src/HOL/Divides.thy
author haftmann
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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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   320
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diff changeset
   321
lemma div_geq: "[| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)"
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   322
  by (simp add: le_div_geq linorder_not_less)
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diff changeset
   323
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   324
lemma div_if: "0<n ==> m div n = (if m<n then 0 else Suc((m-n) div n))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   325
  by (simp add: div_geq)
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parents: 14208
diff changeset
   326
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parents: 14208
diff changeset
   327
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parents: 14208
diff changeset
   328
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   329
(* a simple rearrangement of mod_div_equality: *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff changeset
   330
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   331
  by (cut_tac m = m and n = n in mod_div_equality2, arith)
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diff changeset
   332
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   333
lemma mod_less_divisor [simp]: "0<n ==> m mod n < (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   334
  apply (induct m rule: nat_less_induct)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   335
  apply (rename_tac m)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   336
  apply (case_tac "m<n", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   337
  txt{*case @{term "n \<le> m"}*}
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   338
  apply (simp add: mod_geq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   339
  done
15439
71c0f98e31f1 made diff_less a simp rule
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parents: 15251
diff changeset
   340
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   341
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   342
  apply (drule mod_less_divisor [where m = m])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   343
  apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   344
  done
14267
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parents: 14208
diff changeset
   345
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   346
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   347
  by (simp add: mult_commute div_mult_self_is_m)
14267
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parents: 14208
diff changeset
   348
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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   349
(*mod_mult_distrib2 above is the counterpart for remainder*)
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diff changeset
   350
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   351
25942
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diff changeset
   352
subsubsection {* Proving advancedfacts about Quotient and Remainder *}
a52309ac4a4d added class semiring_div
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diff changeset
   353
a52309ac4a4d added class semiring_div
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parents: 25571
diff changeset
   354
definition
a52309ac4a4d added class semiring_div
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parents: 25571
diff changeset
   355
  quorem :: "(nat*nat) * (nat*nat) => bool" where
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   356
  (*This definition helps prove the harder properties of div and mod.
a52309ac4a4d added class semiring_div
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parents: 25571
diff changeset
   357
    It is copied from IntDiv.thy; should it be overloaded?*)
a52309ac4a4d added class semiring_div
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parents: 25571
diff changeset
   358
  "quorem = (%((a,b), (q,r)).
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   359
                    a = b*q + r &
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   360
                    (if 0<b then 0\<le>r & r<b else b<r & r \<le>0))"
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diff changeset
   361
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diff changeset
   362
lemma unique_quotient_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   363
     "[| b*q' + r'  \<le> b*q + r;  x < b;  r < b |]
14267
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parents: 14208
diff changeset
   364
      ==> q' \<le> (q::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   365
  apply (rule leI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   366
  apply (subst less_iff_Suc_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   367
  apply (auto simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   368
  done
14267
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parents: 14208
diff changeset
   369
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parents: 14208
diff changeset
   370
lemma unique_quotient:
22718
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wenzelm
parents: 22473
diff changeset
   371
     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
14267
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diff changeset
   372
      ==> q = q'"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   373
  apply (simp add: split_ifs quorem_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   374
  apply (blast intro: order_antisym
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   375
    dest: order_eq_refl [THEN unique_quotient_lemma] sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   376
  done
14267
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parents: 14208
diff changeset
   377
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   378
lemma unique_remainder:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   379
     "[| quorem ((a,b), (q,r));  quorem ((a,b), (q',r'));  0 < b |]
14267
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parents: 14208
diff changeset
   380
      ==> r = r'"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   381
  apply (subgoal_tac "q = q'")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   382
   prefer 2 apply (blast intro: unique_quotient)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   383
  apply (simp add: quorem_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   384
  done
14267
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paulson
parents: 14208
diff changeset
   385
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   386
lemma quorem_div_mod: "b > 0 ==> quorem ((a, b), (a div b, a mod b))"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   387
unfolding quorem_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   388
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   389
lemma quorem_div: "[| quorem((a,b),(q,r));  b > 0 |] ==> a div b = q"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   390
by (simp add: quorem_div_mod [THEN unique_quotient])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   391
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   392
lemma quorem_mod: "[| quorem((a,b),(q,r));  b > 0 |] ==> a mod b = r"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   393
by (simp add: quorem_div_mod [THEN unique_remainder])
14267
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parents: 14208
diff changeset
   394
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents: 14208
diff changeset
   395
(** A dividend of zero **)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   396
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   397
lemmas div_0 [simp] = semiring_div_class.div_0 [of "n\<Colon>nat", standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   398
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   399
lemmas mod_0 [simp] = semiring_div_class.mod_0 [of "n\<Colon>nat", standard]
14267
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parents: 14208
diff changeset
   400
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   401
(** proving (a*b) div c = a * (b div c) + a * (b mod c) **)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   402
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   403
lemma quorem_mult1_eq:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   404
  "[| quorem((b,c),(q,r)); c > 0 |]
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   405
   ==> quorem ((a*b, c), (a*q + a*r div c, a*r mod c))"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   406
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
14267
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paulson
parents: 14208
diff changeset
   407
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   408
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
   409
apply (cases "c = 0", simp)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   410
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_div])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   411
done
14267
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paulson
parents: 14208
diff changeset
   412
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   413
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
   414
apply (cases "c = 0", simp)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   415
apply (blast intro: quorem_div_mod [THEN quorem_mult1_eq, THEN quorem_mod])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   416
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   417
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   418
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   419
  apply (rule trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   420
   apply (rule_tac s = "b*a mod c" in trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   421
    apply (rule_tac [2] mod_mult1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   422
   apply (simp_all add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   423
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   424
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   425
lemma mod_mult_distrib_mod:
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   426
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   427
apply (rule mod_mult1_eq' [THEN trans])
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   428
apply (rule mod_mult1_eq)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   429
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   430
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   431
(** proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) **)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   432
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   433
lemma quorem_add1_eq:
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   434
  "[| quorem((a,c),(aq,ar));  quorem((b,c),(bq,br));  c > 0 |]
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   435
   ==> quorem ((a+b, c), (aq + bq + (ar+br) div c, (ar+br) mod c))"
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   436
by (auto simp add: split_ifs mult_ac quorem_def add_mult_distrib2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   437
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   438
(*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
   439
lemma div_add1_eq:
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   440
  "(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
   441
apply (cases "c = 0", simp)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   442
apply (blast intro: quorem_add1_eq [THEN quorem_div] quorem_div_mod)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   443
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   444
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   445
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
   446
apply (cases "c = 0", simp)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   447
apply (blast intro: quorem_div_mod quorem_add1_eq [THEN quorem_mod])
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   448
done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   449
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   450
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   451
subsubsection {* Proving @{prop "a div (b*c) = (a div b) div c"} *}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   452
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   453
(** first, a lemma to bound the remainder **)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   454
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   455
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   456
  apply (cut_tac m = q and n = c in mod_less_divisor)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   457
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   458
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   459
  apply (simp add: add_mult_distrib2)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   460
  done
10559
d3fd54fc659b many new div and mod properties (borrowed from Integ/IntDiv)
paulson
parents: 10214
diff changeset
   461
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   462
lemma quorem_mult2_eq: "[| quorem ((a,b), (q,r));  0 < b;  0 < c |]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   463
      ==> quorem ((a, b*c), (q div c, b*(q mod c) + r))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   464
  by (auto simp add: mult_ac quorem_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
   465
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   466
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   467
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   468
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   469
  apply (force simp add: quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_div])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   470
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   471
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   472
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
   473
  apply (cases "b = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   474
  apply (cases "c = 0", simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   475
  apply (auto simp add: mult_commute quorem_div_mod [THEN quorem_mult2_eq, THEN quorem_mod])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   476
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   477
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   478
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   479
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
   480
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   481
lemma div_mult_mult_lemma:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   482
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   483
  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
   484
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   485
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
   486
  apply (cases "b = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   487
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   488
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   489
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   490
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
   491
  apply (drule div_mult_mult1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   492
  apply (auto simp add: mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   493
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   494
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   495
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   496
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
   497
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   498
lemma div_1 [simp]: "m div Suc 0 = m"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   499
  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
   500
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   501
lemmas div_self [simp] = semiring_div_class.div_self [of "n\<Colon>nat", standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   502
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   503
lemma div_add_self2: "0<n ==> (m+n) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   504
  apply (subgoal_tac "(n + m) div n = Suc ((n+m-n) div n) ")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   505
   apply (simp add: add_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   506
  apply (subst div_geq [symmetric], simp_all)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   507
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   508
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   509
lemma div_add_self1: "0<n ==> (n+m) div n = Suc (m div n)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   510
  by (simp add: add_commute div_add_self2)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   511
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   512
lemma div_mult_self1 [simp]: "!!n::nat. 0<n ==> (m + k*n) div n = k + m div n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   513
  apply (subst div_add1_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   514
  apply (subst div_mult1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   515
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   516
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   517
lemma div_mult_self2 [simp]: "0<n ==> (m + n*k) div n = k + m div (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   518
  by (simp add: mult_commute div_mult_self1)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   519
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
(* Monotonicity of div in first argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   522
lemma div_le_mono [rule_format (no_asm)]:
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   523
    "\<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
   524
apply (case_tac "k=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   525
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
   526
apply (case_tac "n<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   527
(* 1  case n<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   528
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   529
(* 2  case n >= k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   530
apply (case_tac "m<k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   531
(* 2.1  case m<k *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   532
apply simp
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   533
(* 2.2  case m>=k *)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   534
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
   535
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   536
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   537
(* Antimonotonicity of div in second argument *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   538
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
   539
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   540
 prefer 2 apply simp
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   541
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
   542
apply (rename_tac "k")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   543
apply (case_tac "k<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   544
apply (subgoal_tac "~ (k<m) ")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   545
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   546
apply (simp add: div_geq)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   547
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
   548
 prefer 2
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   549
 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
   550
apply (rule le_trans, simp)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   551
apply (simp)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   552
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   553
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   554
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
   555
apply (case_tac "n=0", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   556
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
   557
apply (rule div_le_mono2)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   558
apply (simp_all (no_asm_simp))
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   559
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   560
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   561
(* Similar for "less than" *)
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   562
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
   563
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   564
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
   565
apply (rename_tac "m")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   566
apply (case_tac "m<n", simp)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   567
apply (subgoal_tac "0<n")
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   568
 prefer 2 apply simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   569
apply (simp add: div_geq)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   570
apply (case_tac "n<m")
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   571
 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
   572
  apply (rule impI less_trans_Suc)+
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   573
apply assumption
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   574
  apply (simp_all)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   575
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   576
17085
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   577
declare div_less_dividend [simp]
5b57f995a179 more simprules now have names
paulson
parents: 17084
diff changeset
   578
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   579
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
   580
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
   581
apply (case_tac "n=0", simp)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   582
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
   583
apply (case_tac "Suc (na) <n")
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   584
(* case Suc(na) < n *)
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   585
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
   586
(* case n \<le> Suc(na) *)
16796
140f1e0ea846 generlization of some "nat" theorems
paulson
parents: 16733
diff changeset
   587
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
15439
71c0f98e31f1 made diff_less a simp rule
nipkow
parents: 15251
diff changeset
   588
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
   589
done
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   590
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   591
lemma nat_mod_div_trivial [simp]: "m mod n div n = (0 :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   592
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   593
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   594
lemma nat_mod_mod_trivial [simp]: "m mod n mod n = (m mod n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   595
  by (cases "n = 0") auto
14437
92f6aa05b7bb some new results
paulson
parents: 14430
diff changeset
   596
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   597
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   598
subsubsection{*The Divides Relation*}
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   599
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   600
lemma dvdI [intro?]: "n = m * k ==> m dvd n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   601
  unfolding dvd_def by blast
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   602
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   603
lemma dvdE [elim?]: "!!P. [|m dvd n;  !!k. n = m*k ==> P|] ==> P"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   604
  unfolding dvd_def by blast
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   605
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   606
lemma dvd_0_right [iff]: "m dvd (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   607
  unfolding dvd_def by (blast intro: mult_0_right [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   608
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   609
lemma dvd_0_left: "0 dvd m ==> m = (0::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   610
  by (force simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   611
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   612
lemma dvd_0_left_iff [iff]: "(0 dvd (m::nat)) = (m = 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   613
  by (blast intro: dvd_0_left)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   614
24286
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   615
declare dvd_0_left_iff [noatp]
7619080e49f0 ATP blacklisting is now in theory data, attribute noatp
paulson
parents: 24268
diff changeset
   616
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   617
lemma dvd_1_left [iff]: "Suc 0 dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   618
  unfolding dvd_def by simp
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   619
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   620
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   621
  by (simp add: dvd_def)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   622
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   623
lemmas dvd_refl [simp] = semiring_div_class.dvd_refl [of "m\<Colon>nat", standard]
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   624
lemmas dvd_trans [trans] = semiring_div_class.dvd_trans [of "m\<Colon>nat" n p, standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   625
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   626
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   627
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   628
  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
   629
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   630
text {* @{term "op dvd"} is a partial order *}
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   631
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   632
interpretation dvd: order ["op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> n \<noteq> m"]
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   633
  by unfold_locales (auto intro: dvd_trans dvd_anti_sym)
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
   634
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   635
lemma dvd_add: "[| k dvd m; k dvd n |] ==> k dvd (m+n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   636
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   637
  by (blast intro: add_mult_distrib2 [symmetric])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   638
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   639
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   640
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   641
  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
   642
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   643
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
   644
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   645
  apply (blast intro: dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   646
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   647
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   648
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
   649
  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
   650
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   651
lemma dvd_mult: "k dvd n ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   652
  unfolding dvd_def by (blast intro: mult_left_commute)
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_mult2: "k dvd m ==> k dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   655
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   656
  apply (erule dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   657
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   658
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   659
lemma dvd_triv_right [iff]: "k dvd (m*k :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   660
  by (rule dvd_refl [THEN dvd_mult])
17084
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   661
fb0a80aef0be classical rules must have names for ATP integration
paulson
parents: 16796
diff changeset
   662
lemma dvd_triv_left [iff]: "k dvd (k*m :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   663
  by (rule dvd_refl [THEN dvd_mult2])
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   664
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   665
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   666
  apply (rule iffI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   667
   apply (erule_tac [2] dvd_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   668
   apply (rule_tac [2] dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   669
  apply (subgoal_tac "n = (n+k) -k")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   670
   prefer 2 apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   671
  apply (erule ssubst)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   672
  apply (erule dvd_diff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   673
  apply (rule dvd_refl)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   674
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   675
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   676
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
   677
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   678
  apply (case_tac "n = 0", auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   679
  apply (blast intro: mod_mult_distrib2 [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   680
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   681
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   682
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
   683
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   684
   apply (simp add: mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   685
  apply (simp only: dvd_add dvd_mult)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   686
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   687
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   688
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
   689
  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
   690
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   691
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
   692
  unfolding dvd_def
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   693
  apply (erule exE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   694
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   695
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   696
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   697
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   698
  apply auto
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   699
   apply (subgoal_tac "m*n dvd m*1")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   700
   apply (drule dvd_mult_cancel, auto)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   701
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   702
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   703
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   704
  apply (subst mult_commute)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   705
  apply (erule dvd_mult_cancel1)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   706
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   707
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   708
lemma mult_dvd_mono: "[| i dvd m; j dvd n|] ==> i*j dvd (m*n :: nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   709
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   710
  apply (rule_tac x = "k*ka" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   711
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   712
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   713
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   714
lemma dvd_mult_left: "(i*j :: nat) dvd k ==> i dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   715
  by (simp add: dvd_def mult_assoc, blast)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   716
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   717
lemma dvd_mult_right: "(i*j :: nat) dvd k ==> j dvd k"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   718
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   719
  apply (rule_tac x = "i*k" in exI)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   720
  apply (simp add: mult_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   721
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   722
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   723
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   724
  apply (unfold dvd_def, clarify)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   725
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   726
  apply (erule conjE)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   727
  apply (rule le_trans)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   728
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   729
   apply (erule_tac [2] Suc_leI, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   730
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   731
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   732
lemmas dvd_eq_mod_eq_0 = dvd_def_mod [of "k\<Colon>nat" n, standard]
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   733
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   734
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   735
  apply (subgoal_tac "m mod n = 0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   736
   apply (simp add: mult_div_cancel)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   737
  apply (simp only: dvd_eq_mod_eq_0)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   738
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   739
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   740
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   741
  apply (unfold dvd_def)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   742
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   743
  apply (simp add: power_add)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   744
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   745
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   746
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   747
  by (induct n) auto
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   748
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   749
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
   750
  apply (induct j)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   751
   apply (simp_all add: le_Suc_eq)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   752
  apply (blast dest!: dvd_mult_right)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   753
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   754
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   755
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
   756
  apply (rule power_le_imp_le_exp, assumption)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   757
  apply (erule dvd_imp_le, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   758
  done
21408
fff1731da03b div is now a class
haftmann
parents: 21191
diff changeset
   759
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   760
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   761
  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
   762
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   763
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
   764
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   765
(*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
   766
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   767
  apply (cut_tac m = m in mod_div_equality)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   768
  apply (simp only: add_ac)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   769
  apply (blast intro: sym)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   770
  done
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   771
14131
a4fc8b1af5e7 declarations moved from PreList.thy
paulson
parents: 13882
diff changeset
   772
13152
2a54f99b44b3 Divides.ML -> Divides_lemmas.ML
nipkow
parents: 12338
diff changeset
   773
lemma split_div:
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   774
 "P(n div k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   775
 ((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
   776
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   777
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   778
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   779
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   780
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   781
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   782
    with P show ?Q by(simp add:DIVISION_BY_ZERO_DIV)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   783
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   784
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   785
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   786
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   787
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   788
      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
   789
      show "P i"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   790
      proof (cases)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   791
        assume "i = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   792
        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
   793
      next
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   794
        assume "i \<noteq> 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   795
        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
   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
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   799
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   800
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   801
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   802
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   803
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   804
    with Q show ?P by(simp add:DIVISION_BY_ZERO_DIV)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   805
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   806
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   807
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   808
    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
   809
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   810
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   811
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   812
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   813
lemma split_div_lemma:
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
   814
  "0 < n \<Longrightarrow> (n * q \<le> m \<and> m < n * (Suc q)) = (q = ((m::nat) div n))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   815
apply (rule iffI)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   816
 apply (rule_tac a=m and r = "m - n * q" and r' = "m mod n" in unique_quotient)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   817
   prefer 3; apply assumption
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   818
  apply (simp_all add: quorem_def)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   819
 apply arith
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   820
apply (rule conjI)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   821
 apply (rule_tac P="%x. n * (m div n) \<le> x" in
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   822
    subst [OF mod_div_equality [of _ n]])
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   823
 apply (simp only: add: mult_ac)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   824
 apply (rule_tac P="%x. x < n + n * (m div n)" in
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   825
    subst [OF mod_div_equality [of _ n]])
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   826
apply (simp only: add: mult_ac add_ac)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   827
apply (rule add_less_mono1, simp)
ad4d5365d9d8 went back to >0
nipkow
parents: 25134
diff changeset
   828
done
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   829
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   830
theorem split_div':
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   831
  "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
   832
   (\<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
   833
  apply (case_tac "0 < n")
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   834
  apply (simp only: add: split_div_lemma)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   835
  apply (simp_all add: DIVISION_BY_ZERO_DIV)
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   836
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   837
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   838
lemma split_mod:
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   839
 "P(n mod k :: nat) =
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   840
 ((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
   841
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   842
proof
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   843
  assume P: ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   844
  show ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   845
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   846
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   847
    with P show ?Q by(simp add:DIVISION_BY_ZERO_MOD)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   848
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   849
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   850
    thus ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   851
    proof (simp, intro allI impI)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   852
      fix i j
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   853
      assume "n = k*i + j" "j < k"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   854
      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
   855
    qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   856
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   857
next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   858
  assume Q: ?Q
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   859
  show ?P
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   860
  proof (cases)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   861
    assume "k = 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   862
    with Q show ?P by(simp add:DIVISION_BY_ZERO_MOD)
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   863
  next
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   864
    assume not0: "k \<noteq> 0"
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   865
    with Q have R: ?R by simp
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   866
    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
   867
    show ?P by simp
13189
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   868
  qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   869
qed
81ed5c6de890 Now arith can deal with div/mod arbitrary nat numerals.
nipkow
parents: 13152
diff changeset
   870
13882
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   871
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
   872
  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
   873
    subst [OF mod_div_equality [of _ n]])
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   874
  apply arith
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   875
  done
2266550ab316 New theorems split_div' and mod_div_equality'.
berghofe
parents: 13517
diff changeset
   876
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   877
lemma div_mod_equality':
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   878
  fixes m n :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   879
  shows "m div n * n = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   880
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   881
  have "m mod n \<le> m mod n" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   882
  from div_mod_equality have 
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   883
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   884
  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
   885
    "m div n * n + (m mod n - m mod n) = m - m mod n"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   886
    by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   887
  then show ?thesis by simp
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   888
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   889
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   890
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
   891
subsubsection {*An ``induction'' law for modulus arithmetic.*}
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   892
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   893
lemma mod_induct_0:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   894
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   895
  and base: "P i" and i: "i<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   896
  shows "P 0"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   897
proof (rule ccontr)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   898
  assume contra: "\<not>(P 0)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   899
  from i have p: "0<p" by simp
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   900
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   901
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   902
    fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   903
    show "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   904
    proof (induct k)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   905
      show "?A 0" by simp  -- "by contradiction"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   906
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   907
      fix n
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   908
      assume ih: "?A n"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   909
      show "?A (Suc n)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   910
      proof (clarsimp)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   911
        assume y: "P (p - Suc n)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   912
        have n: "Suc n < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   913
        proof (rule ccontr)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   914
          assume "\<not>(Suc n < p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   915
          hence "p - Suc n = 0"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   916
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   917
          with y contra show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   918
            by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   919
        qed
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   920
        hence n2: "Suc (p - Suc n) = p-n" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   921
        from p have "p - Suc n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   922
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   923
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   924
        show "False"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   925
        proof (cases "n=0")
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   926
          case True
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   927
          with z n2 contra show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   928
        next
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   929
          case False
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   930
          with p have "p-n < p" by arith
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   931
          with z n2 False ih show ?thesis by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   932
        qed
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   933
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   934
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   935
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   936
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   937
  from i obtain k where "0<k \<and> i+k=p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   938
    by (blast dest: less_imp_add_positive)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   939
  hence "0<k \<and> i=p-k" by auto
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   940
  moreover
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   941
  note base
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   942
  ultimately
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   943
  show "False" by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   944
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   945
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   946
lemma mod_induct:
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   947
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   948
  and base: "P i" and i: "i<p" and j: "j<p"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   949
  shows "P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   950
proof -
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   951
  have "\<forall>j<p. P j"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   952
  proof
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   953
    fix j
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   954
    show "j<p \<longrightarrow> P j" (is "?A j")
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   955
    proof (induct j)
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   956
      from step base i show "?A 0"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   957
        by (auto elim: mod_induct_0)
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   958
    next
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   959
      fix k
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   960
      assume ih: "?A k"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   961
      show "?A (Suc k)"
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   962
      proof
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   963
        assume suc: "Suc k < p"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   964
        hence k: "k<p" by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   965
        with ih have "P k" ..
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   966
        with step k have "P (Suc k mod p)"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   967
          by blast
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   968
        moreover
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   969
        from suc have "Suc k mod p = Suc k"
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   970
          by simp
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   971
        ultimately
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   972
        show "P (Suc k)" by simp
14640
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   973
      qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   974
    qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   975
  qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   976
  with j show ?thesis by blast
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   977
qed
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   978
b31870c50c68 new lemmas
paulson
parents: 14437
diff changeset
   979
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   980
lemma mod_add_left_eq: "((a::nat) + b) mod c = (a mod c + b) mod c"
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   981
  apply (rule trans [symmetric])
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   982
   apply (rule mod_add1_eq, simp)
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   983
  apply (rule mod_add1_eq [symmetric])
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   984
  done
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   985
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   986
lemma mod_add_right_eq: "(a+b) mod (c::nat) = (a + (b mod c)) mod c"
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   987
  apply (rule trans [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   988
   apply (rule mod_add1_eq, simp)
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   989
  apply (rule mod_add1_eq [symmetric])
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
   990
  done
18202
46af82efd311 presburger method updated to deal better with mod and div, tweo lemmas added to Divides.thy
chaieb
parents: 18154
diff changeset
   991
22800
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   992
lemma mod_div_decomp:
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   993
  fixes n k :: nat
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   994
  obtains m q where "m = n div k" and "q = n mod k"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   995
    and "n = m * k + q"
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   996
proof -
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   997
  from mod_div_equality have "n = n div k * k + n mod k" by auto
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   998
  moreover have "n div k = n div k" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
   999
  moreover have "n mod k = n mod k" ..
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1000
  note that ultimately show thesis by blast
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1001
qed
eaf5e7ef35d9 added lemmatas
haftmann
parents: 22744
diff changeset
  1002
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1003
25942
a52309ac4a4d added class semiring_div
haftmann
parents: 25571
diff changeset
  1004
subsubsection {* Code generation for div, mod and dvd on nat *}
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1005
22845
5f9138bcb3d7 changed code generator invocation syntax
haftmann
parents: 22800
diff changeset
  1006
definition [code func del]:
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1007
  "divmod (m\<Colon>nat) n = (m div n, m mod n)"
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1008
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1009
lemma divmod_zero [code]: "divmod m 0 = (0, m)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1010
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1011
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1012
lemma divmod_succ [code]:
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1013
  "divmod m (Suc k) = (if m < Suc k then (0, m) else
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1014
    let
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1015
      (p, q) = divmod (m - Suc k) (Suc k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1016
    in (Suc p, q))"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1017
  unfolding divmod_def Let_def split_def
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1018
  by (auto intro: div_geq mod_geq)
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1019
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1020
lemma div_divmod [code]: "m div n = fst (divmod m n)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1021
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1022
22718
936f7580937d tuned proofs;
wenzelm
parents: 22473
diff changeset
  1023
lemma mod_divmod [code]: "m mod n = snd (divmod m n)"
20589
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1024
  unfolding divmod_def by simp
24ecf9bc1a0a explicit divmod algorithm for code generation
haftmann
parents: 20432
diff changeset
  1025
21191
c00161fbf990 code generator module naming improved
haftmann
parents: 20640
diff changeset
  1026
code_modulename SML
23017
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
  1027
  Divides Nat
20640
05e6042394b9 name shifts
haftmann
parents: 20589
diff changeset
  1028
21911
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
  1029
code_modulename OCaml
23017
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
  1030
  Divides Nat
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
  1031
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
  1032
code_modulename Haskell
00c0e4c42396 uniform module names for code generation
haftmann
parents: 22993
diff changeset
  1033
  Divides Nat
21911
e29bcab0c81c added OCaml code generation (without dictionaries)
haftmann
parents: 21408
diff changeset
  1034
23684
8c508c4dc53b introduced (auxiliary) class dvd_mod for more convenient code generation
haftmann
parents: 23162
diff changeset
  1035
hide (open) const divmod
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14208
diff changeset
  1036
3366
2402c6ab1561 Moving div and mod from Arith to Divides
paulson
parents:
diff changeset
  1037
end