src/HOL/Divides.thy
author haftmann
Mon Oct 20 07:45:58 2014 +0200 (2014-10-20)
changeset 58710 7216a10d69ba
parent 58646 cd63a4b12a33
child 58778 e29cae8eab1f
permissions -rw-r--r--
augmented and tuned facts on even/odd and division
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(*  Title:      HOL/Divides.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div and mod *}
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theory Divides
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imports Nat_Transfer
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begin
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subsection {* Syntactic division operations *}
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class div = dvd +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract division in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0 [simp]: "a div 0 = 0"
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    and div_0 [simp]: "0 div a = 0"
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    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
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begin
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text {* @{const div} and @{const mod} *}
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult.commute [of b]
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  by (rule mod_div_equality)
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lemma mod_div_equality': "a mod b + a div b * b = a"
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  using mod_div_equality [of a b]
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  by (simp only: ac_simps)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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lemma mod_by_0 [simp]: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mod_0 [simp]: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp
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lemma div_mult_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b * c) div b = c + a div b"
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  using assms div_mult_self1 [of b a c] by (simp add: mult.commute)
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lemma div_mult_self3 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(c * b + a) div b = c + a div b"
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  using assms by (simp add: add.commute)
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lemma div_mult_self4 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b * c + a) div b = c + a div b"
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  using assms by (simp add: add.commute)
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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proof (cases "b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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    by (simp add: mod_div_equality)
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  also from False div_mult_self1 [of b a c] have
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    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
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      by (simp add: algebra_simps)
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  finally have "a = a div b * b + (a + c * b) mod b"
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    by (simp add: add.commute [of a] add.assoc distrib_right)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: 
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  "(a + b * c) mod b = a mod b"
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  by (simp add: mult.commute [of b])
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lemma mod_mult_self3 [simp]:
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  "(c * b + a) mod b = a mod b"
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  by (simp add: add.commute)
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lemma mod_mult_self4 [simp]:
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  "(b * c + a) mod b = a mod b"
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  by (simp add: add.commute)
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  using div_mult_self2 [of b 0 a] by simp
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  using div_mult_self1 [of b 0 a] by simp
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
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  using mod_mult_self2 [of 0 b a] by simp
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma div_by_1 [simp]: "a div 1 = a"
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  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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lemma mod_by_1 [simp]: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self [simp]: "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using div_mult_self2_is_id [of _ 1] by simp
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lemma div_add_self1 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b + a) div b = a div b + 1"
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  using assms div_mult_self1 [of b a 1] by (simp add: add.commute)
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lemma div_add_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b) div b = a div b + 1"
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  using assms div_add_self1 [of b a] by (simp add: add.commute)
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lemma mod_add_self1 [simp]:
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  "(b + a) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by (simp add: add.commute)
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lemma mod_add_self2 [simp]:
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  "(a + b) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by simp
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lemma mod_div_decomp:
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  fixes a b
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  obtains q r where "q = a div b" and "r = a mod b"
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    and "a = q * b + r"
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proof -
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  from mod_div_equality have "a = a div b * b + a mod b" by simp
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  moreover have "a div b = a div b" ..
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  moreover have "a mod b = a mod b" ..
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  note that ultimately show thesis by blast
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qed
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lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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  with mod_div_equality [of b a] have "b div a * a = b" by simp
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  then have "b = a * (b div a)" unfolding mult.commute ..
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  then have "\<exists>c. b = a * c" ..
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  then show "a dvd b" unfolding dvd_def .
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next
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  assume "a dvd b"
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  then have "\<exists>c. b = a * c" unfolding dvd_def .
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  then obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then have "b mod a = c * a mod a" by (simp add: mult.commute)
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  then show "b mod a = 0" by simp
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qed
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lemma mod_div_trivial [simp]: "a mod b div b = 0"
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proof (cases "b = 0")
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  assume "b = 0"
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  thus ?thesis by simp
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next
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  assume "b \<noteq> 0"
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  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
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    by (rule div_mult_self1 [symmetric])
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  also have "\<dots> = a div b"
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    by (simp only: mod_div_equality')
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  also have "\<dots> = a div b + 0"
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    by simp
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  finally show ?thesis
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    by (rule add_left_imp_eq)
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qed
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lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
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proof -
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  have "a mod b mod b = (a mod b + a div b * b) mod b"
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    by (simp only: mod_mult_self1)
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  also have "\<dots> = a mod b"
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    by (simp only: mod_div_equality')
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  finally show ?thesis .
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qed
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lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
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by (rule dvd_eq_mod_eq_0[THEN iffD1])
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lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
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by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
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lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
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by (drule dvd_div_mult_self) (simp add: mult.commute)
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lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
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apply (cases "a = 0")
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 apply simp
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apply (auto simp: dvd_def mult.assoc)
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done
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lemma div_dvd_div[simp]:
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  "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
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apply (cases "a = 0")
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 apply simp
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apply (unfold dvd_def)
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apply auto
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 apply(blast intro:mult.assoc[symmetric])
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apply(fastforce simp add: mult.assoc)
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done
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lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
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  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
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   apply (simp add: mod_div_equality)
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  apply (simp only: dvd_add dvd_mult)
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  done
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text {* Addition respects modular equivalence. *}
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lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
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proof -
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  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a mod c + b + a div c * c) mod c"
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    by (simp only: ac_simps)
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  also have "\<dots> = (a mod c + b) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
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proof -
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  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a + b mod c + b div c * c) mod c"
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    by (simp only: ac_simps)
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  also have "\<dots> = (a + b mod c) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
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by (rule trans [OF mod_add_left_eq mod_add_right_eq])
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lemma mod_add_cong:
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  assumes "a mod c = a' mod c"
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  assumes "b mod c = b' mod c"
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  shows "(a + b) mod c = (a' + b') mod c"
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proof -
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  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
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    unfolding assms ..
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  thus ?thesis
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    by (simp only: mod_add_eq [symmetric])
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qed
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lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
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  \<Longrightarrow> (x + y) div z = x div z + y div z"
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by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
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text {* Multiplication respects modular equivalence. *}
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lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
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proof -
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  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
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    by (simp only: algebra_simps)
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  also have "\<dots> = (a mod c * b) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
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proof -
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  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
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    by (simp only: algebra_simps)
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  also have "\<dots> = (a * (b mod c)) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
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by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
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lemma mod_mult_cong:
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  assumes "a mod c = a' mod c"
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  assumes "b mod c = b' mod c"
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  shows "(a * b) mod c = (a' * b') mod c"
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proof -
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  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
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    unfolding assms ..
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  thus ?thesis
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    by (simp only: mod_mult_eq [symmetric])
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qed
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text {* Exponentiation respects modular equivalence. *}
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lemma power_mod: "(a mod b)^n mod b = a^n mod b"
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apply (induct n, simp_all)
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apply (rule mod_mult_right_eq [THEN trans])
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apply (simp (no_asm_simp))
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apply (rule mod_mult_eq [symmetric])
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done
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lemma mod_mod_cancel:
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  assumes "c dvd b"
huffman@29404
   315
  shows "a mod b mod c = a mod c"
huffman@29404
   316
proof -
huffman@29404
   317
  from `c dvd b` obtain k where "b = c * k"
huffman@29404
   318
    by (rule dvdE)
huffman@29404
   319
  have "a mod b mod c = a mod (c * k) mod c"
huffman@29404
   320
    by (simp only: `b = c * k`)
huffman@29404
   321
  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
huffman@29404
   322
    by (simp only: mod_mult_self1)
huffman@29404
   323
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
haftmann@57514
   324
    by (simp only: ac_simps ac_simps)
huffman@29404
   325
  also have "\<dots> = a mod c"
huffman@29404
   326
    by (simp only: mod_div_equality)
huffman@29404
   327
  finally show ?thesis .
huffman@29404
   328
qed
huffman@29404
   329
haftmann@30930
   330
lemma div_mult_div_if_dvd:
haftmann@30930
   331
  "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
haftmann@30930
   332
  apply (cases "y = 0", simp)
haftmann@30930
   333
  apply (cases "z = 0", simp)
haftmann@30930
   334
  apply (auto elim!: dvdE simp add: algebra_simps)
haftmann@57512
   335
  apply (subst mult.assoc [symmetric])
nipkow@30476
   336
  apply (simp add: no_zero_divisors)
haftmann@30930
   337
  done
haftmann@30930
   338
haftmann@35367
   339
lemma div_mult_swap:
haftmann@35367
   340
  assumes "c dvd b"
haftmann@35367
   341
  shows "a * (b div c) = (a * b) div c"
haftmann@35367
   342
proof -
haftmann@35367
   343
  from assms have "b div c * (a div 1) = b * a div (c * 1)"
haftmann@35367
   344
    by (simp only: div_mult_div_if_dvd one_dvd)
haftmann@57512
   345
  then show ?thesis by (simp add: mult.commute)
haftmann@35367
   346
qed
haftmann@35367
   347
   
haftmann@30930
   348
lemma div_mult_mult2 [simp]:
haftmann@30930
   349
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@57512
   350
  by (drule div_mult_mult1) (simp add: mult.commute)
haftmann@30930
   351
haftmann@30930
   352
lemma div_mult_mult1_if [simp]:
haftmann@30930
   353
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   354
  by simp_all
nipkow@30476
   355
haftmann@30930
   356
lemma mod_mult_mult1:
haftmann@30930
   357
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   358
proof (cases "c = 0")
haftmann@30930
   359
  case True then show ?thesis by simp
haftmann@30930
   360
next
haftmann@30930
   361
  case False
haftmann@30930
   362
  from mod_div_equality
haftmann@30930
   363
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   364
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   365
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@30930
   366
  with mod_div_equality show ?thesis by simp 
haftmann@30930
   367
qed
haftmann@30930
   368
  
haftmann@30930
   369
lemma mod_mult_mult2:
haftmann@30930
   370
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@57512
   371
  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
haftmann@30930
   372
huffman@47159
   373
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159
   374
  by (fact mod_mult_mult2 [symmetric])
huffman@47159
   375
huffman@47159
   376
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159
   377
  by (fact mod_mult_mult1 [symmetric])
huffman@47159
   378
huffman@31662
   379
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   380
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   381
huffman@31662
   382
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   383
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   384
haftmann@31009
   385
lemma div_power:
huffman@31661
   386
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
nipkow@30476
   387
apply (induct n)
nipkow@30476
   388
 apply simp
nipkow@30476
   389
apply(simp add: div_mult_div_if_dvd dvd_power_same)
nipkow@30476
   390
done
nipkow@30476
   391
haftmann@35367
   392
lemma dvd_div_eq_mult:
haftmann@35367
   393
  assumes "a \<noteq> 0" and "a dvd b"  
haftmann@35367
   394
  shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@35367
   395
proof
haftmann@35367
   396
  assume "b = c * a"
haftmann@35367
   397
  then show "b div a = c" by (simp add: assms)
haftmann@35367
   398
next
haftmann@35367
   399
  assume "b div a = c"
haftmann@35367
   400
  then have "b div a * a = c * a" by simp
haftmann@35367
   401
  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
haftmann@35367
   402
  ultimately show "b = c * a" by simp
haftmann@35367
   403
qed
haftmann@35367
   404
   
haftmann@35367
   405
lemma dvd_div_div_eq_mult:
haftmann@35367
   406
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@35367
   407
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
haftmann@57512
   408
  using assms by (auto simp add: mult.commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
haftmann@35367
   409
huffman@31661
   410
end
huffman@31661
   411
haftmann@35673
   412
class ring_div = semiring_div + comm_ring_1
huffman@29405
   413
begin
huffman@29405
   414
haftmann@36634
   415
subclass ring_1_no_zero_divisors ..
haftmann@36634
   416
huffman@29405
   417
text {* Negation respects modular equivalence. *}
huffman@29405
   418
huffman@29405
   419
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   420
proof -
huffman@29405
   421
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405
   422
    by (simp only: mod_div_equality)
huffman@29405
   423
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
haftmann@57514
   424
    by (simp add: ac_simps)
huffman@29405
   425
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   426
    by (rule mod_mult_self1)
huffman@29405
   427
  finally show ?thesis .
huffman@29405
   428
qed
huffman@29405
   429
huffman@29405
   430
lemma mod_minus_cong:
huffman@29405
   431
  assumes "a mod b = a' mod b"
huffman@29405
   432
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   433
proof -
huffman@29405
   434
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   435
    unfolding assms ..
huffman@29405
   436
  thus ?thesis
huffman@29405
   437
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   438
qed
huffman@29405
   439
huffman@29405
   440
text {* Subtraction respects modular equivalence. *}
huffman@29405
   441
haftmann@54230
   442
lemma mod_diff_left_eq:
haftmann@54230
   443
  "(a - b) mod c = (a mod c - b) mod c"
haftmann@54230
   444
  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp
haftmann@54230
   445
haftmann@54230
   446
lemma mod_diff_right_eq:
haftmann@54230
   447
  "(a - b) mod c = (a - b mod c) mod c"
haftmann@54230
   448
  using mod_add_cong [of a c a "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
haftmann@54230
   449
haftmann@54230
   450
lemma mod_diff_eq:
haftmann@54230
   451
  "(a - b) mod c = (a mod c - b mod c) mod c"
haftmann@54230
   452
  using mod_add_cong [of a c "a mod c" "- b" "- (b mod c)"] mod_minus_cong [of "b mod c" c b] by simp
huffman@29405
   453
huffman@29405
   454
lemma mod_diff_cong:
huffman@29405
   455
  assumes "a mod c = a' mod c"
huffman@29405
   456
  assumes "b mod c = b' mod c"
huffman@29405
   457
  shows "(a - b) mod c = (a' - b') mod c"
haftmann@54230
   458
  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp
huffman@29405
   459
nipkow@30180
   460
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   461
apply (case_tac "y = 0") apply simp
nipkow@30180
   462
apply (auto simp add: dvd_def)
nipkow@30180
   463
apply (subgoal_tac "-(y * k) = y * - k")
thomas@57492
   464
 apply (simp only:)
nipkow@30180
   465
 apply (erule div_mult_self1_is_id)
nipkow@30180
   466
apply simp
nipkow@30180
   467
done
nipkow@30180
   468
nipkow@30180
   469
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   470
apply (case_tac "y = 0") apply simp
nipkow@30180
   471
apply (auto simp add: dvd_def)
nipkow@30180
   472
apply (subgoal_tac "y * k = -y * -k")
thomas@57492
   473
 apply (erule ssubst, rule div_mult_self1_is_id)
nipkow@30180
   474
 apply simp
nipkow@30180
   475
apply simp
nipkow@30180
   476
done
nipkow@30180
   477
huffman@47159
   478
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159
   479
  using div_mult_mult1 [of "- 1" a b]
huffman@47159
   480
  unfolding neg_equal_0_iff_equal by simp
huffman@47159
   481
huffman@47159
   482
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159
   483
  using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159
   484
huffman@47159
   485
lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159
   486
  using div_minus_minus [of "-a" b] by simp
huffman@47159
   487
huffman@47159
   488
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159
   489
  using mod_minus_minus [of "-a" b] by simp
huffman@47159
   490
huffman@47160
   491
lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160
   492
  using div_minus_right [of a 1] by simp
huffman@47160
   493
huffman@47160
   494
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160
   495
  using mod_minus_right [of a 1] by simp
huffman@47160
   496
haftmann@54221
   497
lemma minus_mod_self2 [simp]: 
haftmann@54221
   498
  "(a - b) mod b = a mod b"
haftmann@54221
   499
  by (simp add: mod_diff_right_eq)
haftmann@54221
   500
haftmann@54221
   501
lemma minus_mod_self1 [simp]: 
haftmann@54221
   502
  "(b - a) mod b = - a mod b"
haftmann@54230
   503
  using mod_add_self2 [of "- a" b] by simp
haftmann@54221
   504
huffman@29405
   505
end
huffman@29405
   506
haftmann@54226
   507
class semiring_div_parity = semiring_div + semiring_numeral +
haftmann@54226
   508
  assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@58646
   509
  assumes one_mod_two_eq_one: "1 mod 2 = 1"
haftmann@58710
   510
  assumes zero_not_eq_two: "0 \<noteq> 2"
haftmann@54226
   511
begin
haftmann@54226
   512
haftmann@54226
   513
lemma parity_cases [case_names even odd]:
haftmann@54226
   514
  assumes "a mod 2 = 0 \<Longrightarrow> P"
haftmann@54226
   515
  assumes "a mod 2 = 1 \<Longrightarrow> P"
haftmann@54226
   516
  shows P
haftmann@54226
   517
  using assms parity by blast
haftmann@54226
   518
haftmann@54226
   519
lemma not_mod_2_eq_0_eq_1 [simp]:
haftmann@54226
   520
  "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
haftmann@54226
   521
  by (cases a rule: parity_cases) simp_all
haftmann@54226
   522
haftmann@54226
   523
lemma not_mod_2_eq_1_eq_0 [simp]:
haftmann@54226
   524
  "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
haftmann@54226
   525
  by (cases a rule: parity_cases) simp_all
haftmann@54226
   526
haftmann@54226
   527
end
haftmann@54226
   528
haftmann@25942
   529
haftmann@53067
   530
subsection {* Generic numeral division with a pragmatic type class *}
haftmann@53067
   531
haftmann@53067
   532
text {*
haftmann@53067
   533
  The following type class contains everything necessary to formulate
haftmann@53067
   534
  a division algorithm in ring structures with numerals, restricted
haftmann@53067
   535
  to its positive segments.  This is its primary motiviation, and it
haftmann@53067
   536
  could surely be formulated using a more fine-grained, more algebraic
haftmann@53067
   537
  and less technical class hierarchy.
haftmann@53067
   538
*}
haftmann@53067
   539
haftmann@53067
   540
class semiring_numeral_div = linordered_semidom + minus + semiring_div +
haftmann@53067
   541
  assumes diff_invert_add1: "a + b = c \<Longrightarrow> a = c - b"
haftmann@53067
   542
    and le_add_diff_inverse2: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@53067
   543
  assumes mult_div_cancel: "b * (a div b) = a - a mod b"
haftmann@53067
   544
    and div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067
   545
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067
   546
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067
   547
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067
   548
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067
   549
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067
   550
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067
   551
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067
   552
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@53067
   553
begin
haftmann@53067
   554
haftmann@53067
   555
lemma diff_zero [simp]:
haftmann@53067
   556
  "a - 0 = a"
haftmann@53067
   557
  by (rule diff_invert_add1 [symmetric]) simp
haftmann@53067
   558
haftmann@54226
   559
subclass semiring_div_parity
haftmann@54226
   560
proof
haftmann@54226
   561
  fix a
haftmann@54226
   562
  show "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@54226
   563
  proof (rule ccontr)
haftmann@54226
   564
    assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@54226
   565
    then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@54226
   566
    have "0 < 2" by simp
haftmann@54226
   567
    with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
haftmann@54226
   568
    with `a mod 2 \<noteq> 0` have "0 < a mod 2" by simp
haftmann@54226
   569
    with discrete have "1 \<le> a mod 2" by simp
haftmann@54226
   570
    with `a mod 2 \<noteq> 1` have "1 < a mod 2" by simp
haftmann@54226
   571
    with discrete have "2 \<le> a mod 2" by simp
haftmann@54226
   572
    with `a mod 2 < 2` show False by simp
haftmann@54226
   573
  qed
haftmann@58646
   574
next
haftmann@58646
   575
  show "1 mod 2 = 1"
haftmann@58646
   576
    by (rule mod_less) simp_all
haftmann@58710
   577
next
haftmann@58710
   578
  show "0 \<noteq> 2"
haftmann@58710
   579
    by simp
haftmann@53067
   580
qed
haftmann@53067
   581
haftmann@53067
   582
lemma divmod_digit_1:
haftmann@53067
   583
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067
   584
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067
   585
    and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067
   586
proof -
haftmann@53067
   587
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067
   588
    by (auto intro: trans)
haftmann@53067
   589
  with `0 < b` have "0 < a div b" by (auto intro: div_positive)
haftmann@53067
   590
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
haftmann@53067
   591
  with `0 < b` have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
haftmann@53067
   592
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   593
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   594
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   595
  from assms w_exhaust have "w = 1"
haftmann@53067
   596
    by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067
   597
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067
   598
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   599
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067
   600
  with `w = 1` have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067
   601
  then show ?P and ?Q
haftmann@53067
   602
    by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)
haftmann@53067
   603
qed
haftmann@53067
   604
haftmann@53067
   605
lemma divmod_digit_0:
haftmann@53067
   606
  assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067
   607
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067
   608
    and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067
   609
proof -
haftmann@53067
   610
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   611
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   612
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   613
  moreover have "b \<le> a mod b + b"
haftmann@53067
   614
  proof -
haftmann@53067
   615
    from `0 < b` pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067
   616
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067
   617
    then show ?thesis by simp
haftmann@53067
   618
  qed
haftmann@53067
   619
  moreover note assms w_exhaust
haftmann@53067
   620
  ultimately have "w = 0" by auto
haftmann@53067
   621
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067
   622
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   623
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067
   624
  with `w = 0` have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067
   625
  then show ?P and ?Q
haftmann@53067
   626
    by (simp_all add: div mod)
haftmann@53067
   627
qed
haftmann@53067
   628
haftmann@53067
   629
definition divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@53067
   630
where
haftmann@53067
   631
  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@53067
   632
haftmann@53067
   633
lemma fst_divmod [simp]:
haftmann@53067
   634
  "fst (divmod m n) = numeral m div numeral n"
haftmann@53067
   635
  by (simp add: divmod_def)
haftmann@53067
   636
haftmann@53067
   637
lemma snd_divmod [simp]:
haftmann@53067
   638
  "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067
   639
  by (simp add: divmod_def)
haftmann@53067
   640
haftmann@53067
   641
definition divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@53067
   642
where
haftmann@53067
   643
  "divmod_step l qr = (let (q, r) = qr
haftmann@53067
   644
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@53067
   645
    else (2 * q, r))"
haftmann@53067
   646
haftmann@53067
   647
text {*
haftmann@53067
   648
  This is a formulation of one step (referring to one digit position)
haftmann@53067
   649
  in school-method division: compare the dividend at the current
haftmann@53070
   650
  digit position with the remainder from previous division steps
haftmann@53067
   651
  and evaluate accordingly.
haftmann@53067
   652
*}
haftmann@53067
   653
haftmann@53067
   654
lemma divmod_step_eq [code]:
haftmann@53067
   655
  "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067
   656
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067
   657
  by (simp add: divmod_step_def)
haftmann@53067
   658
haftmann@53067
   659
lemma divmod_step_simps [simp]:
haftmann@53067
   660
  "r < numeral l \<Longrightarrow> divmod_step l (q, r) = (2 * q, r)"
haftmann@53067
   661
  "numeral l \<le> r \<Longrightarrow> divmod_step l (q, r) = (2 * q + 1, r - numeral l)"
haftmann@53067
   662
  by (auto simp add: divmod_step_eq not_le)
haftmann@53067
   663
haftmann@53067
   664
text {*
haftmann@53067
   665
  This is a formulation of school-method division.
haftmann@53067
   666
  If the divisor is smaller than the dividend, terminate.
haftmann@53067
   667
  If not, shift the dividend to the right until termination
haftmann@53067
   668
  occurs and then reiterate single division steps in the
haftmann@53067
   669
  opposite direction.
haftmann@53067
   670
*}
haftmann@53067
   671
haftmann@53067
   672
lemma divmod_divmod_step [code]:
haftmann@53067
   673
  "divmod m n = (if m < n then (0, numeral m)
haftmann@53067
   674
    else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067
   675
proof (cases "m < n")
haftmann@53067
   676
  case True then have "numeral m < numeral n" by simp
haftmann@53067
   677
  then show ?thesis
haftmann@53067
   678
    by (simp add: prod_eq_iff div_less mod_less)
haftmann@53067
   679
next
haftmann@53067
   680
  case False
haftmann@53067
   681
  have "divmod m n =
haftmann@53067
   682
    divmod_step n (numeral m div (2 * numeral n),
haftmann@53067
   683
      numeral m mod (2 * numeral n))"
haftmann@53067
   684
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067
   685
    case True
haftmann@53067
   686
    with divmod_step_simps
haftmann@53067
   687
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   688
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@53067
   689
        by blast
haftmann@53067
   690
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067
   691
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067
   692
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067
   693
      by simp_all
haftmann@53067
   694
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   695
  next
haftmann@53067
   696
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067
   697
      by (simp add: not_le)
haftmann@53067
   698
    with divmod_step_simps
haftmann@53067
   699
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   700
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@53067
   701
        by blast
haftmann@53067
   702
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067
   703
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067
   704
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067
   705
      by (simp_all only: zero_less_numeral)
haftmann@53067
   706
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   707
  qed
haftmann@53067
   708
  then have "divmod m n =
haftmann@53067
   709
    divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067
   710
      numeral m mod numeral (Num.Bit0 n))"
haftmann@53067
   711
    by (simp only: numeral.simps distrib mult_1) 
haftmann@53067
   712
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067
   713
    by (simp add: divmod_def)
haftmann@53067
   714
  with False show ?thesis by simp
haftmann@53067
   715
qed
haftmann@53067
   716
haftmann@53069
   717
lemma divmod_cancel [code]:
haftmann@53069
   718
  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
haftmann@53069
   719
  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
haftmann@53069
   720
proof -
haftmann@53069
   721
  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069
   722
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069
   723
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069
   724
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069
   725
  then show ?P and ?Q
haftmann@53069
   726
    by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@53069
   727
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral)
haftmann@53069
   728
 qed
haftmann@53069
   729
haftmann@53067
   730
end
haftmann@53067
   731
haftmann@53067
   732
hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero
haftmann@53067
   733
  -- {* restore simple accesses for more general variants of theorems *}
haftmann@53067
   734
haftmann@53067
   735
  
haftmann@26100
   736
subsection {* Division on @{typ nat} *}
haftmann@26100
   737
haftmann@26100
   738
text {*
haftmann@26100
   739
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   740
  of a characteristic relation with two input arguments
haftmann@26100
   741
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   742
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   743
*}
haftmann@26100
   744
haftmann@33340
   745
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   746
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   747
    m = fst qr * n + snd qr \<and>
haftmann@30923
   748
      (if n = 0 then fst qr = 0 else if n > 0 then 0 \<le> snd qr \<and> snd qr < n else n < snd qr \<and> snd qr \<le> 0)"
haftmann@26100
   749
haftmann@33340
   750
text {* @{const divmod_nat_rel} is total: *}
haftmann@26100
   751
haftmann@33340
   752
lemma divmod_nat_rel_ex:
haftmann@33340
   753
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   754
proof (cases "n = 0")
haftmann@30923
   755
  case True  with that show thesis
haftmann@33340
   756
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   757
next
haftmann@26100
   758
  case False
haftmann@26100
   759
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   760
  proof (induct m)
haftmann@26100
   761
    case 0 with `n \<noteq> 0`
haftmann@26100
   762
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   763
    then show ?case by blast
haftmann@26100
   764
  next
haftmann@26100
   765
    case (Suc m) then obtain q' r'
haftmann@26100
   766
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   767
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   768
      case True
haftmann@26100
   769
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   770
      with True show ?thesis by blast
haftmann@26100
   771
    next
haftmann@26100
   772
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   773
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   774
      ultimately have "n = Suc r'" by auto
haftmann@26100
   775
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   776
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   777
    qed
haftmann@26100
   778
  qed
haftmann@26100
   779
  with that show thesis
haftmann@33340
   780
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   781
qed
haftmann@26100
   782
haftmann@33340
   783
text {* @{const divmod_nat_rel} is injective: *}
haftmann@26100
   784
haftmann@33340
   785
lemma divmod_nat_rel_unique:
haftmann@33340
   786
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   787
    and "divmod_nat_rel m n qr'"
haftmann@30923
   788
  shows "qr = qr'"
haftmann@26100
   789
proof (cases "n = 0")
haftmann@26100
   790
  case True with assms show ?thesis
haftmann@30923
   791
    by (cases qr, cases qr')
haftmann@33340
   792
      (simp add: divmod_nat_rel_def)
haftmann@26100
   793
next
haftmann@26100
   794
  case False
haftmann@26100
   795
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   796
  apply (rule leI)
haftmann@26100
   797
  apply (subst less_iff_Suc_add)
haftmann@26100
   798
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   799
  done
wenzelm@53374
   800
  from `n \<noteq> 0` assms have *: "fst qr = fst qr'"
haftmann@33340
   801
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
wenzelm@53374
   802
  with assms have "snd qr = snd qr'"
haftmann@33340
   803
    by (simp add: divmod_nat_rel_def)
wenzelm@53374
   804
  with * show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   805
qed
haftmann@26100
   806
haftmann@26100
   807
text {*
haftmann@26100
   808
  We instantiate divisibility on the natural numbers by
haftmann@33340
   809
  means of @{const divmod_nat_rel}:
haftmann@26100
   810
*}
haftmann@25942
   811
haftmann@33340
   812
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767
   813
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   814
haftmann@33340
   815
lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   816
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   817
proof -
haftmann@33340
   818
  from divmod_nat_rel_ex
haftmann@33340
   819
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   820
  then show ?thesis
haftmann@33340
   821
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   822
qed
haftmann@30923
   823
huffman@47135
   824
lemma divmod_nat_unique:
haftmann@33340
   825
  assumes "divmod_nat_rel m n qr" 
haftmann@33340
   826
  shows "divmod_nat m n = qr"
haftmann@33340
   827
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   828
huffman@46551
   829
instantiation nat :: semiring_div
huffman@46551
   830
begin
huffman@46551
   831
haftmann@26100
   832
definition div_nat where
haftmann@33340
   833
  "m div n = fst (divmod_nat m n)"
haftmann@26100
   834
huffman@46551
   835
lemma fst_divmod_nat [simp]:
huffman@46551
   836
  "fst (divmod_nat m n) = m div n"
huffman@46551
   837
  by (simp add: div_nat_def)
huffman@46551
   838
haftmann@26100
   839
definition mod_nat where
haftmann@33340
   840
  "m mod n = snd (divmod_nat m n)"
haftmann@25571
   841
huffman@46551
   842
lemma snd_divmod_nat [simp]:
huffman@46551
   843
  "snd (divmod_nat m n) = m mod n"
huffman@46551
   844
  by (simp add: mod_nat_def)
huffman@46551
   845
haftmann@33340
   846
lemma divmod_nat_div_mod:
haftmann@33340
   847
  "divmod_nat m n = (m div n, m mod n)"
huffman@46551
   848
  by (simp add: prod_eq_iff)
haftmann@26100
   849
huffman@47135
   850
lemma div_nat_unique:
haftmann@33340
   851
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   852
  shows "m div n = q"
huffman@47135
   853
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   854
huffman@47135
   855
lemma mod_nat_unique:
haftmann@33340
   856
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   857
  shows "m mod n = r"
huffman@47135
   858
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   859
haftmann@33340
   860
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
huffman@46551
   861
  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
paulson@14267
   862
huffman@47136
   863
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
huffman@47136
   864
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
huffman@47136
   865
huffman@47136
   866
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
huffman@47136
   867
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   868
huffman@47137
   869
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
huffman@47137
   870
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   871
haftmann@33340
   872
lemma divmod_nat_step:
haftmann@26100
   873
  assumes "0 < n" and "n \<le> m"
haftmann@33340
   874
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
huffman@47135
   875
proof (rule divmod_nat_unique)
huffman@47134
   876
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
huffman@47134
   877
    by (rule divmod_nat_rel)
huffman@47134
   878
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
huffman@47134
   879
    unfolding divmod_nat_rel_def using assms by auto
haftmann@26100
   880
qed
haftmann@25942
   881
wenzelm@26300
   882
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   883
haftmann@26100
   884
lemma div_less [simp]:
haftmann@26100
   885
  fixes m n :: nat
haftmann@26100
   886
  assumes "m < n"
haftmann@26100
   887
  shows "m div n = 0"
huffman@46551
   888
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   889
haftmann@26100
   890
lemma le_div_geq:
haftmann@26100
   891
  fixes m n :: nat
haftmann@26100
   892
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   893
  shows "m div n = Suc ((m - n) div n)"
huffman@46551
   894
  using assms divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   895
haftmann@26100
   896
lemma mod_less [simp]:
haftmann@26100
   897
  fixes m n :: nat
haftmann@26100
   898
  assumes "m < n"
haftmann@26100
   899
  shows "m mod n = m"
huffman@46551
   900
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   901
haftmann@26100
   902
lemma le_mod_geq:
haftmann@26100
   903
  fixes m n :: nat
haftmann@26100
   904
  assumes "n \<le> m"
haftmann@26100
   905
  shows "m mod n = (m - n) mod n"
huffman@46551
   906
  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   907
huffman@47136
   908
instance proof
huffman@47136
   909
  fix m n :: nat
huffman@47136
   910
  show "m div n * n + m mod n = m"
huffman@47136
   911
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136
   912
next
huffman@47136
   913
  fix m n q :: nat
huffman@47136
   914
  assume "n \<noteq> 0"
huffman@47136
   915
  then show "(q + m * n) div n = m + q div n"
huffman@47136
   916
    by (induct m) (simp_all add: le_div_geq)
huffman@47136
   917
next
huffman@47136
   918
  fix m n q :: nat
huffman@47136
   919
  assume "m \<noteq> 0"
huffman@47136
   920
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136
   921
    unfolding divmod_nat_rel_def
huffman@47136
   922
    by (auto split: split_if_asm, simp_all add: algebra_simps)
huffman@47136
   923
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136
   924
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136
   925
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136
   926
next
huffman@47136
   927
  fix n :: nat show "n div 0 = 0"
haftmann@33340
   928
    by (simp add: div_nat_def divmod_nat_zero)
huffman@47136
   929
next
huffman@47136
   930
  fix n :: nat show "0 div n = 0"
huffman@47136
   931
    by (simp add: div_nat_def divmod_nat_zero_left)
haftmann@25942
   932
qed
haftmann@26100
   933
haftmann@25942
   934
end
paulson@14267
   935
haftmann@33361
   936
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@33361
   937
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
blanchet@55414
   938
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
haftmann@33361
   939
haftmann@26100
   940
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   941
wenzelm@51299
   942
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299
   943
haftmann@30934
   944
ML {*
wenzelm@43594
   945
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
   946
(
haftmann@30934
   947
  val div_name = @{const_name div};
haftmann@30934
   948
  val mod_name = @{const_name mod};
haftmann@30934
   949
  val mk_binop = HOLogic.mk_binop;
huffman@48561
   950
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
   951
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
   952
  fun mk_sum [] = HOLogic.zero
huffman@48561
   953
    | mk_sum [t] = t
huffman@48561
   954
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
   955
  fun dest_sum tm =
huffman@48561
   956
    if HOLogic.is_zero tm then []
huffman@48561
   957
    else
huffman@48561
   958
      (case try HOLogic.dest_Suc tm of
huffman@48561
   959
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
   960
      | NONE =>
huffman@48561
   961
          (case try dest_plus tm of
huffman@48561
   962
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
   963
          | NONE => [tm]));
haftmann@25942
   964
haftmann@30934
   965
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
   966
haftmann@30934
   967
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@57514
   968
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))
wenzelm@41550
   969
)
haftmann@25942
   970
*}
haftmann@25942
   971
wenzelm@43594
   972
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
wenzelm@43594
   973
haftmann@26100
   974
haftmann@26100
   975
subsubsection {* Quotient *}
haftmann@26100
   976
haftmann@26100
   977
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   978
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   979
haftmann@26100
   980
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   981
by (simp add: div_geq)
haftmann@26100
   982
haftmann@26100
   983
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   984
by simp
haftmann@26100
   985
haftmann@26100
   986
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   987
by simp
haftmann@26100
   988
haftmann@53066
   989
lemma div_positive:
haftmann@53066
   990
  fixes m n :: nat
haftmann@53066
   991
  assumes "n > 0"
haftmann@53066
   992
  assumes "m \<ge> n"
haftmann@53066
   993
  shows "m div n > 0"
haftmann@53066
   994
proof -
haftmann@53066
   995
  from `m \<ge> n` obtain q where "m = n + q"
haftmann@53066
   996
    by (auto simp add: le_iff_add)
haftmann@53066
   997
  with `n > 0` show ?thesis by simp
haftmann@53066
   998
qed
haftmann@53066
   999
haftmann@25942
  1000
haftmann@25942
  1001
subsubsection {* Remainder *}
haftmann@25942
  1002
haftmann@26100
  1003
lemma mod_less_divisor [simp]:
haftmann@26100
  1004
  fixes m n :: nat
haftmann@26100
  1005
  assumes "n > 0"
haftmann@26100
  1006
  shows "m mod n < (n::nat)"
haftmann@33340
  1007
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
  1008
haftmann@51173
  1009
lemma mod_Suc_le_divisor [simp]:
haftmann@51173
  1010
  "m mod Suc n \<le> n"
haftmann@51173
  1011
  using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173
  1012
haftmann@26100
  1013
lemma mod_less_eq_dividend [simp]:
haftmann@26100
  1014
  fixes m n :: nat
haftmann@26100
  1015
  shows "m mod n \<le> m"
haftmann@26100
  1016
proof (rule add_leD2)
haftmann@26100
  1017
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
  1018
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
  1019
qed
haftmann@26100
  1020
haftmann@26100
  1021
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
  1022
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
  1023
haftmann@26100
  1024
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
  1025
by (simp add: le_mod_geq)
haftmann@26100
  1026
paulson@14267
  1027
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
  1028
by (induct m) (simp_all add: mod_geq)
paulson@14267
  1029
paulson@14267
  1030
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
  1031
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
huffman@47138
  1032
  using mod_div_equality2 [of n m] by arith
paulson@14267
  1033
nipkow@15439
  1034
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
  1035
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
  1036
  apply simp
wenzelm@22718
  1037
  done
paulson@14267
  1038
haftmann@26100
  1039
subsubsection {* Quotient and Remainder *}
paulson@14267
  1040
haftmann@33340
  1041
lemma divmod_nat_rel_mult1_eq:
bulwahn@46552
  1042
  "divmod_nat_rel b c (q, r)
haftmann@33340
  1043
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
  1044
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1045
haftmann@30923
  1046
lemma div_mult1_eq:
haftmann@30923
  1047
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135
  1048
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1049
haftmann@33340
  1050
lemma divmod_nat_rel_add1_eq:
bulwahn@46552
  1051
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340
  1052
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
  1053
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1054
paulson@14267
  1055
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
  1056
lemma div_add1_eq:
nipkow@25134
  1057
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135
  1058
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1059
paulson@14267
  1060
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
  1061
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
  1062
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
  1063
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
  1064
  apply (simp add: add_mult_distrib2)
wenzelm@22718
  1065
  done
paulson@10559
  1066
haftmann@33340
  1067
lemma divmod_nat_rel_mult2_eq:
bulwahn@46552
  1068
  "divmod_nat_rel a b (q, r)
haftmann@33340
  1069
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@57514
  1070
by (auto simp add: mult.commute mult.left_commute divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
  1071
blanchet@55085
  1072
lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
huffman@47135
  1073
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267
  1074
blanchet@55085
  1075
lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
haftmann@57512
  1076
by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267
  1077
paulson@14267
  1078
huffman@46551
  1079
subsubsection {* Further Facts about Quotient and Remainder *}
paulson@14267
  1080
paulson@14267
  1081
lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667
  1082
by (induct m) (simp_all add: div_geq)
paulson@14267
  1083
paulson@14267
  1084
(* Monotonicity of div in first argument *)
haftmann@30923
  1085
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
  1086
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
  1087
apply (case_tac "k=0", simp)
paulson@15251
  1088
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
  1089
apply (case_tac "n<k")
paulson@14267
  1090
(* 1  case n<k *)
paulson@14267
  1091
apply simp
paulson@14267
  1092
(* 2  case n >= k *)
paulson@14267
  1093
apply (case_tac "m<k")
paulson@14267
  1094
(* 2.1  case m<k *)
paulson@14267
  1095
apply simp
paulson@14267
  1096
(* 2.2  case m>=k *)
nipkow@15439
  1097
apply (simp add: div_geq diff_le_mono)
paulson@14267
  1098
done
paulson@14267
  1099
paulson@14267
  1100
(* Antimonotonicity of div in second argument *)
paulson@14267
  1101
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
  1102
apply (subgoal_tac "0<n")
wenzelm@22718
  1103
 prefer 2 apply simp
paulson@15251
  1104
apply (induct_tac k rule: nat_less_induct)
paulson@14267
  1105
apply (rename_tac "k")
paulson@14267
  1106
apply (case_tac "k<n", simp)
paulson@14267
  1107
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
  1108
 prefer 2 apply simp
paulson@14267
  1109
apply (simp add: div_geq)
paulson@15251
  1110
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
  1111
 prefer 2
paulson@14267
  1112
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
  1113
apply (rule le_trans, simp)
nipkow@15439
  1114
apply (simp)
paulson@14267
  1115
done
paulson@14267
  1116
paulson@14267
  1117
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
  1118
apply (case_tac "n=0", simp)
paulson@14267
  1119
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
  1120
apply (rule div_le_mono2)
paulson@14267
  1121
apply (simp_all (no_asm_simp))
paulson@14267
  1122
done
paulson@14267
  1123
wenzelm@22718
  1124
(* Similar for "less than" *)
huffman@47138
  1125
lemma div_less_dividend [simp]:
huffman@47138
  1126
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
  1127
apply (induct m rule: nat_less_induct)
paulson@14267
  1128
apply (rename_tac "m")
paulson@14267
  1129
apply (case_tac "m<n", simp)
paulson@14267
  1130
apply (subgoal_tac "0<n")
wenzelm@22718
  1131
 prefer 2 apply simp
paulson@14267
  1132
apply (simp add: div_geq)
paulson@14267
  1133
apply (case_tac "n<m")
paulson@15251
  1134
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
  1135
  apply (rule impI less_trans_Suc)+
paulson@14267
  1136
apply assumption
nipkow@15439
  1137
  apply (simp_all)
paulson@14267
  1138
done
paulson@14267
  1139
paulson@14267
  1140
text{*A fact for the mutilated chess board*}
paulson@14267
  1141
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
  1142
apply (case_tac "n=0", simp)
paulson@15251
  1143
apply (induct "m" rule: nat_less_induct)
paulson@14267
  1144
apply (case_tac "Suc (na) <n")
paulson@14267
  1145
(* case Suc(na) < n *)
paulson@14267
  1146
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
  1147
(* case n \<le> Suc(na) *)
paulson@16796
  1148
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
  1149
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
  1150
done
paulson@14267
  1151
paulson@14267
  1152
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
  1153
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
  1154
wenzelm@22718
  1155
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
  1156
paulson@14267
  1157
(*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@57514
  1158
lemma mod_eqD:
haftmann@57514
  1159
  fixes m d r q :: nat
haftmann@57514
  1160
  assumes "m mod d = r"
haftmann@57514
  1161
  shows "\<exists>q. m = r + q * d"
haftmann@57514
  1162
proof -
haftmann@57514
  1163
  from mod_div_equality obtain q where "q * d + m mod d = m" by blast
haftmann@57514
  1164
  with assms have "m = r + q * d" by simp
haftmann@57514
  1165
  then show ?thesis ..
haftmann@57514
  1166
qed
paulson@14267
  1167
nipkow@13152
  1168
lemma split_div:
nipkow@13189
  1169
 "P(n div k :: nat) =
nipkow@13189
  1170
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
  1171
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1172
proof
nipkow@13189
  1173
  assume P: ?P
nipkow@13189
  1174
  show ?Q
nipkow@13189
  1175
  proof (cases)
nipkow@13189
  1176
    assume "k = 0"
haftmann@27651
  1177
    with P show ?Q by simp
nipkow@13189
  1178
  next
nipkow@13189
  1179
    assume not0: "k \<noteq> 0"
nipkow@13189
  1180
    thus ?Q
nipkow@13189
  1181
    proof (simp, intro allI impI)
nipkow@13189
  1182
      fix i j
nipkow@13189
  1183
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
  1184
      show "P i"
nipkow@13189
  1185
      proof (cases)
wenzelm@22718
  1186
        assume "i = 0"
wenzelm@22718
  1187
        with n j P show "P i" by simp
nipkow@13189
  1188
      next
wenzelm@22718
  1189
        assume "i \<noteq> 0"
haftmann@57514
  1190
        with not0 n j P show "P i" by(simp add:ac_simps)
nipkow@13189
  1191
      qed
nipkow@13189
  1192
    qed
nipkow@13189
  1193
  qed
nipkow@13189
  1194
next
nipkow@13189
  1195
  assume Q: ?Q
nipkow@13189
  1196
  show ?P
nipkow@13189
  1197
  proof (cases)
nipkow@13189
  1198
    assume "k = 0"
haftmann@27651
  1199
    with Q show ?P by simp
nipkow@13189
  1200
  next
nipkow@13189
  1201
    assume not0: "k \<noteq> 0"
nipkow@13189
  1202
    with Q have R: ?R by simp
nipkow@13189
  1203
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1204
    show ?P by simp
nipkow@13189
  1205
  qed
nipkow@13189
  1206
qed
nipkow@13189
  1207
berghofe@13882
  1208
lemma split_div_lemma:
haftmann@26100
  1209
  assumes "0 < n"
haftmann@26100
  1210
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
  1211
proof
haftmann@26100
  1212
  assume ?rhs
haftmann@26100
  1213
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
  1214
  then have A: "n * q \<le> m" by simp
haftmann@26100
  1215
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
  1216
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
  1217
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
  1218
  with nq have "m < n + n * q" by simp
haftmann@26100
  1219
  then have B: "m < n * Suc q" by simp
haftmann@26100
  1220
  from A B show ?lhs ..
haftmann@26100
  1221
next
haftmann@26100
  1222
  assume P: ?lhs
haftmann@33340
  1223
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@57514
  1224
    unfolding divmod_nat_rel_def by (auto simp add: ac_simps)
haftmann@33340
  1225
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
haftmann@30923
  1226
  have "(q, m - n * q) = (m div n, m mod n)" by auto
haftmann@30923
  1227
  then show ?rhs by simp
haftmann@26100
  1228
qed
berghofe@13882
  1229
berghofe@13882
  1230
theorem split_div':
berghofe@13882
  1231
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
  1232
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
  1233
  apply (case_tac "0 < n")
berghofe@13882
  1234
  apply (simp only: add: split_div_lemma)
haftmann@27651
  1235
  apply simp_all
berghofe@13882
  1236
  done
berghofe@13882
  1237
nipkow@13189
  1238
lemma split_mod:
nipkow@13189
  1239
 "P(n mod k :: nat) =
nipkow@13189
  1240
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
  1241
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1242
proof
nipkow@13189
  1243
  assume P: ?P
nipkow@13189
  1244
  show ?Q
nipkow@13189
  1245
  proof (cases)
nipkow@13189
  1246
    assume "k = 0"
haftmann@27651
  1247
    with P show ?Q by simp
nipkow@13189
  1248
  next
nipkow@13189
  1249
    assume not0: "k \<noteq> 0"
nipkow@13189
  1250
    thus ?Q
nipkow@13189
  1251
    proof (simp, intro allI impI)
nipkow@13189
  1252
      fix i j
nipkow@13189
  1253
      assume "n = k*i + j" "j < k"
haftmann@57514
  1254
      thus "P j" using not0 P by(simp add:ac_simps ac_simps)
nipkow@13189
  1255
    qed
nipkow@13189
  1256
  qed
nipkow@13189
  1257
next
nipkow@13189
  1258
  assume Q: ?Q
nipkow@13189
  1259
  show ?P
nipkow@13189
  1260
  proof (cases)
nipkow@13189
  1261
    assume "k = 0"
haftmann@27651
  1262
    with Q show ?P by simp
nipkow@13189
  1263
  next
nipkow@13189
  1264
    assume not0: "k \<noteq> 0"
nipkow@13189
  1265
    with Q have R: ?R by simp
nipkow@13189
  1266
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1267
    show ?P by simp
nipkow@13189
  1268
  qed
nipkow@13189
  1269
qed
nipkow@13189
  1270
berghofe@13882
  1271
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
huffman@47138
  1272
  using mod_div_equality [of m n] by arith
huffman@47138
  1273
huffman@47138
  1274
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
huffman@47138
  1275
  using mod_div_equality [of m n] by arith
huffman@47138
  1276
(* FIXME: very similar to mult_div_cancel *)
haftmann@22800
  1277
noschinl@52398
  1278
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398
  1279
  apply rule
noschinl@52398
  1280
  apply (cases "b = 0")
noschinl@52398
  1281
  apply simp_all
noschinl@52398
  1282
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398
  1283
  done
noschinl@52398
  1284
haftmann@22800
  1285
huffman@46551
  1286
subsubsection {* An ``induction'' law for modulus arithmetic. *}
paulson@14640
  1287
paulson@14640
  1288
lemma mod_induct_0:
paulson@14640
  1289
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1290
  and base: "P i" and i: "i<p"
paulson@14640
  1291
  shows "P 0"
paulson@14640
  1292
proof (rule ccontr)
paulson@14640
  1293
  assume contra: "\<not>(P 0)"
paulson@14640
  1294
  from i have p: "0<p" by simp
paulson@14640
  1295
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1296
  proof
paulson@14640
  1297
    fix k
paulson@14640
  1298
    show "?A k"
paulson@14640
  1299
    proof (induct k)
paulson@14640
  1300
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1301
    next
paulson@14640
  1302
      fix n
paulson@14640
  1303
      assume ih: "?A n"
paulson@14640
  1304
      show "?A (Suc n)"
paulson@14640
  1305
      proof (clarsimp)
wenzelm@22718
  1306
        assume y: "P (p - Suc n)"
wenzelm@22718
  1307
        have n: "Suc n < p"
wenzelm@22718
  1308
        proof (rule ccontr)
wenzelm@22718
  1309
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1310
          hence "p - Suc n = 0"
wenzelm@22718
  1311
            by simp
wenzelm@22718
  1312
          with y contra show "False"
wenzelm@22718
  1313
            by simp
wenzelm@22718
  1314
        qed
wenzelm@22718
  1315
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1316
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1317
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1318
          by blast
wenzelm@22718
  1319
        show "False"
wenzelm@22718
  1320
        proof (cases "n=0")
wenzelm@22718
  1321
          case True
wenzelm@22718
  1322
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1323
        next
wenzelm@22718
  1324
          case False
wenzelm@22718
  1325
          with p have "p-n < p" by arith
wenzelm@22718
  1326
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1327
        qed
paulson@14640
  1328
      qed
paulson@14640
  1329
    qed
paulson@14640
  1330
  qed
paulson@14640
  1331
  moreover
paulson@14640
  1332
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1333
    by (blast dest: less_imp_add_positive)
paulson@14640
  1334
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1335
  moreover
paulson@14640
  1336
  note base
paulson@14640
  1337
  ultimately
paulson@14640
  1338
  show "False" by blast
paulson@14640
  1339
qed
paulson@14640
  1340
paulson@14640
  1341
lemma mod_induct:
paulson@14640
  1342
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1343
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1344
  shows "P j"
paulson@14640
  1345
proof -
paulson@14640
  1346
  have "\<forall>j<p. P j"
paulson@14640
  1347
  proof
paulson@14640
  1348
    fix j
paulson@14640
  1349
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1350
    proof (induct j)
paulson@14640
  1351
      from step base i show "?A 0"
wenzelm@22718
  1352
        by (auto elim: mod_induct_0)
paulson@14640
  1353
    next
paulson@14640
  1354
      fix k
paulson@14640
  1355
      assume ih: "?A k"
paulson@14640
  1356
      show "?A (Suc k)"
paulson@14640
  1357
      proof
wenzelm@22718
  1358
        assume suc: "Suc k < p"
wenzelm@22718
  1359
        hence k: "k<p" by simp
wenzelm@22718
  1360
        with ih have "P k" ..
wenzelm@22718
  1361
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1362
          by blast
wenzelm@22718
  1363
        moreover
wenzelm@22718
  1364
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1365
          by simp
wenzelm@22718
  1366
        ultimately
wenzelm@22718
  1367
        show "P (Suc k)" by simp
paulson@14640
  1368
      qed
paulson@14640
  1369
    qed
paulson@14640
  1370
  qed
paulson@14640
  1371
  with j show ?thesis by blast
paulson@14640
  1372
qed
paulson@14640
  1373
haftmann@33296
  1374
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1375
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1376
huffman@47138
  1377
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1378
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1379
haftmann@33296
  1380
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1381
by (simp add: mult_2 [symmetric])
haftmann@33296
  1382
haftmann@33296
  1383
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1384
proof -
boehmes@35815
  1385
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1386
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1387
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1388
  then show ?thesis by auto
haftmann@33296
  1389
qed
haftmann@33296
  1390
haftmann@33296
  1391
text{*These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1392
    at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@33296
  1393
    We already have some rules to simplify operands smaller than 3.*}
haftmann@33296
  1394
haftmann@33296
  1395
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1396
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1397
haftmann@33296
  1398
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1399
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1400
haftmann@33296
  1401
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1402
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1403
haftmann@33296
  1404
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1405
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1406
huffman@47108
  1407
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1408
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1409
haftmann@33361
  1410
haftmann@33361
  1411
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
haftmann@33361
  1412
apply (induct "m")
haftmann@33361
  1413
apply (simp_all add: mod_Suc)
haftmann@33361
  1414
done
haftmann@33361
  1415
huffman@47108
  1416
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1417
huffman@47138
  1418
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1419
by (simp add: div_le_mono)
haftmann@33361
  1420
haftmann@33361
  1421
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1422
by (cases n) simp_all
haftmann@33361
  1423
boehmes@35815
  1424
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1425
proof -
boehmes@35815
  1426
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
boehmes@35815
  1427
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
boehmes@35815
  1428
qed
haftmann@33361
  1429
haftmann@33361
  1430
  (* Potential use of algebra : Equality modulo n*)
haftmann@33361
  1431
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
haftmann@57514
  1432
by (simp add: ac_simps ac_simps)
haftmann@33361
  1433
haftmann@33361
  1434
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1435
proof -
haftmann@33361
  1436
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
haftmann@33361
  1437
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
haftmann@33361
  1438
  finally show ?thesis .
haftmann@33361
  1439
qed
haftmann@33361
  1440
haftmann@33361
  1441
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
haftmann@33361
  1442
apply (subst mod_Suc [of m]) 
haftmann@33361
  1443
apply (subst mod_Suc [of "m mod n"], simp) 
haftmann@33361
  1444
done
haftmann@33361
  1445
huffman@47108
  1446
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1447
  fixes n :: nat
huffman@47108
  1448
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
huffman@47108
  1449
  by simp
huffman@47108
  1450
haftmann@53067
  1451
instance nat :: semiring_numeral_div
haftmann@53067
  1452
  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)
haftmann@53067
  1453
haftmann@33361
  1454
haftmann@33361
  1455
subsection {* Division on @{typ int} *}
haftmann@33361
  1456
haftmann@33361
  1457
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
haftmann@33361
  1458
    --{*definition of quotient and remainder*}
huffman@47139
  1459
  "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139
  1460
    (if 0 < b then 0 \<le> r \<and> r < b else if b < 0 then b < r \<and> r \<le> 0 else q = 0))"
haftmann@33361
  1461
haftmann@53067
  1462
text {*
haftmann@53067
  1463
  The following algorithmic devlopment actually echos what has already
haftmann@53067
  1464
  been developed in class @{class semiring_numeral_div}.  In the long
haftmann@53067
  1465
  run it seems better to derive division on @{typ int} just from
haftmann@53067
  1466
  division on @{typ nat} and instantiate @{class semiring_numeral_div}
haftmann@53067
  1467
  accordingly.
haftmann@53067
  1468
*}
haftmann@53067
  1469
haftmann@33361
  1470
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1471
    --{*for the division algorithm*}
huffman@47108
  1472
    "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@33361
  1473
                         else (2 * q, r))"
haftmann@33361
  1474
haftmann@33361
  1475
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@33361
  1476
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1477
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@33361
  1478
     else adjust b (posDivAlg a (2 * b)))"
haftmann@33361
  1479
by auto
haftmann@33361
  1480
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
haftmann@33361
  1481
  (auto simp add: mult_2)
haftmann@33361
  1482
haftmann@33361
  1483
text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@33361
  1484
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1485
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@33361
  1486
     else adjust b (negDivAlg a (2 * b)))"
haftmann@33361
  1487
by auto
haftmann@33361
  1488
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
haftmann@33361
  1489
  (auto simp add: mult_2)
haftmann@33361
  1490
haftmann@33361
  1491
text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@33361
  1492
haftmann@33361
  1493
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1494
    --{*The full division algorithm considers all possible signs for a, b
haftmann@33361
  1495
       including the special case @{text "a=0, b<0"} because 
haftmann@33361
  1496
       @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@33361
  1497
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@33361
  1498
                  else if a = 0 then (0, 0)
huffman@46560
  1499
                       else apsnd uminus (negDivAlg (-a) (-b))
haftmann@33361
  1500
               else 
haftmann@33361
  1501
                  if 0 < b then negDivAlg a b
huffman@46560
  1502
                  else apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1503
haftmann@33361
  1504
instantiation int :: Divides.div
haftmann@33361
  1505
begin
haftmann@33361
  1506
huffman@46551
  1507
definition div_int where
haftmann@33361
  1508
  "a div b = fst (divmod_int a b)"
haftmann@33361
  1509
huffman@46551
  1510
lemma fst_divmod_int [simp]:
huffman@46551
  1511
  "fst (divmod_int a b) = a div b"
huffman@46551
  1512
  by (simp add: div_int_def)
huffman@46551
  1513
huffman@46551
  1514
definition mod_int where
huffman@46560
  1515
  "a mod b = snd (divmod_int a b)"
haftmann@33361
  1516
huffman@46551
  1517
lemma snd_divmod_int [simp]:
huffman@46551
  1518
  "snd (divmod_int a b) = a mod b"
huffman@46551
  1519
  by (simp add: mod_int_def)
huffman@46551
  1520
haftmann@33361
  1521
instance ..
haftmann@33361
  1522
paulson@3366
  1523
end
haftmann@33361
  1524
haftmann@33361
  1525
lemma divmod_int_mod_div:
haftmann@33361
  1526
  "divmod_int p q = (p div q, p mod q)"
huffman@46551
  1527
  by (simp add: prod_eq_iff)
haftmann@33361
  1528
haftmann@33361
  1529
text{*
haftmann@33361
  1530
Here is the division algorithm in ML:
haftmann@33361
  1531
haftmann@33361
  1532
\begin{verbatim}
haftmann@33361
  1533
    fun posDivAlg (a,b) =
haftmann@33361
  1534
      if a<b then (0,a)
haftmann@33361
  1535
      else let val (q,r) = posDivAlg(a, 2*b)
haftmann@33361
  1536
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1537
           end
haftmann@33361
  1538
haftmann@33361
  1539
    fun negDivAlg (a,b) =
haftmann@33361
  1540
      if 0\<le>a+b then (~1,a+b)
haftmann@33361
  1541
      else let val (q,r) = negDivAlg(a, 2*b)
haftmann@33361
  1542
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1543
           end;
haftmann@33361
  1544
haftmann@33361
  1545
    fun negateSnd (q,r:int) = (q,~r);
haftmann@33361
  1546
haftmann@33361
  1547
    fun divmod (a,b) = if 0\<le>a then 
haftmann@33361
  1548
                          if b>0 then posDivAlg (a,b) 
haftmann@33361
  1549
                           else if a=0 then (0,0)
haftmann@33361
  1550
                                else negateSnd (negDivAlg (~a,~b))
haftmann@33361
  1551
                       else 
haftmann@33361
  1552
                          if 0<b then negDivAlg (a,b)
haftmann@33361
  1553
                          else        negateSnd (posDivAlg (~a,~b));
haftmann@33361
  1554
\end{verbatim}
haftmann@33361
  1555
*}
haftmann@33361
  1556
haftmann@33361
  1557
huffman@46551
  1558
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
haftmann@33361
  1559
haftmann@33361
  1560
lemma unique_quotient_lemma:
haftmann@33361
  1561
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
haftmann@33361
  1562
      ==> q' \<le> (q::int)"
haftmann@33361
  1563
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1564
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1565
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1566
apply (erule_tac [2] order_le_less_trans)
webertj@49962
  1567
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1568
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
  1569
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1570
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1571
done
haftmann@33361
  1572
haftmann@33361
  1573
lemma unique_quotient_lemma_neg:
haftmann@33361
  1574
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
haftmann@33361
  1575
      ==> q \<le> (q'::int)"
haftmann@33361
  1576
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
haftmann@33361
  1577
    auto)
haftmann@33361
  1578
haftmann@33361
  1579
lemma unique_quotient:
bulwahn@46552
  1580
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1581
      ==> q = q'"
haftmann@33361
  1582
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361
  1583
apply (blast intro: order_antisym
haftmann@33361
  1584
             dest: order_eq_refl [THEN unique_quotient_lemma] 
haftmann@33361
  1585
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1586
done
haftmann@33361
  1587
haftmann@33361
  1588
haftmann@33361
  1589
lemma unique_remainder:
bulwahn@46552
  1590
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1591
      ==> r = r'"
haftmann@33361
  1592
apply (subgoal_tac "q = q'")
haftmann@33361
  1593
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1594
apply (blast intro: unique_quotient)
haftmann@33361
  1595
done
haftmann@33361
  1596
haftmann@33361
  1597
huffman@46551
  1598
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
haftmann@33361
  1599
haftmann@33361
  1600
text{*And positive divisors*}
haftmann@33361
  1601
haftmann@33361
  1602
lemma adjust_eq [simp]:
huffman@47108
  1603
     "adjust b (q, r) = 
huffman@47108
  1604
      (let diff = r - b in  
huffman@47108
  1605
        if 0 \<le> diff then (2 * q + 1, diff)   
haftmann@33361
  1606
                     else (2*q, r))"
huffman@47108
  1607
  by (simp add: Let_def adjust_def)
haftmann@33361
  1608
haftmann@33361
  1609
declare posDivAlg.simps [simp del]
haftmann@33361
  1610
haftmann@33361
  1611
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1612
lemma posDivAlg_eqn:
haftmann@33361
  1613
     "0 < b ==>  
haftmann@33361
  1614
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
haftmann@33361
  1615
by (rule posDivAlg.simps [THEN trans], simp)
haftmann@33361
  1616
haftmann@33361
  1617
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
haftmann@33361
  1618
theorem posDivAlg_correct:
haftmann@33361
  1619
  assumes "0 \<le> a" and "0 < b"
haftmann@33361
  1620
  shows "divmod_int_rel a b (posDivAlg a b)"
wenzelm@41550
  1621
  using assms
wenzelm@41550
  1622
  apply (induct a b rule: posDivAlg.induct)
wenzelm@41550
  1623
  apply auto
wenzelm@41550
  1624
  apply (simp add: divmod_int_rel_def)
webertj@49962
  1625
  apply (subst posDivAlg_eqn, simp add: distrib_left)
wenzelm@41550
  1626
  apply (case_tac "a < b")
wenzelm@41550
  1627
  apply simp_all
wenzelm@41550
  1628
  apply (erule splitE)
haftmann@57514
  1629
  apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)
wenzelm@41550
  1630
  done
haftmann@33361
  1631
haftmann@33361
  1632
huffman@46551
  1633
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
haftmann@33361
  1634
haftmann@33361
  1635
text{*And positive divisors*}
haftmann@33361
  1636
haftmann@33361
  1637
declare negDivAlg.simps [simp del]
haftmann@33361
  1638
haftmann@33361
  1639
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1640
lemma negDivAlg_eqn:
haftmann@33361
  1641
     "0 < b ==>  
haftmann@33361
  1642
      negDivAlg a b =       
haftmann@33361
  1643
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
haftmann@33361
  1644
by (rule negDivAlg.simps [THEN trans], simp)
haftmann@33361
  1645
haftmann@33361
  1646
(*Correctness of negDivAlg: it computes quotients correctly
haftmann@33361
  1647
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
haftmann@33361
  1648
lemma negDivAlg_correct:
haftmann@33361
  1649
  assumes "a < 0" and "b > 0"
haftmann@33361
  1650
  shows "divmod_int_rel a b (negDivAlg a b)"
wenzelm@41550
  1651
  using assms
wenzelm@41550
  1652
  apply (induct a b rule: negDivAlg.induct)
wenzelm@41550
  1653
  apply (auto simp add: linorder_not_le)
wenzelm@41550
  1654
  apply (simp add: divmod_int_rel_def)
wenzelm@41550
  1655
  apply (subst negDivAlg_eqn, assumption)
wenzelm@41550
  1656
  apply (case_tac "a + b < (0\<Colon>int)")
wenzelm@41550
  1657
  apply simp_all
wenzelm@41550
  1658
  apply (erule splitE)
haftmann@57514
  1659
  apply (auto simp add: distrib_left Let_def ac_simps mult_2_right)
wenzelm@41550
  1660
  done
haftmann@33361
  1661
haftmann@33361
  1662
huffman@46551
  1663
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
haftmann@33361
  1664
haftmann@33361
  1665
(*the case a=0*)
huffman@47139
  1666
lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
haftmann@33361
  1667
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1668
haftmann@33361
  1669
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
haftmann@33361
  1670
by (subst posDivAlg.simps, auto)
haftmann@33361
  1671
huffman@47139
  1672
lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
huffman@47139
  1673
by (subst posDivAlg.simps, auto)
huffman@47139
  1674
haftmann@58410
  1675
lemma negDivAlg_minus1 [simp]: "negDivAlg (- 1) b = (- 1, b - 1)"
haftmann@33361
  1676
by (subst negDivAlg.simps, auto)
haftmann@33361
  1677
huffman@46560
  1678
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
huffman@47139
  1679
by (auto simp add: divmod_int_rel_def)
huffman@47139
  1680
huffman@47139
  1681
lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
huffman@47139
  1682
apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
haftmann@33361
  1683
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
haftmann@33361
  1684
                    posDivAlg_correct negDivAlg_correct)
haftmann@33361
  1685
huffman@47141
  1686
lemma divmod_int_unique:
huffman@47141
  1687
  assumes "divmod_int_rel a b qr" 
huffman@47141
  1688
  shows "divmod_int a b = qr"
huffman@47141
  1689
  using assms divmod_int_correct [of a b]
huffman@47141
  1690
  using unique_quotient [of a b] unique_remainder [of a b]
huffman@47141
  1691
  by (metis pair_collapse)
huffman@47141
  1692
huffman@47141
  1693
lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
huffman@47141
  1694
  using divmod_int_correct by (simp add: divmod_int_mod_div)
huffman@47141
  1695
huffman@47141
  1696
lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
huffman@47141
  1697
  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
huffman@47141
  1698
huffman@47141
  1699
lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
huffman@47141
  1700
  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
huffman@47141
  1701
huffman@47141
  1702
instance int :: ring_div
huffman@47141
  1703
proof
huffman@47141
  1704
  fix a b :: int
huffman@47141
  1705
  show "a div b * b + a mod b = a"
huffman@47141
  1706
    using divmod_int_rel_div_mod [of a b]
haftmann@57512
  1707
    unfolding divmod_int_rel_def by (simp add: mult.commute)
huffman@47141
  1708
next
huffman@47141
  1709
  fix a b c :: int
huffman@47141
  1710
  assume "b \<noteq> 0"
huffman@47141
  1711
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
huffman@47141
  1712
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1713
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141
  1714
  thus "(a + c * b) div b = c + a div b"
huffman@47141
  1715
    by (rule div_int_unique)
huffman@47141
  1716
next
huffman@47141
  1717
  fix a b c :: int
huffman@47141
  1718
  assume "c \<noteq> 0"
huffman@47141
  1719
  hence "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141
  1720
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141
  1721
    unfolding divmod_int_rel_def
huffman@47141
  1722
    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
huffman@47141
  1723
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
huffman@47141
  1724
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
huffman@47141
  1725
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
huffman@47141
  1726
    using divmod_int_rel_div_mod [of a b] .
huffman@47141
  1727
  thus "(c * a) div (c * b) = a div b"
huffman@47141
  1728
    by (rule div_int_unique)
huffman@47141
  1729
next
huffman@47141
  1730
  fix a :: int show "a div 0 = 0"
huffman@47141
  1731
    by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141
  1732
next
huffman@47141
  1733
  fix a :: int show "0 div a = 0"
huffman@47141
  1734
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141
  1735
qed
huffman@47141
  1736
haftmann@33361
  1737
text{*Basic laws about division and remainder*}
haftmann@33361
  1738
haftmann@33361
  1739
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
huffman@47141
  1740
  by (fact mod_div_equality2 [symmetric])
haftmann@33361
  1741
haftmann@33361
  1742
text {* Tool setup *}
haftmann@33361
  1743
huffman@47108
  1744
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
huffman@47108
  1745
lemmas add_0s = add_0_left add_0_right
huffman@47108
  1746
haftmann@33361
  1747
ML {*
wenzelm@43594
  1748
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550
  1749
(
haftmann@33361
  1750
  val div_name = @{const_name div};
haftmann@33361
  1751
  val mod_name = @{const_name mod};
haftmann@33361
  1752
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1753
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1754
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1755
huffman@47165
  1756
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361
  1757
haftmann@33361
  1758
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
haftmann@57514
  1759
    (@{thm diff_conv_add_uminus} :: @{thms add_0s} @ @{thms ac_simps}))
wenzelm@41550
  1760
)
haftmann@33361
  1761
*}
haftmann@33361
  1762
wenzelm@43594
  1763
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
wenzelm@43594
  1764
huffman@47141
  1765
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
huffman@47141
  1766
  using divmod_int_correct [of a b]
huffman@47141
  1767
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1768
wenzelm@45607
  1769
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1770
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1771
huffman@47141
  1772
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
huffman@47141
  1773
  using divmod_int_correct [of a b]
huffman@47141
  1774
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1775
wenzelm@45607
  1776
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1777
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1778
haftmann@33361
  1779
huffman@46551
  1780
subsubsection {* General Properties of div and mod *}
haftmann@33361
  1781
haftmann@33361
  1782
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1783
apply (rule div_int_unique)
haftmann@33361
  1784
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1785
done
haftmann@33361
  1786
haftmann@33361
  1787
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1788
apply (rule div_int_unique)
haftmann@33361
  1789
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1790
done
haftmann@33361
  1791
haftmann@33361
  1792
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1793
apply (rule div_int_unique)
haftmann@33361
  1794
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1795
done
haftmann@33361
  1796
haftmann@33361
  1797
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1798
haftmann@33361
  1799
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1800
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1801
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1802
done
haftmann@33361
  1803
haftmann@33361
  1804
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1805
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1806
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1807
done
haftmann@33361
  1808
haftmann@33361
  1809
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1810
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361
  1811
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1812
done
haftmann@33361
  1813
haftmann@33361
  1814
text{*There is no @{text mod_neg_pos_trivial}.*}
haftmann@33361
  1815
haftmann@33361
  1816
huffman@46551
  1817
subsubsection {* Laws for div and mod with Unary Minus *}
haftmann@33361
  1818
haftmann@33361
  1819
lemma zminus1_lemma:
huffman@47139
  1820
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
haftmann@33361
  1821
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@33361
  1822
                          if r=0 then 0 else b-r)"
haftmann@33361
  1823
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1824
haftmann@33361
  1825
haftmann@33361
  1826
lemma zdiv_zminus1_eq_if:
haftmann@33361
  1827
     "b \<noteq> (0::int)  
haftmann@33361
  1828
      ==> (-a) div b =  
haftmann@33361
  1829
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47140
  1830
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1831
haftmann@33361
  1832
lemma zmod_zminus1_eq_if:
haftmann@33361
  1833
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1834
apply (case_tac "b = 0", simp)
huffman@47140
  1835
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1836
done
haftmann@33361
  1837
haftmann@33361
  1838
lemma zmod_zminus1_not_zero:
haftmann@33361
  1839
  fixes k l :: int
haftmann@33361
  1840
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1841
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1842
haftmann@33361
  1843
lemma zdiv_zminus2_eq_if:
haftmann@33361
  1844
     "b \<noteq> (0::int)  
haftmann@33361
  1845
      ==> a div (-b) =  
haftmann@33361
  1846
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1847
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1848
haftmann@33361
  1849
lemma zmod_zminus2_eq_if:
haftmann@33361
  1850
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1851
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1852
haftmann@33361
  1853
lemma zmod_zminus2_not_zero:
haftmann@33361
  1854
  fixes k l :: int
haftmann@33361
  1855
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1856
  unfolding zmod_zminus2_eq_if by auto 
haftmann@33361
  1857
haftmann@33361
  1858
huffman@46551
  1859
subsubsection {* Computation of Division and Remainder *}
haftmann@33361
  1860
haftmann@33361
  1861
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@33361
  1862
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1863
haftmann@33361
  1864
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@33361
  1865
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1866
haftmann@33361
  1867
text{*a positive, b positive *}
haftmann@33361
  1868
haftmann@33361
  1869
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@33361
  1870
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1871
haftmann@33361
  1872
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@33361
  1873
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1874
haftmann@33361
  1875
text{*a negative, b positive *}
haftmann@33361
  1876
haftmann@33361
  1877
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@33361
  1878
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1879
haftmann@33361
  1880
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@33361
  1881
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1882
haftmann@33361
  1883
text{*a positive, b negative *}
haftmann@33361
  1884
haftmann@33361
  1885
lemma div_pos_neg:
huffman@46560
  1886
     "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1887
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1888
haftmann@33361
  1889
lemma mod_pos_neg:
huffman@46560
  1890
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1891
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1892
haftmann@33361
  1893
text{*a negative, b negative *}
haftmann@33361
  1894
haftmann@33361
  1895
lemma div_neg_neg:
huffman@46560
  1896
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1897
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1898
haftmann@33361
  1899
lemma mod_neg_neg:
huffman@46560
  1900
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1901
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1902
haftmann@33361
  1903
text {*Simplify expresions in which div and mod combine numerical constants*}
haftmann@33361
  1904
huffman@45530
  1905
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1906
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
huffman@45530
  1907
huffman@45530
  1908
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1909
  by (rule div_int_unique [of a b q r],
bulwahn@46552
  1910
    simp add: divmod_int_rel_def)
huffman@45530
  1911
huffman@45530
  1912
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1913
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1914
    simp add: divmod_int_rel_def)
huffman@45530
  1915
huffman@45530
  1916
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1917
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1918
    simp add: divmod_int_rel_def)
huffman@45530
  1919
haftmann@53069
  1920
text {*
haftmann@53069
  1921
  numeral simprocs -- high chance that these can be replaced
haftmann@53069
  1922
  by divmod algorithm from @{class semiring_numeral_div}
haftmann@53069
  1923
*}
haftmann@53069
  1924
haftmann@33361
  1925
ML {*
haftmann@33361
  1926
local
huffman@45530
  1927
  val mk_number = HOLogic.mk_number HOLogic.intT
huffman@45530
  1928
  val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1929
  val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1930
  val zero = @{term "0 :: int"}
huffman@45530
  1931
  val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1932
  val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
haftmann@54489
  1933
  val simps = @{thms arith_simps} @ @{thms rel_simps} @ [@{thm numeral_1_eq_1 [symmetric]}]
haftmann@54489
  1934
  fun prove ctxt goal = (writeln "prove"; Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
haftmann@54489
  1935
    (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps)))));
wenzelm@51717
  1936
  fun binary_proc proc ctxt ct =
haftmann@33361
  1937
    (case Thm.term_of ct of
haftmann@33361
  1938
      _ $ t $ u =>
haftmann@33361
  1939
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
wenzelm@51717
  1940
        SOME args => proc ctxt args
haftmann@33361
  1941
      | NONE => NONE)
haftmann@33361
  1942
    | _ => NONE);
haftmann@33361
  1943
in
huffman@45530
  1944
  fun divmod_proc posrule negrule =
huffman@45530
  1945
    binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
huffman@45530
  1946
      if b = 0 then NONE else let
huffman@45530
  1947
        val (q, r) = pairself mk_number (Integer.div_mod a b)
huffman@45530
  1948
        val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
huffman@45530
  1949
        val (goal2, goal3, rule) = if b > 0
huffman@45530
  1950
          then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
huffman@45530
  1951
          else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
huffman@45530
  1952
      in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
haftmann@33361
  1953
end
haftmann@33361
  1954
*}
haftmann@33361
  1955
huffman@47108
  1956
simproc_setup binary_int_div
huffman@47108
  1957
  ("numeral m div numeral n :: int" |
haftmann@54489
  1958
   "numeral m div - numeral n :: int" |
haftmann@54489
  1959
   "- numeral m div numeral n :: int" |
haftmann@54489
  1960
   "- numeral m div - numeral n :: int") =
huffman@45530
  1961
  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
haftmann@33361
  1962
huffman@47108
  1963
simproc_setup binary_int_mod
huffman@47108
  1964
  ("numeral m mod numeral n :: int" |
haftmann@54489
  1965
   "numeral m mod - numeral n :: int" |
haftmann@54489
  1966
   "- numeral m mod numeral n :: int" |
haftmann@54489
  1967
   "- numeral m mod - numeral n :: int") =
huffman@45530
  1968
  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
haftmann@33361
  1969
huffman@47108
  1970
lemmas posDivAlg_eqn_numeral [simp] =
huffman@47108
  1971
    posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
huffman@47108
  1972
huffman@47108
  1973
lemmas negDivAlg_eqn_numeral [simp] =
haftmann@54489
  1974
    negDivAlg_eqn [of "numeral v" "- numeral w", OF zero_less_numeral] for v w
haftmann@33361
  1975
haftmann@33361
  1976
haftmann@55172
  1977
text {* Special-case simplification: @{text "\<plusminus>1 div z"} and @{text "\<plusminus>1 mod z"} *}
haftmann@55172
  1978
haftmann@55172
  1979
lemma [simp]:
haftmann@55172
  1980
  shows div_one_bit0: "1 div numeral (Num.Bit0 v) = (0 :: int)"
haftmann@55172
  1981
    and mod_one_bit0: "1 mod numeral (Num.Bit0 v) = (1 :: int)"
wenzelm@55439
  1982
    and div_one_bit1: "1 div numeral (Num.Bit1 v) = (0 :: int)"
wenzelm@55439
  1983
    and mod_one_bit1: "1 mod numeral (Num.Bit1 v) = (1 :: int)"
wenzelm@55439
  1984
    and div_one_neg_numeral: "1 div - numeral v = (- 1 :: int)"
wenzelm@55439
  1985
    and mod_one_neg_numeral: "1 mod - numeral v = (1 :: int) - numeral v"
haftmann@55172
  1986
  by (simp_all del: arith_special
haftmann@55172
  1987
    add: div_pos_pos mod_pos_pos div_pos_neg mod_pos_neg posDivAlg_eqn)
wenzelm@55439
  1988
haftmann@55172
  1989
lemma [simp]:
haftmann@55172
  1990
  shows div_neg_one_numeral: "- 1 div numeral v = (- 1 :: int)"
haftmann@55172
  1991
    and mod_neg_one_numeral: "- 1 mod numeral v = numeral v - (1 :: int)"
haftmann@55172
  1992
    and div_neg_one_neg_bit0: "- 1 div - numeral (Num.Bit0 v) = (0 :: int)"
haftmann@55172
  1993
    and mod_neg_one_neb_bit0: "- 1 mod - numeral (Num.Bit0 v) = (- 1 :: int)"
haftmann@55172
  1994
    and div_neg_one_neg_bit1: "- 1 div - numeral (Num.Bit1 v) = (0 :: int)"
haftmann@55172
  1995
    and mod_neg_one_neb_bit1: "- 1 mod - numeral (Num.Bit1 v) = (- 1 :: int)"
haftmann@55172
  1996
  by (simp_all add: div_eq_minus1 zmod_minus1)
haftmann@33361
  1997
haftmann@33361
  1998
huffman@46551
  1999
subsubsection {* Monotonicity in the First Argument (Dividend) *}
haftmann@33361
  2000
haftmann@33361
  2001
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361
  2002
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2003
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  2004
apply (rule unique_quotient_lemma)
haftmann@33361
  2005
apply (erule subst)
haftmann@33361
  2006
apply (erule subst, simp_all)
haftmann@33361
  2007
done
haftmann@33361
  2008
haftmann@33361
  2009
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361
  2010
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2011
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  2012
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  2013
apply (erule subst)
haftmann@33361
  2014
apply (erule subst, simp_all)
haftmann@33361
  2015
done
haftmann@33361
  2016
haftmann@33361
  2017
huffman@46551
  2018
subsubsection {* Monotonicity in the Second Argument (Divisor) *}
haftmann@33361
  2019
haftmann@33361
  2020
lemma q_pos_lemma:
haftmann@33361
  2021
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  2022
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  2023
 apply (simp add: zero_less_mult_iff)
webertj@49962
  2024
apply (simp add: distrib_left)
haftmann@33361
  2025
done
haftmann@33361
  2026
haftmann@33361
  2027
lemma zdiv_mono2_lemma:
haftmann@33361
  2028
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
haftmann@33361
  2029
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
haftmann@33361
  2030
      ==> q \<le> (q'::int)"
haftmann@33361
  2031
apply (frule q_pos_lemma, assumption+) 
haftmann@33361
  2032
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  2033
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  2034
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  2035
 prefer 2 apply simp
webertj@49962
  2036
apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512
  2037
apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361
  2038
apply (rule mult_right_mono, auto)
haftmann@33361
  2039
done
haftmann@33361
  2040
haftmann@33361
  2041
lemma zdiv_mono2:
haftmann@33361
  2042
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  2043
apply (subgoal_tac "b \<noteq> 0")
haftmann@33361
  2044
 prefer 2 apply arith
haftmann@33361
  2045
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2046
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  2047
apply (rule zdiv_mono2_lemma)
haftmann@33361
  2048
apply (erule subst)
haftmann@33361
  2049
apply (erule subst, simp_all)
haftmann@33361
  2050
done
haftmann@33361
  2051
haftmann@33361
  2052
lemma q_neg_lemma:
haftmann@33361
  2053
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  2054
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  2055
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  2056
done
haftmann@33361
  2057
haftmann@33361
  2058
lemma zdiv_mono2_neg_lemma:
haftmann@33361
  2059
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
haftmann@33361
  2060
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
haftmann@33361
  2061
      ==> q' \<le> (q::int)"
haftmann@33361
  2062
apply (frule q_neg_lemma, assumption+) 
haftmann@33361
  2063
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  2064
 apply (simp add: mult_less_cancel_left)
webertj@49962
  2065
apply (simp add: distrib_left)
haftmann@33361
  2066
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  2067
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  2068
done
haftmann@33361
  2069
haftmann@33361
  2070
lemma zdiv_mono2_neg:
haftmann@33361
  2071
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361
  2072
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2073
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  2074
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  2075
apply (erule subst)
haftmann@33361
  2076
apply (erule subst, simp_all)
haftmann@33361
  2077
done
haftmann@33361
  2078
haftmann@33361
  2079
huffman@46551
  2080
subsubsection {* More Algebraic Laws for div and mod *}
haftmann@33361
  2081
haftmann@33361
  2082
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
haftmann@33361
  2083
haftmann@33361
  2084
lemma zmult1_lemma:
bulwahn@46552
  2085
     "[| divmod_int_rel b c (q, r) |]  
haftmann@33361
  2086
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@57514
  2087
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)
haftmann@33361
  2088
haftmann@33361
  2089
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  2090
apply (case_tac "c = 0", simp)
huffman@47140
  2091
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  2092
done
haftmann@33361
  2093
haftmann@33361
  2094
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@33361
  2095
haftmann@33361
  2096
lemma zadd1_lemma:
bulwahn@46552
  2097
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
haftmann@33361
  2098
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
webertj@49962
  2099
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
haftmann@33361
  2100
haftmann@33361
  2101
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  2102
lemma zdiv_zadd1_eq:
haftmann@33361
  2103
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  2104
apply (case_tac "c = 0", simp)
huffman@47140
  2105
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
haftmann@33361
  2106
done
haftmann@33361
  2107
haftmann@33361
  2108
lemma posDivAlg_div_mod:
haftmann@33361
  2109
  assumes "k \<ge> 0"
haftmann@33361
  2110
  and "l \<ge> 0"
haftmann@33361
  2111
  shows "posDivAlg k l = (k div l, k mod l)"
haftmann@33361
  2112
proof (cases "l = 0")
haftmann@33361
  2113
  case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@33361
  2114
next
haftmann@33361
  2115
  case False with assms posDivAlg_correct
haftmann@33361
  2116
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@33361
  2117
    by simp
huffman@47140
  2118
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  2119
  show ?thesis by simp
haftmann@33361
  2120
qed
haftmann@33361
  2121
haftmann@33361
  2122
lemma negDivAlg_div_mod:
haftmann@33361
  2123
  assumes "k < 0"
haftmann@33361
  2124
  and "l > 0"
haftmann@33361
  2125
  shows "negDivAlg k l = (k div l, k mod l)"
haftmann@33361
  2126
proof -
haftmann@33361
  2127
  from assms have "l \<noteq> 0" by simp
haftmann@33361
  2128
  from assms negDivAlg_correct
haftmann@33361
  2129
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@33361
  2130
    by simp
huffman@47140
  2131
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  2132
  show ?thesis by simp
haftmann@33361
  2133
qed
haftmann@33361
  2134
haftmann@33361
  2135
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  2136
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  2137
haftmann@33361
  2138
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  2139
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  2140
huffman@47108
  2141
lemma zmod_zdiv_equality':
huffman@47108
  2142
  "(m\<Colon>int) mod n = m - (m div n) * n"
huffman@47141
  2143
  using mod_div_equality [of m n] by arith
huffman@47108
  2144
haftmann@33361
  2145
blanchet@55085
  2146
subsubsection {* Proving  @{term "a div (b * c) = (a div b) div c"} *}
haftmann@33361
  2147
haftmann@33361
  2148
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  2149
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  2150
  to cause particular problems.*)
haftmann@33361
  2151
haftmann@33361
  2152
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
haftmann@33361
  2153
blanchet@55085
  2154
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361
  2155
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  2156
 apply (simp add: algebra_simps)
haftmann@33361
  2157
apply (rule order_le_less_trans)
haftmann@33361
  2158
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  2159
 apply (rule mult_left_mono_neg)
huffman@35216
  2160
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  2161
 apply (simp)
haftmann@33361
  2162
apply (simp)
haftmann@33361
  2163
done
haftmann@33361
  2164
haftmann@33361
  2165
lemma zmult2_lemma_aux2:
haftmann@33361
  2166
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  2167
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  2168
 apply arith
haftmann@33361
  2169
apply (simp add: mult_le_0_iff)
haftmann@33361
  2170
done
haftmann@33361
  2171
haftmann@33361
  2172
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  2173
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  2174
apply arith
haftmann@33361
  2175
apply (simp add: zero_le_mult_iff)
haftmann@33361
  2176
done
haftmann@33361
  2177
haftmann@33361
  2178
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  2179
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  2180
 apply (simp add: right_diff_distrib)
haftmann@33361
  2181
apply (rule order_less_le_trans)
haftmann@33361
  2182
 apply (erule mult_strict_right_mono)
haftmann@33361
  2183
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  2184
  apply simp
huffman@35216
  2185
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  2186
apply simp
haftmann@33361
  2187
done
haftmann@33361
  2188
bulwahn@46552
  2189
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
haftmann@33361
  2190
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@57514
  2191
by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff
webertj@49962
  2192
                   zero_less_mult_iff distrib_left [symmetric] 
huffman@47139
  2193
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
haftmann@33361
  2194
haftmann@53068
  2195
lemma zdiv_zmult2_eq:
haftmann@53068
  2196
  fixes a b c :: int
haftmann@53068
  2197
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
  2198
apply (case_tac "b = 0", simp)
haftmann@53068
  2199
apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  2200
done
haftmann@33361
  2201
haftmann@33361
  2202
lemma zmod_zmult2_eq:
haftmann@53068
  2203
  fixes a b c :: int
haftmann@53068
  2204
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
  2205
apply (case_tac "b = 0", simp)
haftmann@53068
  2206
apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  2207
done
haftmann@33361
  2208
huffman@47108
  2209
lemma div_pos_geq:
huffman@47108
  2210
  fixes k l :: int
huffman@47108
  2211
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2212
  shows "k div l = (k - l) div l + 1"
huffman@47108
  2213
proof -
huffman@47108
  2214
  have "k = (k - l) + l" by simp
huffman@47108
  2215
  then obtain j where k: "k = j + l" ..
huffman@47108
  2216
  with assms show ?thesis by simp
huffman@47108
  2217
qed
huffman@47108
  2218
huffman@47108
  2219
lemma mod_pos_geq:
huffman@47108
  2220
  fixes k l :: int
huffman@47108
  2221
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2222
  shows "k mod l = (k - l) mod l"
huffman@47108
  2223
proof -
huffman@47108
  2224
  have "k = (k - l) + l" by simp
huffman@47108
  2225
  then obtain j where k: "k = j + l" ..
huffman@47108
  2226
  with assms show ?thesis by simp
huffman@47108
  2227
qed
huffman@47108
  2228
haftmann@33361
  2229
huffman@46551
  2230
subsubsection {* Splitting Rules for div and mod *}
haftmann@33361
  2231
haftmann@33361
  2232
text{*The proofs of the two lemmas below are essentially identical*}
haftmann@33361
  2233
haftmann@33361
  2234
lemma split_pos_lemma:
haftmann@33361
  2235
 "0<k ==> 
haftmann@33361
  2236
    P(n div k :: int)(n mod k) = (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i j)"
haftmann@33361
  2237
apply (rule iffI, clarify)
haftmann@33361
  2238
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2239
 apply (subst mod_add_eq) 
haftmann@33361
  2240
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2241
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
haftmann@33361
  2242
txt{*converse direction*}
haftmann@33361
  2243
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2244
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2245
done
haftmann@33361
  2246
haftmann@33361
  2247
lemma split_neg_lemma:
haftmann@33361
  2248
 "k<0 ==>
haftmann@33361
  2249
    P(n div k :: int)(n mod k) = (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i j)"
haftmann@33361
  2250
apply (rule iffI, clarify)
haftmann@33361
  2251
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2252
 apply (subst mod_add_eq) 
haftmann@33361
  2253
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2254
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
haftmann@33361
  2255
txt{*converse direction*}
haftmann@33361
  2256
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2257
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2258
done
haftmann@33361
  2259
haftmann@33361
  2260
lemma split_zdiv:
haftmann@33361
  2261
 "P(n div k :: int) =
haftmann@33361
  2262
  ((k = 0 --> P 0) & 
haftmann@33361
  2263
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
haftmann@33361
  2264
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  2265
apply (case_tac "k=0", simp)
haftmann@33361
  2266
apply (simp only: linorder_neq_iff)
haftmann@33361
  2267
apply (erule disjE) 
haftmann@33361
  2268
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
haftmann@33361
  2269
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  2270
done
haftmann@33361
  2271
haftmann@33361
  2272
lemma split_zmod:
haftmann@33361
  2273
 "P(n mod k :: int) =
haftmann@33361
  2274
  ((k = 0 --> P n) & 
haftmann@33361
  2275
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
haftmann@33361
  2276
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  2277
apply (case_tac "k=0", simp)
haftmann@33361
  2278
apply (simp only: linorder_neq_iff)
haftmann@33361
  2279
apply (erule disjE) 
haftmann@33361
  2280
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
haftmann@33361
  2281
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  2282
done
haftmann@33361
  2283
webertj@33730
  2284
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
webertj@33730
  2285
  when these are applied to some constant that is of the form
huffman@47108
  2286
  @{term "numeral k"}: *}
huffman@47108
  2287
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  2288
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  2289
haftmann@33361
  2290
huffman@47166
  2291
subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
huffman@47166
  2292
huffman@47166
  2293
lemma pos_divmod_int_rel_mult_2:
huffman@47166
  2294
  assumes "0 \<le> b"
huffman@47166
  2295
  assumes "divmod_int_rel a b (q, r)"
huffman@47166
  2296
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166
  2297
  using assms unfolding divmod_int_rel_def by auto
huffman@47166
  2298
haftmann@54489
  2299
declaration {* K (Lin_Arith.add_simps @{thms uminus_numeral_One}) *}
haftmann@54489
  2300
huffman@47166
  2301
lemma neg_divmod_int_rel_mult_2:
huffman@47166
  2302
  assumes "b \<le> 0"
huffman@47166
  2303
  assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166
  2304
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166
  2305
  using assms unfolding divmod_int_rel_def by auto
haftmann@33361
  2306
haftmann@33361
  2307
text{*computing div by shifting *}
haftmann@33361
  2308
haftmann@33361
  2309
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
huffman@47166
  2310
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
huffman@47166
  2311
  by (rule div_int_unique)
haftmann@33361
  2312
boehmes@35815
  2313
lemma neg_zdiv_mult_2: 
boehmes@35815
  2314
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
huffman@47166
  2315
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
huffman@47166
  2316
  by (rule div_int_unique)
haftmann@33361
  2317
huffman@47108
  2318
(* FIXME: add rules for negative numerals *)
huffman@47108
  2319
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  2320
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  2321
    numeral v div (numeral w :: int)"
huffman@47108
  2322
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  2323
  by (rule div_mult_mult1, simp)
huffman@47108
  2324
huffman@47108
  2325
lemma zdiv_numeral_Bit1 [simp]:
huffman@47108
  2326
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
huffman@47108
  2327
    (numeral v div (numeral w :: int))"
huffman@47108
  2328
  unfolding numeral.simps
haftmann@57512
  2329
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  2330
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  2331
haftmann@33361
  2332
lemma pos_zmod_mult_2:
haftmann@33361
  2333
  fixes a b :: int
haftmann@33361
  2334
  assumes "0 \<le> a"
haftmann@33361
  2335
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
huffman@47166
  2336
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2337
  by (rule mod_int_unique)
haftmann@33361
  2338
haftmann@33361
  2339
lemma neg_zmod_mult_2:
haftmann@33361
  2340
  fixes a b :: int
haftmann@33361
  2341
  assumes "a \<le> 0"
haftmann@33361
  2342
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
huffman@47166
  2343
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2344
  by (rule mod_int_unique)
haftmann@33361
  2345
huffman@47108
  2346
(* FIXME: add rules for negative numerals *)
huffman@47108
  2347
lemma zmod_numeral_Bit0 [simp]:
huffman@47108
  2348
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
huffman@47108
  2349
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  2350
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  2351
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  2352
huffman@47108
  2353
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  2354
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2355
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  2356
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512
  2357
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  2358
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  2359
nipkow@39489
  2360
lemma zdiv_eq_0_iff:
nipkow@39489
  2361
 "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
nipkow@39489
  2362
proof
nipkow@39489
  2363
  assume ?L
nipkow@39489
  2364
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
nipkow@39489
  2365
  with `?L` show ?R by blast
nipkow@39489
  2366
next
nipkow@39489
  2367
  assume ?R thus ?L
nipkow@39489
  2368
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  2369
qed
nipkow@39489
  2370
nipkow@39489
  2371
huffman@46551
  2372
subsubsection {* Quotients of Signs *}
haftmann@33361
  2373
haftmann@33361
  2374
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2375
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2376
apply (rule order_trans)
haftmann@33361
  2377
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2378
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2379
done
haftmann@33361
  2380
haftmann@33361
  2381
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2382
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2383
haftmann@33361
  2384
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2385
by (drule zdiv_mono1, auto)
haftmann@33361
  2386
nipkow@33804
  2387
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
nipkow@33804
  2388
conditional upon the sign of @{text a} or @{text b}. There are many more.
nipkow@33804
  2389
They should all be simp rules unless that causes too much search. *}
nipkow@33804
  2390
haftmann@33361
  2391
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2392
apply auto
haftmann@33361
  2393
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2394
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2395
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2396
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2397
done
haftmann@33361
  2398
haftmann@33361
  2399
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2400
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2401
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2402
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2403
done
haftmann@33361
  2404
haftmann@33361
  2405
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2406
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2407
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2408
nipkow@39489
  2409
lemma pos_imp_zdiv_pos_iff:
nipkow@39489
  2410
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
nipkow@39489
  2411
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
nipkow@39489
  2412
by arith
nipkow@39489
  2413
haftmann@33361
  2414
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2415
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2416
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2417
nipkow@33804
  2418
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2419
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2420
apply rule
nipkow@33804
  2421
 apply rule
nipkow@33804
  2422
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2423
 apply(cases "b=0")apply simp
nipkow@33804
  2424
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2425
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2426
done
nipkow@33804
  2427
nipkow@39489
  2428
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2429
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2430
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2431
done
nipkow@39489
  2432
nipkow@39489
  2433
haftmann@33361
  2434
subsubsection {* The Divides Relation *}
haftmann@33361
  2435
huffman@47108
  2436
lemmas dvd_eq_mod_eq_0_numeral [simp] =
huffman@47108
  2437
  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
huffman@47108
  2438
huffman@47108
  2439
huffman@47108
  2440
subsubsection {* Further properties *}
huffman@47108
  2441
haftmann@33361
  2442
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
haftmann@33361
  2443
  using zmod_zdiv_equality[where a="m" and b="n"]
huffman@47142
  2444
  by (simp add: algebra_simps) (* FIXME: generalize *)
haftmann@33361
  2445
haftmann@33361
  2446
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
haftmann@33361
  2447
apply (subst split_div, auto)
haftmann@33361
  2448
apply (subst split_zdiv, auto)
haftmann@33361
  2449
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
haftmann@33361
  2450
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2451
done
haftmann@33361
  2452
haftmann@33361
  2453
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
haftmann@33361
  2454
apply (subst split_mod, auto)
haftmann@33361
  2455
apply (subst split_zmod, auto)
haftmann@33361
  2456
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
haftmann@33361
  2457
       in unique_remainder)
haftmann@33361
  2458
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2459
done
haftmann@33361
  2460
haftmann@33361
  2461
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
haftmann@33361
  2462
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2463
haftmann@33361
  2464
text{*Suggested by Matthias Daum*}
haftmann@33361
  2465
lemma int_power_div_base:
haftmann@33361
  2466
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2467
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2468
 apply (erule ssubst)
haftmann@33361
  2469
 apply (simp only: power_add)
haftmann@33361
  2470
 apply simp_all
haftmann@33361
  2471
done
haftmann@33361
  2472
haftmann@33361
  2473
text {* by Brian Huffman *}
haftmann@33361
  2474
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2475
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2476
haftmann@33361
  2477
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2478
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2479
haftmann@33361
  2480
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2481
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2482
haftmann@33361
  2483
lemmas zmod_simps =
haftmann@33361
  2484
  mod_add_left_eq  [symmetric]
haftmann@33361
  2485
  mod_add_right_eq [symmetric]
huffman@47142
  2486
  mod_mult_right_eq[symmetric]
haftmann@33361
  2487
  mod_mult_left_eq [symmetric]
huffman@47164
  2488
  power_mod
haftmann@33361
  2489
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2490
haftmann@33361
  2491
text {* Distributive laws for function @{text nat}. *}
haftmann@33361
  2492
haftmann@33361
  2493
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2494
apply (rule linorder_cases [of y 0])
haftmann@33361
  2495
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2496
apply simp
haftmann@33361
  2497
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2498
done
haftmann@33361
  2499
haftmann@33361
  2500
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2501
lemma nat_mod_distrib:
haftmann@33361
  2502
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2503
apply (case_tac "y = 0", simp)
haftmann@33361
  2504
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2505
done
haftmann@33361
  2506
haftmann@33361
  2507
text  {* transfer setup *}
haftmann@33361
  2508
haftmann@33361
  2509
lemma transfer_nat_int_functions:
haftmann@33361
  2510
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2511
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2512
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2513
haftmann@33361
  2514
lemma transfer_nat_int_function_closures:
haftmann@33361
  2515
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2516
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2517
  apply (cases "y = 0")
haftmann@33361
  2518
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2519
  apply (cases "y = 0")
haftmann@33361
  2520
  apply auto
haftmann@33361
  2521
done
haftmann@33361
  2522
haftmann@35644
  2523
declare transfer_morphism_nat_int [transfer add return: