@14267
   606
paulson@14267
   607
haftmann@25942
   608
subsubsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   609
paulson@14267
   610
lemma div_1 [simp]: "m div Suc 0 = m"
wenzelm@22718
   611
  by (induct m) (simp_all add: div_geq)
paulson@14267
   612
haftmann@25942
   613
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard]
paulson@14267
   614
paulson@14267
   615
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
wenzelm@22718
   616
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
wenzelm@22718
   617
   apply (simp add: add_commute)
wenzelm@22718
   618
  apply (subst div_geq [symmetric], simp_all)
wenzelm@22718
   619
  done
paulson@14267
   620
paulson@14267
   621
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
wenzelm@22718
   622
  by (simp add: add_commute div_add_self2)
paulson@14267
   623
paulson@14267
   624
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
wenzelm@22718
   625
  apply (subst div_add1_eq)
wenzelm@22718
   626
  apply (subst div_mult1_eq, simp)
wenzelm@22718
   627
  done
paulson@14267
   628
paulson@14267
   629
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
wenzelm@22718
   630
  by (simp add: mult_commute div_mult_self1)
paulson@14267
   631
paulson@14267
   632
paulson@14267
   633
(* Monotonicity of div in first argument *)
paulson@14267
   634
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   635
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   636
apply (case_tac "k=0", simp)
paulson@15251
   637
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   638
apply (case_tac "n<k")
paulson@14267
   639
(* 1  case n<k *)
paulson@14267
   640
apply simp
paulson@14267
   641
(* 2  case n >= k *)
paulson@14267
   642
apply (case_tac "m<k")
paulson@14267
   643
(* 2.1  case m<k *)
paulson@14267
   644
apply simp
paulson@14267
   645
(* 2.2  case m>=k *)
nipkow@15439
   646
apply (simp add: div_geq diff_le_mono)
paulson@14267
   647
done
paulson@14267
   648
paulson@14267
   649
(* Antimonotonicity of div in second argument *)
paulson@14267
   650
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   651
apply (subgoal_tac "0<n")
wenzelm@22718
   652
 prefer 2 apply simp
paulson@15251
   653
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   654
apply (rename_tac "k")
paulson@14267
   655
apply (case_tac "k<n", simp)
paulson@14267
   656
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   657
 prefer 2 apply simp
paulson@14267
   658
apply (simp add: div_geq)
paulson@15251
   659
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   660
 prefer 2
paulson@14267
   661
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   662
apply (rule le_trans, simp)
nipkow@15439
   663
apply (simp)
paulson@14267
   664
done
paulson@14267
   665
paulson@14267
   666
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   667
apply (case_tac "n=0", simp)
paulson@14267
   668
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   669
apply (rule div_le_mono2)
paulson@14267
   670
apply (simp_all (no_asm_simp))
paulson@14267
   671
done
paulson@14267
   672
wenzelm@22718
   673
(* Similar for "less than" *)
paulson@17085
   674
lemma div_less_dividend [rule_format]:
paulson@14267
   675
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   676
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   677
apply (rename_tac "m")
paulson@14267
   678
apply (case_tac "m<n", simp)
paulson@14267
   679
apply (subgoal_tac "0<n")
wenzelm@22718
   680
 prefer 2 apply simp
paulson@14267
   681
apply (simp add: div_geq)
paulson@14267
   682
apply (case_tac "n<m")
paulson@15251
   683
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   684
  apply (rule impI less_trans_Suc)+
paulson@14267
   685
apply assumption
nipkow@15439
   686
  apply (simp_all)
paulson@14267
   687
done
paulson@14267
   688
paulson@17085
   689
declare div_less_dividend [simp]
paulson@17085
   690
paulson@14267
   691
text{*A fact for the mutilated chess board*}
paulson@14267
   692
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   693
apply (case_tac "n=0", simp)
paulson@15251
   694
apply (induct "m" rule: nat_less_induct)
paulson@14267
   695
apply (case_tac "Suc (na) <n")
paulson@14267
   696
(* case Suc(na) < n *)
paulson@14267
   697
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   698
(* case n \<le> Suc(na) *)
paulson@16796
   699
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   700
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   701
done
paulson@14267
   702
paulson@14437
   703
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
wenzelm@22718
   704
  by (cases "n = 0") auto
paulson@14437
   705
paulson@14437
   706
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
wenzelm@22718
   707
  by (cases "n = 0") auto
paulson@14437
   708
paulson@14267
   709
haftmann@25942
   710
subsubsection{*The Divides Relation*}
paulson@14267
   711
paulson@14267
   712
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
wenzelm@22718
   713
  unfolding dvd_def by blast
paulson@14267
   714
paulson@14267
   715
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
wenzelm@22718
   716
  unfolding dvd_def by blast
nipkow@13152
   717
paulson@14267
   718
lemma dvd_0_right [iff]: "m dvd (0::nat)"
wenzelm@22718
   719
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
paulson@14267
   720
paulson@14267
   721
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
wenzelm@22718
   722
  by (force simp add: dvd_def)
paulson@14267
   723
paulson@14267
   724
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
wenzelm@22718
   725
  by (blast intro: dvd_0_left)
paulson@14267
   726
paulson@24286
   727
declare dvd_0_left_iff [noatp]
paulson@24286
   728
paulson@14267
   729
lemma dvd_1_left [iff]: "Suc 0 dvd k"
wenzelm@22718
   730
  unfolding dvd_def by simp
paulson@14267
   731
paulson@14267
   732
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
wenzelm@22718
   733
  by (simp add: dvd_def)
paulson@14267
   734
haftmann@25942
   735
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard]
haftmann@25942
   736
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard]
paulson@14267
   737
paulson@14267
   738
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
wenzelm@22718
   739
  unfolding dvd_def
wenzelm@22718
   740
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
paulson@14267
   741
haftmann@23684
   742
text {* @{term "op dvd"} is a partial order *}
haftmann@23684
   743
haftmann@25942
   744
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
haftmann@23684
   745
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
haftmann@23684
   746
paulson@14267
   747
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
wenzelm@22718
   748
  unfolding dvd_def
wenzelm@22718
   749
  by (blast intro: add_mult_distrib2 [symmetric])
paulson@14267
   750
paulson@14267
   751
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
wenzelm@22718
   752
  unfolding dvd_def
wenzelm@22718
   753
  by (blast intro: diff_mult_distrib2 [symmetric])
paulson@14267
   754
paulson@14267
   755
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
wenzelm@22718
   756
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   757
  apply (blast intro: dvd_add)
wenzelm@22718
   758
  done
paulson@14267
   759
paulson@14267
   760
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
wenzelm@22718
   761
  by (drule_tac m = m in dvd_diff, auto)
paulson@14267
   762
paulson@14267
   763
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
wenzelm@22718
   764
  unfolding dvd_def by (blast intro: mult_left_commute)
paulson@14267
   765
paulson@14267
   766
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
wenzelm@22718
   767
  apply (subst mult_commute)
wenzelm@22718
   768
  apply (erule dvd_mult)
wenzelm@22718
   769
  done
paulson@14267
   770
paulson@17084
   771
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
wenzelm@22718
   772
  by (rule dvd_refl [THEN dvd_mult])
paulson@17084
   773
paulson@17084
   774
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
wenzelm@22718
   775
  by (rule dvd_refl [THEN dvd_mult2])
paulson@14267
   776
paulson@14267
   777
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
wenzelm@22718
   778
  apply (rule iffI)
wenzelm@22718
   779
   apply (erule_tac [2] dvd_add)
wenzelm@22718
   780
   apply (rule_tac [2] dvd_refl)
wenzelm@22718
   781
  apply (subgoal_tac "n = (n+k) -k")
wenzelm@22718
   782
   prefer 2 apply simp
wenzelm@22718
   783
  apply (erule ssubst)
wenzelm@22718
   784
  apply (erule dvd_diff)
wenzelm@22718
   785
  apply (rule dvd_refl)
wenzelm@22718
   786
  done
paulson@14267
   787
paulson@14267
   788
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
wenzelm@22718
   789
  unfolding dvd_def
wenzelm@22718
   790
  apply (case_tac "n = 0", auto)
wenzelm@22718
   791
  apply (blast intro: mod_mult_distrib2 [symmetric])
wenzelm@22718
   792
  done
paulson@14267
   793
paulson@14267
   794
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
wenzelm@22718
   795
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
wenzelm@22718
   796
   apply (simp add: mod_div_equality)
wenzelm@22718
   797
  apply (simp only: dvd_add dvd_mult)
wenzelm@22718
   798
  done
paulson@14267
   799
paulson@14267
   800
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
wenzelm@22718
   801
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
paulson@14267
   802
paulson@14267
   803
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
wenzelm@22718
   804
  unfolding dvd_def
wenzelm@22718
   805
  apply (erule exE)
wenzelm@22718
   806
  apply (simp add: mult_ac)
wenzelm@22718
   807
  done
paulson@14267
   808
paulson@14267
   809
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
wenzelm@22718
   810
  apply auto
wenzelm@22718
   811
   apply (subgoal_tac "m*n dvd m*1")
wenzelm@22718
   812
   apply (drule dvd_mult_cancel, auto)
wenzelm@22718
   813
  done
paulson@14267
   814
paulson@14267
   815
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
wenzelm@22718
   816
  apply (subst mult_commute)
wenzelm@22718
   817
  apply (erule dvd_mult_cancel1)
wenzelm@22718
   818
  done
paulson@14267
   819
paulson@14267
   820
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
wenzelm@22718
   821
  apply (unfold dvd_def, clarify)
wenzelm@22718
   822
  apply (rule_tac x = "k*ka" in exI)
wenzelm@22718
   823
  apply (simp add: mult_ac)
wenzelm@22718
   824
  done
paulson@14267
   825
paulson@14267
   826
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
wenzelm@22718
   827
  by (simp add: dvd_def mult_assoc, blast)
paulson@14267
   828
paulson@14267
   829
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
wenzelm@22718
   830
  apply (unfold dvd_def, clarify)
wenzelm@22718
   831
  apply (rule_tac x = "i*k" in exI)
wenzelm@22718
   832
  apply (simp add: mult_ac)
wenzelm@22718
   833
  done
paulson@14267
   834
paulson@14267
   835
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
wenzelm@22718
   836
  apply (unfold dvd_def, clarify)
wenzelm@22718
   837
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
wenzelm@22718
   838
  apply (erule conjE)
wenzelm@22718
   839
  apply (rule le_trans)
wenzelm@22718
   840
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
wenzelm@22718
   841
   apply (erule_tac [2] Suc_leI, simp)
wenzelm@22718
   842
  done
paulson@14267
   843
haftmann@25942
   844
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard]
paulson@14267
   845
paulson@14267
   846
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
wenzelm@22718
   847
  apply (subgoal_tac "m mod n = 0")
wenzelm@22718
   848
   apply (simp add: mult_div_cancel)
wenzelm@22718
   849
  apply (simp only: dvd_eq_mod_eq_0)
wenzelm@22718
   850
  done
paulson@14267
   851
haftmann@21408
   852
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
wenzelm@22718
   853
  apply (unfold dvd_def)
wenzelm@22718
   854
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   855
  apply (simp add: power_add)
wenzelm@22718
   856
  done
haftmann@21408
   857
haftmann@26100
   858
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
haftmann@26100
   859
  apply (rule trans [symmetric])
haftmann@26100
   860
   apply (rule mod_add1_eq, simp)
haftmann@26100
   861
  apply (rule mod_add1_eq [symmetric])
haftmann@26100
   862
  done
haftmann@26100
   863
haftmann@26100
   864
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
haftmann@26100
   865
  apply (rule trans [symmetric])
haftmann@26100
   866
   apply (rule mod_add1_eq, simp)
haftmann@26100
   867
  apply (rule mod_add1_eq [symmetric])
haftmann@26100
   868
  done
haftmann@26100
   869
nipkow@25162
   870
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
wenzelm@22718
   871
  by (induct n) auto
haftmann@21408
   872
haftmann@21408
   873
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
wenzelm@22718
   874
  apply (induct j)
wenzelm@22718
   875
   apply (simp_all add: le_Suc_eq)
wenzelm@22718
   876
  apply (blast dest!: dvd_mult_right)
wenzelm@22718
   877
  done
haftmann@21408
   878
haftmann@21408
   879
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
wenzelm@22718
   880
  apply (rule power_le_imp_le_exp, assumption)
wenzelm@22718
   881
  apply (erule dvd_imp_le, simp)
wenzelm@22718
   882
  done
haftmann@21408
   883
paulson@14267
   884
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
wenzelm@22718
   885
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   886
wenzelm@22718
   887
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   888
paulson@14267
   889
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   890
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
wenzelm@22718
   891
  apply (cut_tac m = m in mod_div_equality)
wenzelm@22718
   892
  apply (simp only: add_ac)
wenzelm@22718
   893
  apply (blast intro: sym)
wenzelm@22718
   894
  done
paulson@14267
   895
nipkow@13152
   896
lemma split_div:
nipkow@13189
   897
 "P(n div k :: nat) =
nipkow@13189
   898
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   899
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   900
proof
nipkow@13189
   901
  assume P: ?P
nipkow@13189
   902
  show ?Q
nipkow@13189
   903
  proof (cases)
nipkow@13189
   904
    assume "k = 0"
nipkow@13189
   905
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   906
  next
nipkow@13189
   907
    assume not0: "k \<noteq> 0"
nipkow@13189
   908
    thus ?Q
nipkow@13189
   909
    proof (simp, intro allI impI)
nipkow@13189
   910
      fix i j
nipkow@13189
   911
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   912
      show "P i"
nipkow@13189
   913
      proof (cases)
wenzelm@22718
   914
        assume "i = 0"
wenzelm@22718
   915
        with n j P show "P i" by simp
nipkow@13189
   916
      next
wenzelm@22718
   917
        assume "i \<noteq> 0"
wenzelm@22718
   918
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   919
      qed
nipkow@13189
   920
    qed
nipkow@13189
   921
  qed
nipkow@13189
   922
next
nipkow@13189
   923
  assume Q: ?Q
nipkow@13189
   924
  show ?P
nipkow@13189
   925
  proof (cases)
nipkow@13189
   926
    assume "k = 0"
nipkow@13189
   927
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   928
  next
nipkow@13189
   929
    assume not0: "k \<noteq> 0"
nipkow@13189
   930
    with Q have R: ?R by simp
nipkow@13189
   931
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   932
    show ?P by simp
nipkow@13189
   933
  qed
nipkow@13189
   934
qed
nipkow@13189
   935
berghofe@13882
   936
lemma split_div_lemma:
haftmann@26100
   937
  assumes "0 < n"
haftmann@26100
   938
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
   939
proof
haftmann@26100
   940
  assume ?rhs
haftmann@26100
   941
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
   942
  then have A: "n * q \<le> m" by simp
haftmann@26100
   943
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
   944
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
   945
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
   946
  with nq have "m < n + n * q" by simp
haftmann@26100
   947
  then have B: "m < n * Suc q" by simp
haftmann@26100
   948
  from A B show ?lhs ..
haftmann@26100
   949
next
haftmann@26100
   950
  assume P: ?lhs
haftmann@26100
   951
  then have "divmod_rel m n q (m - n * q)"
haftmann@26100
   952
    unfolding divmod_rel_def by (auto simp add: mult_ac)
haftmann@26100
   953
  then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
haftmann@26100
   954
qed
berghofe@13882
   955
berghofe@13882
   956
theorem split_div':
berghofe@13882
   957
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   958
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   959
  apply (case_tac "0 < n")
berghofe@13882
   960
  apply (simp only: add: split_div_lemma)
berghofe@13882
   961
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
berghofe@13882
   962
  done
berghofe@13882
   963
nipkow@13189
   964
lemma split_mod:
nipkow@13189
   965
 "P(n mod k :: nat) =
nipkow@13189
   966
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   967
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   968
proof
nipkow@13189
   969
  assume P: ?P
nipkow@13189
   970
  show ?Q
nipkow@13189
   971
  proof (cases)
nipkow@13189
   972
    assume "k = 0"
nipkow@13189
   973
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   974
  next
nipkow@13189
   975
    assume not0: "k \<noteq> 0"
nipkow@13189
   976
    thus ?Q
nipkow@13189
   977
    proof (simp, intro allI impI)
nipkow@13189
   978
      fix i j
nipkow@13189
   979
      assume "n = k*i + j" "j < k"
nipkow@13189
   980
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   981
    qed
nipkow@13189
   982
  qed
nipkow@13189
   983
next
nipkow@13189
   984
  assume Q: ?Q
nipkow@13189
   985
  show ?P
nipkow@13189
   986
  proof (cases)
nipkow@13189
   987
    assume "k = 0"
nipkow@13189
   988
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   989
  next
nipkow@13189
   990
    assume not0: "k \<noteq> 0"
nipkow@13189
   991
    with Q have R: ?R by simp
nipkow@13189
   992
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   993
    show ?P by simp
nipkow@13189
   994
  qed
nipkow@13189
   995
qed
nipkow@13189
   996
berghofe@13882
   997
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
   998
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
   999
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
  1000
  apply arith
berghofe@13882
  1001
  done
berghofe@13882
  1002
haftmann@22800
  1003
lemma div_mod_equality':
haftmann@22800
  1004
  fixes m n :: nat
haftmann@22800
  1005
  shows "m div n * n = m - m mod n"
haftmann@22800
  1006
proof -
haftmann@22800
  1007
  have "m mod n \<le> m mod n" ..
haftmann@22800
  1008
  from div_mod_equality have 
haftmann@22800
  1009
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
haftmann@22800
  1010
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
haftmann@22800
  1011
    "m div n * n + (m mod n - m mod n) = m - m mod n"
haftmann@22800
  1012
    by simp
haftmann@22800
  1013
  then show ?thesis by simp
haftmann@22800
  1014
qed
haftmann@22800
  1015
haftmann@22800
  1016
haftmann@25942
  1017
subsubsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
  1018
paulson@14640
  1019
lemma mod_induct_0:
paulson@14640
  1020
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1021
  and base: "P i" and i: "i<p"
paulson@14640
  1022
  shows "P 0"
paulson@14640
  1023
proof (rule ccontr)
paulson@14640
  1024
  assume contra: "\<not>(P 0)"
paulson@14640
  1025
  from i have p: "0<p" by simp
paulson@14640
  1026
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1027
  proof
paulson@14640
  1028
    fix k
paulson@14640
  1029
    show "?A k"
paulson@14640
  1030
    proof (induct k)
paulson@14640
  1031
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1032
    next
paulson@14640
  1033
      fix n
paulson@14640
  1034
      assume ih: "?A n"
paulson@14640
  1035
      show "?A (Suc n)"
paulson@14640
  1036
      proof (clarsimp)
wenzelm@22718
  1037
        assume y: "P (p - Suc n)"
wenzelm@22718
  1038
        have n: "Suc n < p"
wenzelm@22718
  1039
        proof (rule ccontr)
wenzelm@22718
  1040
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1041
          hence "p - Suc n = 0"
wenzelm@22718
  1042
            by simp
wenzelm@22718
  1043
          with y contra show "False"
wenzelm@22718
  1044
            by simp
wenzelm@22718
  1045
        qed
wenzelm@22718
  1046
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1047
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1048
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1049
          by blast
wenzelm@22718
  1050
        show "False"
wenzelm@22718
  1051
        proof (cases "n=0")
wenzelm@22718
  1052
          case True
wenzelm@22718
  1053
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1054
        next
wenzelm@22718
  1055
          case False
wenzelm@22718
  1056
          with p have "p-n < p" by arith
wenzelm@22718
  1057
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1058
        qed
paulson@14640
  1059
      qed
paulson@14640
  1060
    qed
paulson@14640
  1061
  qed
paulson@14640
  1062
  moreover
paulson@14640
  1063
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1064
    by (blast dest: less_imp_add_positive)
paulson@14640
  1065
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1066
  moreover
paulson@14640
  1067
  note base
paulson@14640
  1068
  ultimately
paulson@14640
  1069
  show "False" by blast
paulson@14640
  1070
qed
paulson@14640
  1071
paulson@14640
  1072
lemma mod_induct:
paulson@14640
  1073
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1074
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1075
  shows "P j"
paulson@14640
  1076
proof -
paulson@14640
  1077
  have "\<forall>j<p. P j"
paulson@14640
  1078
  proof
paulson@14640
  1079
    fix j
paulson@14640
  1080
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1081
    proof (induct j)
paulson@14640
  1082
      from step base i show "?A 0"
wenzelm@22718
  1083
        by (auto elim: mod_induct_0)
paulson@14640
  1084
    next
paulson@14640
  1085
      fix k
paulson@14640
  1086
      assume ih: "?A k"
paulson@14640
  1087
      show "?A (Suc k)"
paulson@14640
  1088
      proof
wenzelm@22718
  1089
        assume suc: "Suc k < p"
wenzelm@22718
  1090
        hence k: "k<p" by simp
wenzelm@22718
  1091
        with ih have "P k" ..
wenzelm@22718
  1092
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1093
          by blast
wenzelm@22718
  1094
        moreover
wenzelm@22718
  1095
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1096
          by simp
wenzelm@22718
  1097
        ultimately
wenzelm@22718
  1098
        show "P (Suc k)" by simp
paulson@14640
  1099
      qed
paulson@14640
  1100
    qed
paulson@14640
  1101
  qed
paulson@14640
  1102
  with j show ?thesis by blast
paulson@14640
  1103
qed
paulson@14640
  1104
paulson@3366
  1105
end