src/HOL/Divides.thy
author haftmann
Wed Feb 13 09:35:31 2008 +0100 (2008-02-13)
changeset 26062 16f334d7156a
parent 25947 1f2f4d941e9e
child 26072 f65a7fa2da6c
permissions -rw-r--r--
more abstract lemmas
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(*  Title:      HOL/Divides.thy
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div, mod and the divides relation "dvd" *}
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theory Divides
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imports Power
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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 = times +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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begin
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definition
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  dvd  :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "dvd" 50)
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where
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  [code func del]: "m dvd n \<longleftrightarrow> (\<exists>k. n = m * k)"
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end
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subsection {* Abstract divisibility 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: "a div 0 = 0"
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    and mult_div: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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begin
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lemma div_by_1: "a div 1 = a"
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  using mult_div [of 1 a] zero_neq_one by simp
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lemma mod_by_1: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_by_0: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mult_mod: "a * b mod b = 0"
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proof (cases "b = 0")
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  case True then show ?thesis by (simp add: mod_by_0)
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next
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  case False with mult_div have abb: "a * b div b = a" .
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  from mod_div_equality have "a * b div b * b + a * b mod b = a * b" .
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  with abb have "a * b + a * b mod b = a * b + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self: "a mod a = 0"
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  using mult_mod [of one] by simp
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lemma div_self: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using mult_div [of _ one] by simp
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lemma div_0: "0 div a = 0"
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proof (cases "a = 0")
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  case True then show ?thesis by (simp add: div_by_0)
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next
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  case False with mult_div have "0 * a div a = 0" .
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  then show ?thesis by simp
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qed
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lemma mod_0: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp 
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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lemma dvdI [intro?]: "a = b * c \<Longrightarrow> b dvd a"
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  unfolding dvd_def ..
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lemma dvdE [elim?]: "b dvd a \<Longrightarrow> (\<And>c. a = b * c \<Longrightarrow> P) \<Longrightarrow> P"
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  unfolding dvd_def by blast 
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lemma dvd_def_mod [code func]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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  with mod_div_equality [of b a] have "b div a * a = b" by simp
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  then have "b = a * (b div a)" unfolding mult_commute ..
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  then have "\<exists>c. b = a * c" ..
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  then show "a dvd b" unfolding dvd_def .
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next
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  assume "a dvd b"
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  then have "\<exists>c. b = a * c" unfolding dvd_def .
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  then obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then have "b mod a = c * a mod a" by (simp add: mult_commute)
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  then show "b mod a = 0" by (simp add: mult_mod)
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qed
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lemma dvd_refl: "a dvd a"
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  unfolding dvd_def_mod mod_self ..
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lemma dvd_trans:
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  assumes "a dvd b" and "b dvd c"
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  shows "a dvd c"
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proof -
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  from assms obtain v where "b = a * v" unfolding dvd_def by auto
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  moreover from assms obtain w where "c = b * w" unfolding dvd_def by auto
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  ultimately have "c = a * (v * w)" by (simp add: mult_assoc)
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  then show ?thesis unfolding dvd_def ..
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qed
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lemma zero_dvd_iff [noatp]: "0 dvd a \<longleftrightarrow> a = 0"
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  unfolding dvd_def by simp
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lemma dvd_0: "a dvd 0"
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unfolding dvd_def proof
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  show "0 = a * 0" by simp
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qed
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lemma one_dvd: "1 dvd a"
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  unfolding dvd_def by simp
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lemma dvd_mult: "a dvd c \<Longrightarrow> a dvd (b * c)"
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  unfolding dvd_def by (blast intro: mult_left_commute)
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lemma dvd_mult2: "a dvd b \<Longrightarrow> a dvd (b * c)"
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  apply (subst mult_commute)
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  apply (erule dvd_mult)
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  done
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lemma dvd_triv_right: "a dvd b * a"
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  by (rule dvd_mult) (rule dvd_refl)
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lemma dvd_triv_left: "a dvd a * b"
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  by (rule dvd_mult2) (rule dvd_refl)
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lemma mult_dvd_mono: "a dvd c \<Longrightarrow> b dvd d \<Longrightarrow> a * b dvd c * d"
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  apply (unfold dvd_def, clarify)
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  apply (rule_tac x = "k * ka" in exI)
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  apply (simp add: mult_ac)
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  done
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lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
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  by (simp add: dvd_def mult_assoc, blast)
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lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
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  unfolding mult_ac [of a] by (rule dvd_mult_left)
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end
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subsection {* Division on the natural numbers *}
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instantiation nat :: semiring_div
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begin
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definition
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  div_def: "m div n == wfrec (pred_nat^+)
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                          (%f j. if j<n | n=0 then 0 else Suc (f (j-n))) m"
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lemma div_eq: "(%m. m div n) = wfrec (pred_nat^+)
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               (%f j. if j<n | n=0 then 0 else Suc (f (j-n)))"
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by (simp add: div_def)
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definition
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  mod_def: "m mod n == wfrec (pred_nat^+)
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                          (%f j. if j<n | n=0 then j else f (j-n)) m"
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lemma mod_eq: "(%m. m mod n) =
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              wfrec (pred_nat^+) (%f j. if j<n | n=0 then j else f (j-n))"
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by (simp add: mod_def)
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lemmas wf_less_trans = def_wfrec [THEN trans,
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  OF eq_reflection wf_pred_nat [THEN wf_trancl], standard]
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lemma div_less [simp]: "m < n \<Longrightarrow> m div n = (0\<Colon>nat)"
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  by (rule div_eq [THEN wf_less_trans]) simp
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lemma le_div_geq: "0 < n \<Longrightarrow> n \<le> m \<Longrightarrow> m div n = Suc ((m - n) div n)"
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  by (rule div_eq [THEN wf_less_trans]) (simp add: cut_apply less_eq)
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lemma DIVISION_BY_ZERO_MOD [simp]: "a mod 0 = (a\<Colon>nat)"
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  by (rule mod_eq [THEN wf_less_trans]) simp
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lemma mod_less [simp]: "m < n \<Longrightarrow> m mod n = (m\<Colon>nat)"
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  by (rule mod_eq [THEN wf_less_trans]) simp
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lemma le_mod_geq: "(n\<Colon>nat) \<le> m \<Longrightarrow> m mod n = (m - n) mod n"
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  by (cases "n = 0", simp, rule mod_eq [THEN wf_less_trans])
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    (simp add: cut_apply less_eq)
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lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
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  by (simp add: le_mod_geq)
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instance proof
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  fix n m :: nat
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  show "(m div n) * n + m mod n = m"
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    apply (cases "n = 0", simp)
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    apply (induct m rule: less_induct)
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    apply (subst mod_if)
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    apply (simp add: add_assoc add_diff_inverse le_div_geq)
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    done
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next
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  fix n :: nat
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  show "n div 0 = 0"
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    by (rule div_eq [THEN wf_less_trans]) simp
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next
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  fix n m :: nat
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  assume "n \<noteq> 0"
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  then show "m * n div n = m"
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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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subsubsection{*Simproc for Cancelling Div and Mod*}
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lemmas mod_div_equality = 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 = mod_div_equality2 [of "n\<Colon>nat" m, standard]
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lemmas div_mod_equality = div_mod_equality [of "m\<Colon>nat" n k, standard]
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lemmas div_mod_equality2 = div_mod_equality2 [of "m\<Colon>nat" n k, 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 Divides.div};
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val mod_name = @{const_name Divides.mod};
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val mk_binop = HOLogic.mk_binop;
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val mk_sum = NatArithUtils.mk_sum;
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val dest_sum = NatArithUtils.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 NatArithUtils.prove_conv all_tac (NatArithUtils.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 = NatArithUtils.prep_simproc
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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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subsubsection {* Remainder *}
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lemmas DIVISION_BY_ZERO_MOD [simp] = mod_by_0 [of "a\<Colon>nat", standard]
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lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
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  by (induct m) (simp_all add: le_div_geq)
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lemma mod_geq: "~ m < (n::nat) ==> m mod n = (m-n) mod n"
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  by (simp add: le_mod_geq linorder_not_less)
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lemma mod_1 [simp]: "m mod Suc 0 = 0"
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  by (induct m) (simp_all add: mod_geq)
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lemmas mod_self [simp] = semiring_div_class.mod_self [of "n\<Colon>nat", standard]
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lemma mod_add_self2 [simp]: "(m+n) mod n = m mod (n::nat)"
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  apply (subgoal_tac "(n + m) mod n = (n+m-n) mod n")
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   apply (simp add: add_commute)
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  apply (subst le_mod_geq [symmetric], simp_all)
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  done
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lemma mod_add_self1 [simp]: "(n+m) mod n = m mod (n::nat)"
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  by (simp add: add_commute mod_add_self2)
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lemma mod_mult_self1 [simp]: "(m + k*n) mod n = m mod (n::nat)"
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  by (induct k) (simp_all add: add_left_commute [of _ n])
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lemma mod_mult_self2 [simp]: "(m + n*k) mod n = m mod (n::nat)"
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  by (simp add: mult_commute mod_mult_self1)
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lemma mod_mult_distrib: "(m mod n) * (k::nat) = (m*k) mod (n*k)"
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  apply (cases "n = 0", simp)
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  apply (cases "k = 0", simp)
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  apply (induct m rule: nat_less_induct)
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  apply (subst mod_if, simp)
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  apply (simp add: mod_geq diff_mult_distrib)
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  done
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lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
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  by (simp add: mult_commute [of k] mod_mult_distrib)
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lemma mod_mult_self_is_0 [simp]: "(m*n) mod n = (0::nat)"
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  apply (cases "n = 0", simp)
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  apply (induct m, simp)
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  apply (rename_tac k)
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  apply (cut_tac m = "k * n" and n = n in mod_add_self2)
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  apply (simp add: add_commute)
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  done
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lemma mod_mult_self1_is_0 [simp]: "(n*m) mod n = (0::nat)"
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  by (simp add: mult_commute mod_mult_self_is_0)
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subsubsection{*Quotient*}
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lemmas DIVISION_BY_ZERO_DIV [simp] = div_by_0 [of "a\<Colon>nat", standard]
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lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
haftmann@25942
   322
  by (simp add: le_div_geq linorder_not_less)
paulson@14267
   323
paulson@14267
   324
lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
wenzelm@22718
   325
  by (simp add: div_geq)
paulson@14267
   326
paulson@14267
   327
paulson@14267
   328
paulson@14267
   329
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   330
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
wenzelm@22718
   331
  by (cut_tac m = m and n = n in mod_div_equality2, arith)
paulson@14267
   332
paulson@14267
   333
lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
wenzelm@22718
   334
  apply (induct m rule: nat_less_induct)
wenzelm@22718
   335
  apply (rename_tac m)
wenzelm@22718
   336
  apply (case_tac "m<n", simp)
wenzelm@22718
   337
  txt{*case @{term "n \<le> m"}*}
wenzelm@22718
   338
  apply (simp add: mod_geq)
wenzelm@22718
   339
  done
nipkow@15439
   340
nipkow@15439
   341
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   342
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   343
  apply simp
wenzelm@22718
   344
  done
paulson@14267
   345
paulson@14267
   346
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
wenzelm@22718
   347
  by (simp add: mult_commute div_mult_self_is_m)
paulson@14267
   348
paulson@14267
   349
(*mod_mult_distrib2 above is the counterpart for remainder*)
paulson@14267
   350
paulson@14267
   351
haftmann@25942
   352
subsubsection {* Proving advancedfacts about Quotient and Remainder *}
haftmann@25942
   353
haftmann@25942
   354
definition
haftmann@25942
   355
  quorem :: "(nat*nat) * (nat*nat) => bool" where
haftmann@25942
   356
  (*This definition helps prove the harder properties of div and mod.
haftmann@25942
   357
    It is copied from IntDiv.thy; should it be overloaded?*)
haftmann@25942
   358
  "quorem = (%((a,b), (q,r)).
haftmann@25942
   359
                    a = b*q + r &
haftmann@25942
   360
                    (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
paulson@14267
   361
paulson@14267
   362
lemma unique_quotient_lemma:
wenzelm@22718
   363
     "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
paulson@14267
   364
      ==> q' \<le> (q::nat)"
wenzelm@22718
   365
  apply (rule leI)
wenzelm@22718
   366
  apply (subst less_iff_Suc_add)
wenzelm@22718
   367
  apply (auto simp add: add_mult_distrib2)
wenzelm@22718
   368
  done
paulson@14267
   369
paulson@14267
   370
lemma unique_quotient:
wenzelm@22718
   371
     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
paulson@14267
   372
      ==> q = q'"
wenzelm@22718
   373
  apply (simp add: split_ifs quorem_def)
wenzelm@22718
   374
  apply (blast intro: order_antisym
wenzelm@22718
   375
    dest: order_eq_refl [THEN unique_quotient_lemma] sym)
wenzelm@22718
   376
  done
paulson@14267
   377
paulson@14267
   378
lemma unique_remainder:
wenzelm@22718
   379
     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
paulson@14267
   380
      ==> r = r'"
wenzelm@22718
   381
  apply (subgoal_tac "q = q'")
wenzelm@22718
   382
   prefer 2 apply (blast intro: unique_quotient)
wenzelm@22718
   383
  apply (simp add: quorem_def)
wenzelm@22718
   384
  done
paulson@14267
   385
nipkow@25162
   386
lemma quorem_div_mod: "b > 0 ==> quorem ((a, b), (a div b, a mod b))"
nipkow@25162
   387
unfolding quorem_def by simp
paulson@14267
   388
nipkow@25162
   389
lemma quorem_div: "[| quorem((a,b),(q,r));  b > 0 |] ==> a div b = q"
nipkow@25162
   390
by (simp add: quorem_div_mod [THEN unique_quotient])
paulson@14267
   391
nipkow@25162
   392
lemma quorem_mod: "[| quorem((a,b),(q,r));  b > 0 |] ==> a mod b = r"
nipkow@25162
   393
by (simp add: quorem_div_mod [THEN unique_remainder])
paulson@14267
   394
paulson@14267
   395
(** A dividend of zero **)
paulson@14267
   396
haftmann@25942
   397
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard]
paulson@14267
   398
haftmann@25942
   399
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard]
paulson@14267
   400
paulson@14267
   401
(** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
paulson@14267
   402
paulson@14267
   403
lemma quorem_mult1_eq:
nipkow@25162
   404
  "[| quorem((b,c),(q,r)); c > 0 |]
nipkow@25162
   405
   ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
nipkow@25162
   406
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
paulson@14267
   407
paulson@14267
   408
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
nipkow@25134
   409
apply (cases "c = 0", simp)
nipkow@25134
   410
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
nipkow@25134
   411
done
paulson@14267
   412
paulson@14267
   413
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
nipkow@25134
   414
apply (cases "c = 0", simp)
nipkow@25134
   415
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
nipkow@25134
   416
done
paulson@14267
   417
paulson@14267
   418
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
wenzelm@22718
   419
  apply (rule trans)
wenzelm@22718
   420
   apply (rule_tac s = "b*a mod c" in trans)
wenzelm@22718
   421
    apply (rule_tac [2] mod_mult1_eq)
wenzelm@22718
   422
   apply (simp_all add: mult_commute)
wenzelm@22718
   423
  done
paulson@14267
   424
nipkow@25162
   425
lemma mod_mult_distrib_mod:
nipkow@25162
   426
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
nipkow@25162
   427
apply (rule mod_mult1_eq' [THEN trans])
nipkow@25162
   428
apply (rule mod_mult1_eq)
nipkow@25162
   429
done
paulson@14267
   430
paulson@14267
   431
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
paulson@14267
   432
paulson@14267
   433
lemma quorem_add1_eq:
nipkow@25162
   434
  "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c > 0 |]
nipkow@25162
   435
   ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
nipkow@25162
   436
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
paulson@14267
   437
paulson@14267
   438
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   439
lemma div_add1_eq:
nipkow@25134
   440
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
nipkow@25134
   441
apply (cases "c = 0", simp)
nipkow@25134
   442
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod)
nipkow@25134
   443
done
paulson@14267
   444
paulson@14267
   445
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
nipkow@25134
   446
apply (cases "c = 0", simp)
nipkow@25134
   447
apply (blast intro: quorem_div_mod quorem_add1_eq [THEN quorem_mod])
nipkow@25134
   448
done
paulson@14267
   449
paulson@14267
   450
haftmann@25942
   451
subsubsection {* Proving @{prop "a div (b*c) = (a div b) div c"} *}
paulson@14267
   452
paulson@14267
   453
(** first, a lemma to bound the remainder **)
paulson@14267
   454
paulson@14267
   455
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   456
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   457
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   458
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   459
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   460
  done
paulson@10559
   461
wenzelm@22718
   462
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
paulson@14267
   463
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
wenzelm@22718
   464
  by (auto simp add: mult_ac quorem_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   465
paulson@14267
   466
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
wenzelm@22718
   467
  apply (cases "b = 0", simp)
wenzelm@22718
   468
  apply (cases "c = 0", simp)
wenzelm@22718
   469
  apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
wenzelm@22718
   470
  done
paulson@14267
   471
paulson@14267
   472
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
wenzelm@22718
   473
  apply (cases "b = 0", simp)
wenzelm@22718
   474
  apply (cases "c = 0", simp)
wenzelm@22718
   475
  apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
wenzelm@22718
   476
  done
paulson@14267
   477
paulson@14267
   478
haftmann@25942
   479
subsubsection{*Cancellation of Common Factors in Division*}
paulson@14267
   480
paulson@14267
   481
lemma div_mult_mult_lemma:
wenzelm@22718
   482
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   483
  by (auto simp add: div_mult2_eq)
paulson@14267
   484
paulson@14267
   485
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   486
  apply (cases "b = 0")
wenzelm@22718
   487
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
wenzelm@22718
   488
  done
paulson@14267
   489
paulson@14267
   490
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
wenzelm@22718
   491
  apply (drule div_mult_mult1)
wenzelm@22718
   492
  apply (auto simp add: mult_commute)
wenzelm@22718
   493
  done
paulson@14267
   494
paulson@14267
   495
haftmann@25942
   496
subsubsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   497
paulson@14267
   498
lemma div_1 [simp]: "m div Suc 0 = m"
wenzelm@22718
   499
  by (induct m) (simp_all add: div_geq)
paulson@14267
   500
haftmann@25942
   501
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard]
paulson@14267
   502
paulson@14267
   503
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
wenzelm@22718
   504
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
wenzelm@22718
   505
   apply (simp add: add_commute)
wenzelm@22718
   506
  apply (subst div_geq [symmetric], simp_all)
wenzelm@22718
   507
  done
paulson@14267
   508
paulson@14267
   509
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
wenzelm@22718
   510
  by (simp add: add_commute div_add_self2)
paulson@14267
   511
paulson@14267
   512
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
wenzelm@22718
   513
  apply (subst div_add1_eq)
wenzelm@22718
   514
  apply (subst div_mult1_eq, simp)
wenzelm@22718
   515
  done
paulson@14267
   516
paulson@14267
   517
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
wenzelm@22718
   518
  by (simp add: mult_commute div_mult_self1)
paulson@14267
   519
paulson@14267
   520
paulson@14267
   521
(* Monotonicity of div in first argument *)
paulson@14267
   522
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   523
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   524
apply (case_tac "k=0", simp)
paulson@15251
   525
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   526
apply (case_tac "n<k")
paulson@14267
   527
(* 1  case n<k *)
paulson@14267
   528
apply simp
paulson@14267
   529
(* 2  case n >= k *)
paulson@14267
   530
apply (case_tac "m<k")
paulson@14267
   531
(* 2.1  case m<k *)
paulson@14267
   532
apply simp
paulson@14267
   533
(* 2.2  case m>=k *)
nipkow@15439
   534
apply (simp add: div_geq diff_le_mono)
paulson@14267
   535
done
paulson@14267
   536
paulson@14267
   537
(* Antimonotonicity of div in second argument *)
paulson@14267
   538
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   539
apply (subgoal_tac "0<n")
wenzelm@22718
   540
 prefer 2 apply simp
paulson@15251
   541
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   542
apply (rename_tac "k")
paulson@14267
   543
apply (case_tac "k<n", simp)
paulson@14267
   544
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   545
 prefer 2 apply simp
paulson@14267
   546
apply (simp add: div_geq)
paulson@15251
   547
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   548
 prefer 2
paulson@14267
   549
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   550
apply (rule le_trans, simp)
nipkow@15439
   551
apply (simp)
paulson@14267
   552
done
paulson@14267
   553
paulson@14267
   554
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   555
apply (case_tac "n=0", simp)
paulson@14267
   556
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   557
apply (rule div_le_mono2)
paulson@14267
   558
apply (simp_all (no_asm_simp))
paulson@14267
   559
done
paulson@14267
   560
wenzelm@22718
   561
(* Similar for "less than" *)
paulson@17085
   562
lemma div_less_dividend [rule_format]:
paulson@14267
   563
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   564
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   565
apply (rename_tac "m")
paulson@14267
   566
apply (case_tac "m<n", simp)
paulson@14267
   567
apply (subgoal_tac "0<n")
wenzelm@22718
   568
 prefer 2 apply simp
paulson@14267
   569
apply (simp add: div_geq)
paulson@14267
   570
apply (case_tac "n<m")
paulson@15251
   571
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   572
  apply (rule impI less_trans_Suc)+
paulson@14267
   573
apply assumption
nipkow@15439
   574
  apply (simp_all)
paulson@14267
   575
done
paulson@14267
   576
paulson@17085
   577
declare div_less_dividend [simp]
paulson@17085
   578
paulson@14267
   579
text{*A fact for the mutilated chess board*}
paulson@14267
   580
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   581
apply (case_tac "n=0", simp)
paulson@15251
   582
apply (induct "m" rule: nat_less_induct)
paulson@14267
   583
apply (case_tac "Suc (na) <n")
paulson@14267
   584
(* case Suc(na) < n *)
paulson@14267
   585
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   586
(* case n \<le> Suc(na) *)
paulson@16796
   587
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   588
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   589
done
paulson@14267
   590
paulson@14437
   591
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
wenzelm@22718
   592
  by (cases "n = 0") auto
paulson@14437
   593
paulson@14437
   594
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
wenzelm@22718
   595
  by (cases "n = 0") auto
paulson@14437
   596
paulson@14267
   597
haftmann@25942
   598
subsubsection{*The Divides Relation*}
paulson@14267
   599
paulson@14267
   600
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
wenzelm@22718
   601
  unfolding dvd_def by blast
paulson@14267
   602
paulson@14267
   603
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
wenzelm@22718
   604
  unfolding dvd_def by blast
nipkow@13152
   605
paulson@14267
   606
lemma dvd_0_right [iff]: "m dvd (0::nat)"
wenzelm@22718
   607
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
paulson@14267
   608
paulson@14267
   609
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
wenzelm@22718
   610
  by (force simp add: dvd_def)
paulson@14267
   611
paulson@14267
   612
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
wenzelm@22718
   613
  by (blast intro: dvd_0_left)
paulson@14267
   614
paulson@24286
   615
declare dvd_0_left_iff [noatp]
paulson@24286
   616
paulson@14267
   617
lemma dvd_1_left [iff]: "Suc 0 dvd k"
wenzelm@22718
   618
  unfolding dvd_def by simp
paulson@14267
   619
paulson@14267
   620
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
wenzelm@22718
   621
  by (simp add: dvd_def)
paulson@14267
   622
haftmann@25942
   623
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard]
haftmann@25942
   624
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard]
paulson@14267
   625
paulson@14267
   626
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
wenzelm@22718
   627
  unfolding dvd_def
wenzelm@22718
   628
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
paulson@14267
   629
haftmann@23684
   630
text {* @{term "op dvd"} is a partial order *}
haftmann@23684
   631
haftmann@25942
   632
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
haftmann@23684
   633
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
haftmann@23684
   634
paulson@14267
   635
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
wenzelm@22718
   636
  unfolding dvd_def
wenzelm@22718
   637
  by (blast intro: add_mult_distrib2 [symmetric])
paulson@14267
   638
paulson@14267
   639
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
wenzelm@22718
   640
  unfolding dvd_def
wenzelm@22718
   641
  by (blast intro: diff_mult_distrib2 [symmetric])
paulson@14267
   642
paulson@14267
   643
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
wenzelm@22718
   644
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   645
  apply (blast intro: dvd_add)
wenzelm@22718
   646
  done
paulson@14267
   647
paulson@14267
   648
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
wenzelm@22718
   649
  by (drule_tac m = m in dvd_diff, auto)
paulson@14267
   650
paulson@14267
   651
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
wenzelm@22718
   652
  unfolding dvd_def by (blast intro: mult_left_commute)
paulson@14267
   653
paulson@14267
   654
lemma dvd_mult2: "k dvd m ==> k dvd (m*n :: nat)"
wenzelm@22718
   655
  apply (subst mult_commute)
wenzelm@22718
   656
  apply (erule dvd_mult)
wenzelm@22718
   657
  done
paulson@14267
   658
paulson@17084
   659
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
wenzelm@22718
   660
  by (rule dvd_refl [THEN dvd_mult])
paulson@17084
   661
paulson@17084
   662
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
wenzelm@22718
   663
  by (rule dvd_refl [THEN dvd_mult2])
paulson@14267
   664
paulson@14267
   665
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
wenzelm@22718
   666
  apply (rule iffI)
wenzelm@22718
   667
   apply (erule_tac [2] dvd_add)
wenzelm@22718
   668
   apply (rule_tac [2] dvd_refl)
wenzelm@22718
   669
  apply (subgoal_tac "n = (n+k) -k")
wenzelm@22718
   670
   prefer 2 apply simp
wenzelm@22718
   671
  apply (erule ssubst)
wenzelm@22718
   672
  apply (erule dvd_diff)
wenzelm@22718
   673
  apply (rule dvd_refl)
wenzelm@22718
   674
  done
paulson@14267
   675
paulson@14267
   676
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
wenzelm@22718
   677
  unfolding dvd_def
wenzelm@22718
   678
  apply (case_tac "n = 0", auto)
wenzelm@22718
   679
  apply (blast intro: mod_mult_distrib2 [symmetric])
wenzelm@22718
   680
  done
paulson@14267
   681
paulson@14267
   682
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
wenzelm@22718
   683
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
wenzelm@22718
   684
   apply (simp add: mod_div_equality)
wenzelm@22718
   685
  apply (simp only: dvd_add dvd_mult)
wenzelm@22718
   686
  done
paulson@14267
   687
paulson@14267
   688
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
wenzelm@22718
   689
  by (blast intro: dvd_mod_imp_dvd dvd_mod)
paulson@14267
   690
paulson@14267
   691
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
wenzelm@22718
   692
  unfolding dvd_def
wenzelm@22718
   693
  apply (erule exE)
wenzelm@22718
   694
  apply (simp add: mult_ac)
wenzelm@22718
   695
  done
paulson@14267
   696
paulson@14267
   697
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
wenzelm@22718
   698
  apply auto
wenzelm@22718
   699
   apply (subgoal_tac "m*n dvd m*1")
wenzelm@22718
   700
   apply (drule dvd_mult_cancel, auto)
wenzelm@22718
   701
  done
paulson@14267
   702
paulson@14267
   703
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
wenzelm@22718
   704
  apply (subst mult_commute)
wenzelm@22718
   705
  apply (erule dvd_mult_cancel1)
wenzelm@22718
   706
  done
paulson@14267
   707
paulson@14267
   708
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
wenzelm@22718
   709
  apply (unfold dvd_def, clarify)
wenzelm@22718
   710
  apply (rule_tac x = "k*ka" in exI)
wenzelm@22718
   711
  apply (simp add: mult_ac)
wenzelm@22718
   712
  done
paulson@14267
   713
paulson@14267
   714
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
wenzelm@22718
   715
  by (simp add: dvd_def mult_assoc, blast)
paulson@14267
   716
paulson@14267
   717
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
wenzelm@22718
   718
  apply (unfold dvd_def, clarify)
wenzelm@22718
   719
  apply (rule_tac x = "i*k" in exI)
wenzelm@22718
   720
  apply (simp add: mult_ac)
wenzelm@22718
   721
  done
paulson@14267
   722
paulson@14267
   723
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
wenzelm@22718
   724
  apply (unfold dvd_def, clarify)
wenzelm@22718
   725
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
wenzelm@22718
   726
  apply (erule conjE)
wenzelm@22718
   727
  apply (rule le_trans)
wenzelm@22718
   728
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
wenzelm@22718
   729
   apply (erule_tac [2] Suc_leI, simp)
wenzelm@22718
   730
  done
paulson@14267
   731
haftmann@25942
   732
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard]
paulson@14267
   733
paulson@14267
   734
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
wenzelm@22718
   735
  apply (subgoal_tac "m mod n = 0")
wenzelm@22718
   736
   apply (simp add: mult_div_cancel)
wenzelm@22718
   737
  apply (simp only: dvd_eq_mod_eq_0)
wenzelm@22718
   738
  done
paulson@14267
   739
haftmann@21408
   740
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
wenzelm@22718
   741
  apply (unfold dvd_def)
wenzelm@22718
   742
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   743
  apply (simp add: power_add)
wenzelm@22718
   744
  done
haftmann@21408
   745
nipkow@25162
   746
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
wenzelm@22718
   747
  by (induct n) auto
haftmann@21408
   748
haftmann@21408
   749
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
wenzelm@22718
   750
  apply (induct j)
wenzelm@22718
   751
   apply (simp_all add: le_Suc_eq)
wenzelm@22718
   752
  apply (blast dest!: dvd_mult_right)
wenzelm@22718
   753
  done
haftmann@21408
   754
haftmann@21408
   755
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
wenzelm@22718
   756
  apply (rule power_le_imp_le_exp, assumption)
wenzelm@22718
   757
  apply (erule dvd_imp_le, simp)
wenzelm@22718
   758
  done
haftmann@21408
   759
paulson@14267
   760
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
wenzelm@22718
   761
  by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   762
wenzelm@22718
   763
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   764
paulson@14267
   765
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   766
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
wenzelm@22718
   767
  apply (cut_tac m = m in mod_div_equality)
wenzelm@22718
   768
  apply (simp only: add_ac)
wenzelm@22718
   769
  apply (blast intro: sym)
wenzelm@22718
   770
  done
paulson@14267
   771
paulson@14131
   772
nipkow@13152
   773
lemma split_div:
nipkow@13189
   774
 "P(n div k :: nat) =
nipkow@13189
   775
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   776
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   777
proof
nipkow@13189
   778
  assume P: ?P
nipkow@13189
   779
  show ?Q
nipkow@13189
   780
  proof (cases)
nipkow@13189
   781
    assume "k = 0"
nipkow@13189
   782
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   783
  next
nipkow@13189
   784
    assume not0: "k \<noteq> 0"
nipkow@13189
   785
    thus ?Q
nipkow@13189
   786
    proof (simp, intro allI impI)
nipkow@13189
   787
      fix i j
nipkow@13189
   788
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   789
      show "P i"
nipkow@13189
   790
      proof (cases)
wenzelm@22718
   791
        assume "i = 0"
wenzelm@22718
   792
        with n j P show "P i" by simp
nipkow@13189
   793
      next
wenzelm@22718
   794
        assume "i \<noteq> 0"
wenzelm@22718
   795
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   796
      qed
nipkow@13189
   797
    qed
nipkow@13189
   798
  qed
nipkow@13189
   799
next
nipkow@13189
   800
  assume Q: ?Q
nipkow@13189
   801
  show ?P
nipkow@13189
   802
  proof (cases)
nipkow@13189
   803
    assume "k = 0"
nipkow@13189
   804
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
nipkow@13189
   805
  next
nipkow@13189
   806
    assume not0: "k \<noteq> 0"
nipkow@13189
   807
    with Q have R: ?R by simp
nipkow@13189
   808
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   809
    show ?P by simp
nipkow@13189
   810
  qed
nipkow@13189
   811
qed
nipkow@13189
   812
berghofe@13882
   813
lemma split_div_lemma:
paulson@14267
   814
  "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
nipkow@25162
   815
apply (rule iffI)
nipkow@25162
   816
 apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
nipkow@25162
   817
   prefer 3; apply assumption
nipkow@25162
   818
  apply (simp_all add: quorem_def)
nipkow@25162
   819
 apply arith
nipkow@25162
   820
apply (rule conjI)
nipkow@25162
   821
 apply (rule_tac P="%x. n * (m div n) \<le> x" in
berghofe@13882
   822
    subst [OF mod_div_equality [of _ n]])
nipkow@25162
   823
 apply (simp only: add: mult_ac)
nipkow@25162
   824
 apply (rule_tac P="%x. x < n + n * (m div n)" in
berghofe@13882
   825
    subst [OF mod_div_equality [of _ n]])
nipkow@25162
   826
apply (simp only: add: mult_ac add_ac)
nipkow@25162
   827
apply (rule add_less_mono1, simp)
nipkow@25162
   828
done
berghofe@13882
   829
berghofe@13882
   830
theorem split_div':
berghofe@13882
   831
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   832
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   833
  apply (case_tac "0 < n")
berghofe@13882
   834
  apply (simp only: add: split_div_lemma)
berghofe@13882
   835
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
berghofe@13882
   836
  done
berghofe@13882
   837
nipkow@13189
   838
lemma split_mod:
nipkow@13189
   839
 "P(n mod k :: nat) =
nipkow@13189
   840
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   841
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   842
proof
nipkow@13189
   843
  assume P: ?P
nipkow@13189
   844
  show ?Q
nipkow@13189
   845
  proof (cases)
nipkow@13189
   846
    assume "k = 0"
nipkow@13189
   847
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   848
  next
nipkow@13189
   849
    assume not0: "k \<noteq> 0"
nipkow@13189
   850
    thus ?Q
nipkow@13189
   851
    proof (simp, intro allI impI)
nipkow@13189
   852
      fix i j
nipkow@13189
   853
      assume "n = k*i + j" "j < k"
nipkow@13189
   854
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   855
    qed
nipkow@13189
   856
  qed
nipkow@13189
   857
next
nipkow@13189
   858
  assume Q: ?Q
nipkow@13189
   859
  show ?P
nipkow@13189
   860
  proof (cases)
nipkow@13189
   861
    assume "k = 0"
nipkow@13189
   862
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
nipkow@13189
   863
  next
nipkow@13189
   864
    assume not0: "k \<noteq> 0"
nipkow@13189
   865
    with Q have R: ?R by simp
nipkow@13189
   866
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   867
    show ?P by simp
nipkow@13189
   868
  qed
nipkow@13189
   869
qed
nipkow@13189
   870
berghofe@13882
   871
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
   872
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
   873
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
   874
  apply arith
berghofe@13882
   875
  done
berghofe@13882
   876
haftmann@22800
   877
lemma div_mod_equality':
haftmann@22800
   878
  fixes m n :: nat
haftmann@22800
   879
  shows "m div n * n = m - m mod n"
haftmann@22800
   880
proof -
haftmann@22800
   881
  have "m mod n \<le> m mod n" ..
haftmann@22800
   882
  from div_mod_equality have 
haftmann@22800
   883
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
haftmann@22800
   884
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
haftmann@22800
   885
    "m div n * n + (m mod n - m mod n) = m - m mod n"
haftmann@22800
   886
    by simp
haftmann@22800
   887
  then show ?thesis by simp
haftmann@22800
   888
qed
haftmann@22800
   889
haftmann@22800
   890
haftmann@25942
   891
subsubsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
   892
paulson@14640
   893
lemma mod_induct_0:
paulson@14640
   894
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
   895
  and base: "P i" and i: "i<p"
paulson@14640
   896
  shows "P 0"
paulson@14640
   897
proof (rule ccontr)
paulson@14640
   898
  assume contra: "\<not>(P 0)"
paulson@14640
   899
  from i have p: "0<p" by simp
paulson@14640
   900
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
   901
  proof
paulson@14640
   902
    fix k
paulson@14640
   903
    show "?A k"
paulson@14640
   904
    proof (induct k)
paulson@14640
   905
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
   906
    next
paulson@14640
   907
      fix n
paulson@14640
   908
      assume ih: "?A n"
paulson@14640
   909
      show "?A (Suc n)"
paulson@14640
   910
      proof (clarsimp)
wenzelm@22718
   911
        assume y: "P (p - Suc n)"
wenzelm@22718
   912
        have n: "Suc n < p"
wenzelm@22718
   913
        proof (rule ccontr)
wenzelm@22718
   914
          assume "\<not>(Suc n < p)"
wenzelm@22718
   915
          hence "p - Suc n = 0"
wenzelm@22718
   916
            by simp
wenzelm@22718
   917
          with y contra show "False"
wenzelm@22718
   918
            by simp
wenzelm@22718
   919
        qed
wenzelm@22718
   920
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
   921
        from p have "p - Suc n < p" by arith
wenzelm@22718
   922
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
   923
          by blast
wenzelm@22718
   924
        show "False"
wenzelm@22718
   925
        proof (cases "n=0")
wenzelm@22718
   926
          case True
wenzelm@22718
   927
          with z n2 contra show ?thesis by simp
wenzelm@22718
   928
        next
wenzelm@22718
   929
          case False
wenzelm@22718
   930
          with p have "p-n < p" by arith
wenzelm@22718
   931
          with z n2 False ih show ?thesis by simp
wenzelm@22718
   932
        qed
paulson@14640
   933
      qed
paulson@14640
   934
    qed
paulson@14640
   935
  qed
paulson@14640
   936
  moreover
paulson@14640
   937
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
   938
    by (blast dest: less_imp_add_positive)
paulson@14640
   939
  hence "0<k \<and> i=p-k" by auto
paulson@14640
   940
  moreover
paulson@14640
   941
  note base
paulson@14640
   942
  ultimately
paulson@14640
   943
  show "False" by blast
paulson@14640
   944
qed
paulson@14640
   945
paulson@14640
   946
lemma mod_induct:
paulson@14640
   947
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
   948
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
   949
  shows "P j"
paulson@14640
   950
proof -
paulson@14640
   951
  have "\<forall>j<p. P j"
paulson@14640
   952
  proof
paulson@14640
   953
    fix j
paulson@14640
   954
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
   955
    proof (induct j)
paulson@14640
   956
      from step base i show "?A 0"
wenzelm@22718
   957
        by (auto elim: mod_induct_0)
paulson@14640
   958
    next
paulson@14640
   959
      fix k
paulson@14640
   960
      assume ih: "?A k"
paulson@14640
   961
      show "?A (Suc k)"
paulson@14640
   962
      proof
wenzelm@22718
   963
        assume suc: "Suc k < p"
wenzelm@22718
   964
        hence k: "k<p" by simp
wenzelm@22718
   965
        with ih have "P k" ..
wenzelm@22718
   966
        with step k have "P (Suc k mod p)"
wenzelm@22718
   967
          by blast
wenzelm@22718
   968
        moreover
wenzelm@22718
   969
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
   970
          by simp
wenzelm@22718
   971
        ultimately
wenzelm@22718
   972
        show "P (Suc k)" by simp
paulson@14640
   973
      qed
paulson@14640
   974
    qed
paulson@14640
   975
  qed
paulson@14640
   976
  with j show ?thesis by blast
paulson@14640
   977
qed
paulson@14640
   978
paulson@14640
   979
chaieb@18202
   980
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
chaieb@18202
   981
  apply (rule trans [symmetric])
wenzelm@22718
   982
   apply (rule mod_add1_eq, simp)
chaieb@18202
   983
  apply (rule mod_add1_eq [symmetric])
chaieb@18202
   984
  done
chaieb@18202
   985
chaieb@18202
   986
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
wenzelm@22718
   987
  apply (rule trans [symmetric])
wenzelm@22718
   988
   apply (rule mod_add1_eq, simp)
wenzelm@22718
   989
  apply (rule mod_add1_eq [symmetric])
wenzelm@22718
   990
  done
chaieb@18202
   991
haftmann@22800
   992
lemma mod_div_decomp:
haftmann@22800
   993
  fixes n k :: nat
haftmann@22800
   994
  obtains m q where "m = n div k" and "q = n mod k"
haftmann@22800
   995
    and "n = m * k + q"
haftmann@22800
   996
proof -
haftmann@22800
   997
  from mod_div_equality have "n = n div k * k + n mod k" by auto
haftmann@22800
   998
  moreover have "n div k = n div k" ..
haftmann@22800
   999
  moreover have "n mod k = n mod k" ..
haftmann@22800
  1000
  note that ultimately show thesis by blast
haftmann@22800
  1001
qed
haftmann@22800
  1002
haftmann@20589
  1003
haftmann@25942
  1004
subsubsection {* Code generation for div, mod and dvd on nat *}
haftmann@20589
  1005
haftmann@22845
  1006
definition [code func del]:
haftmann@20589
  1007
  "divmod (m\<Colon>nat) n = (m div n, m mod n)"
haftmann@20589
  1008
wenzelm@22718
  1009
lemma divmod_zero [code]: "divmod m 0 = (0, m)"
haftmann@20589
  1010
  unfolding divmod_def by simp
haftmann@20589
  1011
haftmann@20589
  1012
lemma divmod_succ [code]:
haftmann@20589
  1013
  "divmod m (Suc k) = (if m < Suc k then (0, m) else
haftmann@20589
  1014
    let
haftmann@20589
  1015
      (p, q) = divmod (m - Suc k) (Suc k)
wenzelm@22718
  1016
    in (Suc p, q))"
haftmann@20589
  1017
  unfolding divmod_def Let_def split_def
haftmann@20589
  1018
  by (auto intro: div_geq mod_geq)
haftmann@20589
  1019
wenzelm@22718
  1020
lemma div_divmod [code]: "m div n = fst (divmod m n)"
haftmann@20589
  1021
  unfolding divmod_def by simp
haftmann@20589
  1022
wenzelm@22718
  1023
lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
haftmann@20589
  1024
  unfolding divmod_def by simp
haftmann@20589
  1025
haftmann@21191
  1026
code_modulename SML
haftmann@23017
  1027
  Divides Nat
haftmann@20640
  1028
haftmann@21911
  1029
code_modulename OCaml
haftmann@23017
  1030
  Divides Nat
haftmann@23017
  1031
haftmann@23017
  1032
code_modulename Haskell
haftmann@23017
  1033
  Divides Nat
haftmann@21911
  1034
haftmann@23684
  1035
hide (open) const divmod
paulson@14267
  1036
paulson@3366
  1037
end