src/HOL/Divides.thy
author haftmann
Fri, 18 Jul 2008 18:25:53 +0200
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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 and mod *}
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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 div = dvd +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract division in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + 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 [simp]: "a div 0 = 0"
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    and div_0 [simp]: "0 div a = 0"
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    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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begin
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text {* @{const div} and @{const mod} *}
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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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lemma mod_by_0 [simp]: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mod_0 [simp]: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp 
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lemma div_mult_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b * c) div b = c + a div b"
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  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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proof (cases "b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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    by (simp add: mod_div_equality)
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  also from False div_mult_self1 [of b a c] have
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    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
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      by (simp add: left_distrib add_ac)
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  finally have "a = a div b * b + (a + c * b) mod b"
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    by (simp add: add_commute [of a] add_assoc left_distrib)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
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  by (simp add: mult_commute [of b])
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  using div_mult_self2 [of b 0 a] by simp
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  using div_mult_self1 [of b 0 a] by simp
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
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  using mod_mult_self2 [of 0 b a] by simp
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma div_by_1 [simp]: "a div 1 = a"
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  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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lemma mod_by_1 [simp]: "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_self [simp]: "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using div_mult_self2_is_id [of _ 1] by simp
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lemma div_add_self1:
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  assumes "b \<noteq> 0"
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  shows "(b + a) div b = a div b + 1"
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  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
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lemma div_add_self2:
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  assumes "b \<noteq> 0"
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  shows "(a + b) div b = a div b + 1"
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  using assms div_add_self1 [of b a] by (simp add: add_commute)
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lemma mod_add_self1:
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  "(b + a) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
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lemma mod_add_self2:
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  "(a + b) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by simp
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lemma mod_div_decomp:
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  fixes a b
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  obtains q r where "q = a div b" and "r = a mod b"
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    and "a = q * b + r"
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proof -
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  from mod_div_equality have "a = a div b * b + a mod b" by simp
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  moreover have "a div b = a div b" ..
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  moreover have "a mod b = a mod b" ..
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  note that ultimately show thesis by blast
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qed
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lemma dvd_eq_mod_eq_0 [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
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qed
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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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a52309ac4a4d added class semiring_div
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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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   243
  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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   247
  shows "divmod m n = (q, r)"
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   248
  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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   250
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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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   254
  using assms by (auto dest: divmod_eq simp add: div_nat_def)
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   255
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   256
lemma mod_eq:
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   257
  assumes "divmod_rel m n q r" 
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   258
  shows "m mod n = r"
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   259
  using assms by (auto dest: divmod_eq simp add: mod_nat_def)
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   261
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
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   262
proof -
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   263
  from divmod_rel_ex
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   264
    obtain q r where rel: "divmod_rel m n q r" .
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   265
  moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
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   266
    by simp_all
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   267
  ultimately show ?thesis by simp
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qed
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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   269
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lemma divmod_zero:
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  "divmod m 0 = (0, m)"
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   272
proof -
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   273
  from divmod_rel [of m 0] show ?thesis
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   274
    unfolding divmod_div_mod divmod_rel_def by simp
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qed
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   276
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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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   281
  from divmod_rel [of m n] show ?thesis
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   282
    unfolding divmod_div_mod divmod_rel_def
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   283
    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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   285
qed
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   286
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lemma divmod_step:
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   288
  assumes "0 < n" and "n \<le> m"
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   289
  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
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   290
proof -
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   291
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
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   292
  with assms have m_div_n: "m div n \<ge> 1"
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   293
    by (cases "m div n") (auto simp add: divmod_rel_def)
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   294
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
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   295
    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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lemma le_div_geq:
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  fixes m n :: nat
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  assumes "0 < n" and "n \<le> m"
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  shows "m div n = Suc ((m - n) div n)"
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  using assms divmod_step divmod_div_mod by simp
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lemma mod_less [simp]:
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  fixes m n :: nat
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  assumes "m < n"
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  shows "m mod n = m"
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  using assms divmod_base divmod_div_mod by simp
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lemma le_mod_geq:
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  fixes m n :: nat
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  assumes "n \<le> m"
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  shows "m mod n = (m - n) mod n"
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  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
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instance proof
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  fix m n :: nat show "m div n * n + m mod n = m"
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    using divmod_rel [of m n] by (simp add: divmod_rel_def)
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next
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  fix n :: nat show "n div 0 = 0"
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    using divmod_zero divmod_div_mod [of n 0] by simp
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next
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  fix n :: nat show "0 div n = 0"
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    using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def)
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next
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  fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n"
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    by (induct m) (simp_all add: le_div_geq)
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qed
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end
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text {* Simproc for cancelling @{const div} and @{const mod} *}
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(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
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lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*)
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ML {*
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structure CancelDivModData =
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struct
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val div_name = @{const_name div};
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val mod_name = @{const_name mod};
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val mk_binop = HOLogic.mk_binop;
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val mk_sum = ArithData.mk_sum;
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val dest_sum = ArithData.dest_sum;
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(*logic*)
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val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
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val trans = trans
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val prove_eq_sums =
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  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
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  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
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end;
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structure CancelDivMod = CancelDivModFun(CancelDivModData);
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val cancel_div_mod_proc = Simplifier.simproc @{theory}
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  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
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Addsimprocs[cancel_div_mod_proc];
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*}
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text {* code generator setup *}
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lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
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   383
  let (q, r) = divmod (m - n) n in (Suc q, r))"
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   384
  by (simp add: divmod_zero divmod_base divmod_step)
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    (simp add: divmod_div_mod)
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   387
code_modulename SML
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   388
  Divides Nat
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   389
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   390
code_modulename OCaml
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   391
  Divides Nat
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   392
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   393
code_modulename Haskell
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   394
  Divides Nat
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   395
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subsubsection {* Quotient *}
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   398
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   399
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
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   400
  by (simp add: le_div_geq linorder_not_less)
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   401
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   402
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
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   403
  by (simp add: div_geq)
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   404
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   405
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
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   406
  by simp
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   407
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   408
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
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   409
  by simp
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diff changeset
   410
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   411
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   412
subsubsection {* Remainder *}
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   413
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   414
lemma mod_less_divisor [simp]:
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   415
  fixes m n :: nat
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   416
  assumes "n > 0"
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   417
  shows "m mod n < (n::nat)"
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   418
  using assms divmod_rel unfolding divmod_rel_def by auto
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   419
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   420
lemma mod_less_eq_dividend [simp]:
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   421
  fixes m n :: nat
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diff changeset
   422
  shows "m mod n \<le> m"
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   423
proof (rule add_leD2)
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   424
  from mod_div_equality have "m div n * n + m mod n = m" .
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   425
  then show "m div n * n + m mod n \<le> m" by auto
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   426
qed
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   427
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   428
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
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   429
  by (simp add: le_mod_geq linorder_not_less)
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   430
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   431
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
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   432
  by (simp add: le_mod_geq)
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   433
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   434
lemma mod_1 [simp]: "m mod Suc 0 = 0"
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   435
  by (induct m) (simp_all add: mod_geq)
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diff changeset
   436
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   437
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
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936f7580937d tuned proofs;
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diff changeset
   438
  apply (cases "n = 0", simp)
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diff changeset
   439
  apply (cases "k = 0", simp)
936f7580937d tuned proofs;
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diff changeset
   440
  apply (induct m rule: nat_less_induct)
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diff changeset
   441
  apply (subst mod_if, simp)
936f7580937d tuned proofs;
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diff changeset
   442
  apply (simp add: mod_geq diff_mult_distrib)
936f7580937d tuned proofs;
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diff changeset
   443
  done
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   444
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   445
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
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diff changeset
   446
  by (simp add: mult_commute [of k] mod_mult_distrib)
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   447
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   448
(* a simple rearrangement of mod_div_equality: *)
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parents: 14208
diff changeset
   449
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   450
  by (cut_tac a = m and b = n in mod_div_equality2, arith)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   451
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   452
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   453
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   454
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   455
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   456
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   457
subsubsection {* Quotient and Remainder *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   458
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   459
lemma divmod_rel_mult1_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   460
  "[| divmod_rel b c q r; c > 0 |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   461
   ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   462
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   463
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   464
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   465
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   466
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   467
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   468
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   469
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   470
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   471
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN mod_eq])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   472
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   473
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   474
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   475
  apply (rule trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   476
   apply (rule_tac s = "b*a mod c" in trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   477
    apply (rule_tac [2] mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   478
   apply (simp_all add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   479
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   480
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   481
lemma mod_mult_distrib_mod:
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   482
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   483
apply (rule mod_mult1_eq' [THEN trans])
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   484
apply (rule mod_mult1_eq)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   485
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   486
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   487
lemma divmod_rel_add1_eq:
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   488
  "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   489
   ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   490
by (auto simp add: split_ifs mult_ac divmod_rel_def add_mult_distrib2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   491
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   492
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   493
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   494
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   495
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   496
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   497
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   498
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   499
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   500
apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   501
apply (blast intro: divmod_rel_add1_eq [THEN mod_eq] divmod_rel)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   502
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   503
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   504
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   505
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   506
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   507
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   508
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   509
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   510
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   511
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   512
      ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   513
  by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   514
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   515
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   516
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   517
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   518
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   519
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   520
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   521
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   522
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   523
  apply (cases "c = 0", simp)
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   524
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   525
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   526
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   527
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   528
subsubsection{*Cancellation of Common Factors in Division*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   529
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   530
lemma div_mult_mult_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   531
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   532
  by (auto simp add: div_mult2_eq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   533
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   534
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   535
  apply (cases "b = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   536
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   537
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   538
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   539
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   540
  apply (drule div_mult_mult1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   541
  apply (auto simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   542
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   543
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   544
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   545
subsubsection{*Further Facts about Quotient and Remainder*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   546
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   547
lemma div_1 [simp]: "m div Suc 0 = m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   548
  by (induct m) (simp_all add: div_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   549
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   550
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   551
(* Monotonicity of div in first argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   552
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   553
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   554
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   555
apply (induct "n" rule: nat_less_induct, clarify)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   556
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   557
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   558
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   559
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   560
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   561
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   562
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   563
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   564
apply (simp add: div_geq diff_le_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   565
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   566
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   567
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   568
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   569
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   570
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   571
apply (induct_tac k rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   572
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   573
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   574
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   575
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   576
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   577
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   578
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   579
 apply (blast intro: div_le_mono diff_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   580
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   581
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   582
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   583
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   584
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   585
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   586
apply (subgoal_tac "m div n \<le> m div 1", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   587
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   588
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   589
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   590
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   591
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   592
lemma div_less_dividend [rule_format]:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   593
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   594
apply (induct_tac m rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   595
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   596
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   597
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   598
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   599
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   600
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   601
 apply (subgoal_tac "(m-n) div n < (m-n) ")
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   602
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   603
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   604
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   605
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   606
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   607
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   608
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   609
text{*A fact for the mutilated chess board*}
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   610
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   611
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   612
apply (induct "m" rule: nat_less_induct)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   613
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   614
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   615
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   616
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   617
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   618
apply (auto simp add: Suc_diff_le le_mod_geq)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   619
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   620
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   621
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   622
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   623
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   624
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   625
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   626
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   627
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   628
subsubsection {* The Divides Relation *}
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   629
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   630
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   631
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   632
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   633
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   634
  by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   635
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   636
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   637
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   638
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   639
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   640
text {* @{term "op dvd"} is a partial order *}
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   641
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   642
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   643
  by unfold_locales (auto intro: dvd_refl dvd_trans dvd_anti_sym)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   644
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   645
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   646
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   647
  by (blast intro: diff_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   648
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   649
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   650
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   651
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   652
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   653
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   654
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   655
  by (drule_tac m = m in dvd_diff, auto)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   656
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   657
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   658
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   659
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   660
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   661
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   662
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   663
  apply (erule ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   664
  apply (erule dvd_diff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   665
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   666
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   667
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   668
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   669
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   670
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   671
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   672
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   673
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   674
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   675
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   676
   apply (simp add: mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   677
  apply (simp only: dvd_add dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   678
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   679
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   680
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   681
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   682
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   683
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   684
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   685
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   686
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   687
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   688
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   689
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   690
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   691
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   692
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   693
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   694
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   695
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   696
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   697
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   698
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   699
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   700
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   701
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   702
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   703
  apply (erule conjE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   704
  apply (rule le_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   705
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   706
   apply (erule_tac [2] Suc_leI, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   707
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   708
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   709
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   710
  apply (subgoal_tac "m mod n = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   711
   apply (simp add: mult_div_cancel)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   712
  apply (simp only: dvd_eq_mod_eq_0)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   713
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   714
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   715
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   716
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   717
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   718
  apply (simp add: power_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   719
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   720
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   721
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   722
  apply (rule trans [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   723
   apply (rule mod_add1_eq, simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   724
  apply (rule mod_add1_eq [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   725
  done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   726
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   727
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   728
  apply (rule trans [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   729
   apply (rule mod_add1_eq, simp)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   730
  apply (rule mod_add1_eq [symmetric])
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   731
  done
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   732
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   733
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   734
  by (induct n) auto
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   735
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   736
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   737
  apply (induct j)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   738
   apply (simp_all add: le_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   739
  apply (blast dest!: dvd_mult_right)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   740
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   741
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   742
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   743
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   744
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   745
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   746
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   747
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   748
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   749
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   750
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   751
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   752
(*Loses information, namely we also have r<d provided d is nonzero*)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   753
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   754
  apply (cut_tac a = m in mod_div_equality)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   755
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   756
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   757
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   758
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   759
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   760
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   761
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   762
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   763
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   764
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   765
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   766
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   767
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   768
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   769
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   770
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   771
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   772
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   773
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   774
      assume n: "n = k*i + j" and j: "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   775
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   776
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   777
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   778
        with n j P show "P i" by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   779
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   780
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   781
        with not0 n j P show "P i" by(simp add:add_ac)
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   782
      qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   783
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   784
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   785
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   786
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   787
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   788
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   789
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   790
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   791
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   792
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   793
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   794
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
   795
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   796
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   797
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   798
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   799
lemma split_div_lemma:
26100
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   800
  assumes "0 < n"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   801
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   802
proof
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   803
  assume ?rhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   804
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   805
  then have A: "n * q \<le> m" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   806
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   807
  then have "m < m + (n - (m mod n))" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   808
  then have "m < n + (m - (m mod n))" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   809
  with nq have "m < n + n * q" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   810
  then have B: "m < n * Suc q" by simp
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   811
  from A B show ?lhs ..
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   812
next
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   813
  assume P: ?lhs
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   814
  then have "divmod_rel m n q (m - n * q)"
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   815
    unfolding divmod_rel_def by (auto simp add: mult_ac)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   816
  then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
fbc60cd02ae2 using only an relation predicate to construct div and mod
haftmann
parents: 26072
diff changeset
   817
qed
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   818
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   819
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   820
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   821
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   822
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   823
  apply (simp only: add: split_div_lemma)
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   824
  apply simp_all
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   825
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   826
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   827
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   828
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   829
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   830
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   831
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   832
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   833
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   834
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   835
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   836
    with P show ?Q by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   837
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   838
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   839
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   840
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   841
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   842
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   843
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   844
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   845
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   846
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   847
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   848
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   849
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   850
    assume "k = 0"
27651
16a26996c30e moved op dvd to theory Ring_and_Field; generalized a couple of lemmas
haftmann
parents: 27540
diff changeset
   851
    with Q show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   852
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   853
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   854
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   855
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
13517
42efec18f5b2 Added div+mod cancelling simproc
nipkow
parents: 13189
diff changeset
   856
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   857
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   858
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   859
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   860
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   861
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   862
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   863
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   864
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   865
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   866
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   867
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   868
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   869
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   870
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   871
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   872
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   873
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   874
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   875
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   876
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   877
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   878
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   879
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   880
subsubsection {*An ``induction'' law for modulus arithmetic.*}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   881
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   882
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   883
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   884
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   885
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   886
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   887
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   888
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   889
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   890
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   891
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   892
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   893
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   894
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   895
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   896
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   897
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   898
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   899
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   900
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   901
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   902
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   903
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   904
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   905
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   906
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   907
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   908
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   909
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   910
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   911
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   912
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   913
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   914
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   915
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   916
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   917
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   918
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   919
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   920
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   921
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   922
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   923
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   924
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   925
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   926
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   927
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   928
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   929
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   930
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   931
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   932
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   933
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   934
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   935
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   936
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   937
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   938
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   939
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   940
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   941
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   942
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   943
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   944
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   945
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   946
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   947
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   948
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   949
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   950
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   951
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   952
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   953
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   954
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   955
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   956
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   957
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   958
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   959
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   960
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   961
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   962
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   963
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   964
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   965
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   966
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   967
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
   968
end