src/HOL/Divides.thy
author hoelzl
Fri Feb 19 13:40:50 2016 +0100 (2016-02-19)
changeset 62378 85ed00c1fe7c
parent 61944 5d06ecfdb472
child 62390 842917225d56
permissions -rw-r--r--
generalize more theorems to support enat and ennreal
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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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section \<open>The division operators div and mod\<close>
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theory Divides
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imports Parity
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begin
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subsection \<open>Abstract division in commutative semirings.\<close>
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class div = dvd + divide +
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  fixes mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "mod" 70)
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class semiring_div = semidom + 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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subclass algebraic_semidom
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proof
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  fix b a
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  assume "b \<noteq> 0"
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  then show "a * b div b = a"
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    using div_mult_self1 [of b 0 a] by (simp add: ac_simps)
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qed simp
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lemma div_by_1:
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  "a div 1 = a"
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  by (fact divide_1)
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lemma div_mult_self1_is_id:
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  "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  by (fact nonzero_mult_divide_cancel_left)
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lemma div_mult_self2_is_id:
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  "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  by (fact nonzero_mult_divide_cancel_right)
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text \<open>@{const divide} and @{const mod}\<close>
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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 mod_mult_self1_is_0 [simp]:
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  "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]:
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  "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma mod_by_1 [simp]:
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  "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]:
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  "a mod a = 0"
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  using mod_mult_self2_is_0 [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_imp_mod_0 [simp]:
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  assumes "a dvd b"
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  shows "b mod a = 0"
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proof -
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  from assms 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 show "b mod a = 0" by simp
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qed
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lemma mod_eq_0_iff_dvd:
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  "a mod b = 0 \<longleftrightarrow> b dvd a"
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proof
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  assume "b dvd a"
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  then show "a mod b = 0" by simp
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next
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  assume "a mod b = 0"
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  with mod_div_equality [of a b] have "a div b * b = a" by simp
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  then have "a = b * (a div b)" by (simp add: ac_simps)
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  then show "b dvd a" ..
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qed
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lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
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  "a dvd b \<longleftrightarrow> b mod a = 0"
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  by (simp add: mod_eq_0_iff_dvd)
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lemma mod_div_trivial [simp]:
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  "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]:
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  "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_mod_imp_dvd:
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  assumes "k dvd m mod n" and "k dvd n"
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  shows "k dvd m"
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proof -
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  from assms have "k dvd (m div n) * n + m mod n"
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    by (simp only: dvd_add dvd_mult)
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  then show ?thesis by (simp add: mod_div_equality)
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qed
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text \<open>Addition respects modular equivalence.\<close>
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lemma mod_add_left_eq: \<comment> \<open>FIXME reorient\<close>
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  "(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: \<comment> \<open>FIXME reorient\<close>
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  "(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: \<comment> \<open>FIXME reorient\<close>
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  "(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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text \<open>Multiplication respects modular equivalence.\<close>
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lemma mod_mult_left_eq: \<comment> \<open>FIXME reorient\<close>
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  "(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: \<comment> \<open>FIXME reorient\<close>
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  "(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: \<comment> \<open>FIXME reorient\<close>
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  "(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 \<open>Exponentiation respects modular equivalence.\<close>
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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"
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  shows "a mod b mod c = a mod c"
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proof -
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  from \<open>c dvd b\<close> obtain k where "b = c * k"
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    by (rule dvdE)
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  have "a mod b mod c = a mod (c * k) mod c"
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    by (simp only: \<open>b = c * k\<close>)
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  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
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    by (simp only: mod_mult_self1)
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   322
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
haftmann@58786
   323
    by (simp only: ac_simps)
huffman@29404
   324
  also have "\<dots> = a mod c"
huffman@29404
   325
    by (simp only: mod_div_equality)
huffman@29404
   326
  finally show ?thesis .
huffman@29404
   327
qed
huffman@29404
   328
haftmann@30930
   329
lemma div_mult_mult2 [simp]:
haftmann@30930
   330
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@57512
   331
  by (drule div_mult_mult1) (simp add: mult.commute)
haftmann@30930
   332
haftmann@30930
   333
lemma div_mult_mult1_if [simp]:
haftmann@30930
   334
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   335
  by simp_all
nipkow@30476
   336
haftmann@30930
   337
lemma mod_mult_mult1:
haftmann@30930
   338
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   339
proof (cases "c = 0")
haftmann@30930
   340
  case True then show ?thesis by simp
haftmann@30930
   341
next
haftmann@30930
   342
  case False
haftmann@30930
   343
  from mod_div_equality
haftmann@30930
   344
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   345
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   346
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
lp15@60562
   347
  with mod_div_equality show ?thesis by simp
haftmann@30930
   348
qed
lp15@60562
   349
haftmann@30930
   350
lemma mod_mult_mult2:
haftmann@30930
   351
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@57512
   352
  using mod_mult_mult1 [of c a b] by (simp add: mult.commute)
haftmann@30930
   353
huffman@47159
   354
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159
   355
  by (fact mod_mult_mult2 [symmetric])
huffman@47159
   356
huffman@47159
   357
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159
   358
  by (fact mod_mult_mult1 [symmetric])
huffman@47159
   359
huffman@31662
   360
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   361
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   362
huffman@31662
   363
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   364
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   365
huffman@31661
   366
end
huffman@31661
   367
haftmann@59833
   368
class ring_div = comm_ring_1 + semiring_div
huffman@29405
   369
begin
huffman@29405
   370
haftmann@60353
   371
subclass idom_divide ..
haftmann@36634
   372
wenzelm@60758
   373
text \<open>Negation respects modular equivalence.\<close>
huffman@29405
   374
huffman@29405
   375
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   376
proof -
huffman@29405
   377
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405
   378
    by (simp only: mod_div_equality)
huffman@29405
   379
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
haftmann@57514
   380
    by (simp add: ac_simps)
huffman@29405
   381
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   382
    by (rule mod_mult_self1)
huffman@29405
   383
  finally show ?thesis .
huffman@29405
   384
qed
huffman@29405
   385
huffman@29405
   386
lemma mod_minus_cong:
huffman@29405
   387
  assumes "a mod b = a' mod b"
huffman@29405
   388
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   389
proof -
huffman@29405
   390
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   391
    unfolding assms ..
huffman@29405
   392
  thus ?thesis
huffman@29405
   393
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   394
qed
huffman@29405
   395
wenzelm@60758
   396
text \<open>Subtraction respects modular equivalence.\<close>
huffman@29405
   397
haftmann@54230
   398
lemma mod_diff_left_eq:
haftmann@54230
   399
  "(a - b) mod c = (a mod c - b) mod c"
haftmann@54230
   400
  using mod_add_cong [of a c "a mod c" "- b" "- b"] by simp
haftmann@54230
   401
haftmann@54230
   402
lemma mod_diff_right_eq:
haftmann@54230
   403
  "(a - b) mod c = (a - b mod c) mod c"
haftmann@54230
   404
  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
   405
haftmann@54230
   406
lemma mod_diff_eq:
haftmann@54230
   407
  "(a - b) mod c = (a mod c - b mod c) mod c"
haftmann@54230
   408
  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
   409
huffman@29405
   410
lemma mod_diff_cong:
huffman@29405
   411
  assumes "a mod c = a' mod c"
huffman@29405
   412
  assumes "b mod c = b' mod c"
huffman@29405
   413
  shows "(a - b) mod c = (a' - b') mod c"
haftmann@54230
   414
  using assms mod_add_cong [of a c a' "- b" "- b'"] mod_minus_cong [of b c "b'"] by simp
huffman@29405
   415
nipkow@30180
   416
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   417
apply (case_tac "y = 0") apply simp
nipkow@30180
   418
apply (auto simp add: dvd_def)
nipkow@30180
   419
apply (subgoal_tac "-(y * k) = y * - k")
thomas@57492
   420
 apply (simp only:)
nipkow@30180
   421
 apply (erule div_mult_self1_is_id)
nipkow@30180
   422
apply simp
nipkow@30180
   423
done
nipkow@30180
   424
nipkow@30180
   425
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   426
apply (case_tac "y = 0") apply simp
nipkow@30180
   427
apply (auto simp add: dvd_def)
nipkow@30180
   428
apply (subgoal_tac "y * k = -y * -k")
thomas@57492
   429
 apply (erule ssubst, rule div_mult_self1_is_id)
nipkow@30180
   430
 apply simp
nipkow@30180
   431
apply simp
nipkow@30180
   432
done
nipkow@30180
   433
haftmann@60867
   434
lemma div_diff [simp]:
haftmann@60867
   435
  "z dvd x \<Longrightarrow> z dvd y \<Longrightarrow> (x - y) div z = x div z - y div z"
haftmann@60867
   436
  using div_add [of _ _ "- y"] by (simp add: dvd_neg_div)
nipkow@59380
   437
huffman@47159
   438
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159
   439
  using div_mult_mult1 [of "- 1" a b]
huffman@47159
   440
  unfolding neg_equal_0_iff_equal by simp
huffman@47159
   441
huffman@47159
   442
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159
   443
  using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159
   444
huffman@47159
   445
lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159
   446
  using div_minus_minus [of "-a" b] by simp
huffman@47159
   447
huffman@47159
   448
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159
   449
  using mod_minus_minus [of "-a" b] by simp
huffman@47159
   450
huffman@47160
   451
lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160
   452
  using div_minus_right [of a 1] by simp
huffman@47160
   453
huffman@47160
   454
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160
   455
  using mod_minus_right [of a 1] by simp
huffman@47160
   456
lp15@60562
   457
lemma minus_mod_self2 [simp]:
haftmann@54221
   458
  "(a - b) mod b = a mod b"
haftmann@54221
   459
  by (simp add: mod_diff_right_eq)
haftmann@54221
   460
lp15@60562
   461
lemma minus_mod_self1 [simp]:
haftmann@54221
   462
  "(b - a) mod b = - a mod b"
haftmann@54230
   463
  using mod_add_self2 [of "- a" b] by simp
haftmann@54221
   464
huffman@29405
   465
end
huffman@29405
   466
haftmann@58778
   467
wenzelm@60758
   468
subsubsection \<open>Parity and division\<close>
haftmann@58778
   469
lp15@60562
   470
class semiring_div_parity = semiring_div + comm_semiring_1_cancel + numeral +
haftmann@54226
   471
  assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@58786
   472
  assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
haftmann@58710
   473
  assumes zero_not_eq_two: "0 \<noteq> 2"
haftmann@54226
   474
begin
haftmann@54226
   475
haftmann@54226
   476
lemma parity_cases [case_names even odd]:
haftmann@54226
   477
  assumes "a mod 2 = 0 \<Longrightarrow> P"
haftmann@54226
   478
  assumes "a mod 2 = 1 \<Longrightarrow> P"
haftmann@54226
   479
  shows P
haftmann@54226
   480
  using assms parity by blast
haftmann@54226
   481
haftmann@58786
   482
lemma one_div_two_eq_zero [simp]:
haftmann@58778
   483
  "1 div 2 = 0"
haftmann@58778
   484
proof (cases "2 = 0")
haftmann@58778
   485
  case True then show ?thesis by simp
haftmann@58778
   486
next
haftmann@58778
   487
  case False
haftmann@58778
   488
  from mod_div_equality have "1 div 2 * 2 + 1 mod 2 = 1" .
haftmann@58778
   489
  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
haftmann@58953
   490
  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
haftmann@58953
   491
  then have "1 div 2 = 0 \<or> 2 = 0" by simp
haftmann@58778
   492
  with False show ?thesis by auto
haftmann@58778
   493
qed
haftmann@58778
   494
haftmann@58786
   495
lemma not_mod_2_eq_0_eq_1 [simp]:
haftmann@58786
   496
  "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
haftmann@58786
   497
  by (cases a rule: parity_cases) simp_all
haftmann@58786
   498
haftmann@58786
   499
lemma not_mod_2_eq_1_eq_0 [simp]:
haftmann@58786
   500
  "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
haftmann@58786
   501
  by (cases a rule: parity_cases) simp_all
haftmann@58786
   502
haftmann@58778
   503
subclass semiring_parity
haftmann@58778
   504
proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
haftmann@58778
   505
  show "1 mod 2 = 1"
haftmann@58778
   506
    by (fact one_mod_two_eq_one)
haftmann@58778
   507
next
haftmann@58778
   508
  fix a b
haftmann@58778
   509
  assume "a mod 2 = 1"
haftmann@58778
   510
  moreover assume "b mod 2 = 1"
haftmann@58778
   511
  ultimately show "(a + b) mod 2 = 0"
haftmann@58778
   512
    using mod_add_eq [of a b 2] by simp
haftmann@58778
   513
next
haftmann@58778
   514
  fix a b
haftmann@58778
   515
  assume "(a * b) mod 2 = 0"
haftmann@58778
   516
  then have "(a mod 2) * (b mod 2) = 0"
haftmann@58778
   517
    by (cases "a mod 2 = 0") (simp_all add: mod_mult_eq [of a b 2])
haftmann@58778
   518
  then show "a mod 2 = 0 \<or> b mod 2 = 0"
haftmann@58778
   519
    by (rule divisors_zero)
haftmann@58778
   520
next
haftmann@58778
   521
  fix a
haftmann@58778
   522
  assume "a mod 2 = 1"
haftmann@58778
   523
  then have "a = a div 2 * 2 + 1" using mod_div_equality [of a 2] by simp
haftmann@58778
   524
  then show "\<exists>b. a = b + 1" ..
haftmann@58778
   525
qed
haftmann@58778
   526
haftmann@58778
   527
lemma even_iff_mod_2_eq_zero:
haftmann@58778
   528
  "even a \<longleftrightarrow> a mod 2 = 0"
haftmann@58778
   529
  by (fact dvd_eq_mod_eq_0)
haftmann@58778
   530
haftmann@58778
   531
lemma even_succ_div_two [simp]:
haftmann@58778
   532
  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
haftmann@58778
   533
  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
haftmann@58778
   534
haftmann@58778
   535
lemma odd_succ_div_two [simp]:
haftmann@58778
   536
  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
haftmann@58778
   537
  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
haftmann@58778
   538
haftmann@58778
   539
lemma even_two_times_div_two:
haftmann@58778
   540
  "even a \<Longrightarrow> 2 * (a div 2) = a"
haftmann@58778
   541
  by (fact dvd_mult_div_cancel)
haftmann@58778
   542
haftmann@58834
   543
lemma odd_two_times_div_two_succ [simp]:
haftmann@58778
   544
  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
haftmann@58778
   545
  using mod_div_equality2 [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
haftmann@60868
   546
 
haftmann@54226
   547
end
haftmann@54226
   548
haftmann@25942
   549
wenzelm@60758
   550
subsection \<open>Generic numeral division with a pragmatic type class\<close>
wenzelm@60758
   551
wenzelm@60758
   552
text \<open>
haftmann@53067
   553
  The following type class contains everything necessary to formulate
haftmann@53067
   554
  a division algorithm in ring structures with numerals, restricted
haftmann@53067
   555
  to its positive segments.  This is its primary motiviation, and it
haftmann@53067
   556
  could surely be formulated using a more fine-grained, more algebraic
haftmann@53067
   557
  and less technical class hierarchy.
wenzelm@60758
   558
\<close>
haftmann@53067
   559
lp15@60562
   560
class semiring_numeral_div = semiring_div + comm_semiring_1_cancel + linordered_semidom +
haftmann@59816
   561
  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067
   562
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067
   563
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067
   564
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067
   565
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067
   566
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067
   567
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067
   568
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067
   569
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@61275
   570
  fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@61275
   571
    and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@61275
   572
  assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
   573
    and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
haftmann@61275
   574
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
   575
    else (2 * q, r))"
wenzelm@61799
   576
    \<comment> \<open>These are conceptually definitions but force generated code
haftmann@61275
   577
    to be monomorphic wrt. particular instances of this class which
haftmann@61275
   578
    yields a significant speedup.\<close>
haftmann@61275
   579
haftmann@53067
   580
begin
haftmann@53067
   581
haftmann@59816
   582
lemma mult_div_cancel:
haftmann@59816
   583
  "b * (a div b) = a - a mod b"
haftmann@59816
   584
proof -
haftmann@59816
   585
  have "b * (a div b) + a mod b = a"
haftmann@59816
   586
    using mod_div_equality [of a b] by (simp add: ac_simps)
haftmann@59816
   587
  then have "b * (a div b) + a mod b - a mod b = a - a mod b"
haftmann@59816
   588
    by simp
haftmann@59816
   589
  then show ?thesis
haftmann@59816
   590
    by simp
haftmann@59816
   591
qed
haftmann@53067
   592
haftmann@54226
   593
subclass semiring_div_parity
haftmann@54226
   594
proof
haftmann@54226
   595
  fix a
haftmann@54226
   596
  show "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@54226
   597
  proof (rule ccontr)
haftmann@54226
   598
    assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@54226
   599
    then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@54226
   600
    have "0 < 2" by simp
haftmann@54226
   601
    with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
wenzelm@60758
   602
    with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
haftmann@54226
   603
    with discrete have "1 \<le> a mod 2" by simp
wenzelm@60758
   604
    with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
haftmann@54226
   605
    with discrete have "2 \<le> a mod 2" by simp
wenzelm@60758
   606
    with \<open>a mod 2 < 2\<close> show False by simp
haftmann@54226
   607
  qed
haftmann@58646
   608
next
haftmann@58646
   609
  show "1 mod 2 = 1"
haftmann@58646
   610
    by (rule mod_less) simp_all
haftmann@58710
   611
next
haftmann@58710
   612
  show "0 \<noteq> 2"
haftmann@58710
   613
    by simp
haftmann@53067
   614
qed
haftmann@53067
   615
haftmann@53067
   616
lemma divmod_digit_1:
haftmann@53067
   617
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067
   618
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067
   619
    and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067
   620
proof -
haftmann@53067
   621
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067
   622
    by (auto intro: trans)
wenzelm@60758
   623
  with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
haftmann@53067
   624
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
wenzelm@60758
   625
  with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
haftmann@53067
   626
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   627
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   628
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   629
  from assms w_exhaust have "w = 1"
haftmann@53067
   630
    by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067
   631
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067
   632
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   633
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
wenzelm@60758
   634
  with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067
   635
  then show ?P and ?Q
haftmann@60867
   636
    by (simp_all add: div mod add_implies_diff [symmetric])
haftmann@53067
   637
qed
haftmann@53067
   638
haftmann@53067
   639
lemma divmod_digit_0:
haftmann@53067
   640
  assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067
   641
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067
   642
    and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067
   643
proof -
haftmann@53067
   644
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   645
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   646
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   647
  moreover have "b \<le> a mod b + b"
haftmann@53067
   648
  proof -
wenzelm@60758
   649
    from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067
   650
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067
   651
    then show ?thesis by simp
haftmann@53067
   652
  qed
haftmann@53067
   653
  moreover note assms w_exhaust
haftmann@53067
   654
  ultimately have "w = 0" by auto
haftmann@53067
   655
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067
   656
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   657
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
wenzelm@60758
   658
  with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067
   659
  then show ?P and ?Q
haftmann@53067
   660
    by (simp_all add: div mod)
haftmann@53067
   661
qed
haftmann@53067
   662
haftmann@60867
   663
lemma fst_divmod:
haftmann@53067
   664
  "fst (divmod m n) = numeral m div numeral n"
haftmann@53067
   665
  by (simp add: divmod_def)
haftmann@53067
   666
haftmann@60867
   667
lemma snd_divmod:
haftmann@53067
   668
  "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067
   669
  by (simp add: divmod_def)
haftmann@53067
   670
wenzelm@60758
   671
text \<open>
haftmann@53067
   672
  This is a formulation of one step (referring to one digit position)
haftmann@53067
   673
  in school-method division: compare the dividend at the current
haftmann@53070
   674
  digit position with the remainder from previous division steps
haftmann@53067
   675
  and evaluate accordingly.
wenzelm@60758
   676
\<close>
haftmann@53067
   677
haftmann@61275
   678
lemma divmod_step_eq [simp]:
haftmann@53067
   679
  "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067
   680
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067
   681
  by (simp add: divmod_step_def)
haftmann@53067
   682
wenzelm@60758
   683
text \<open>
haftmann@53067
   684
  This is a formulation of school-method division.
haftmann@53067
   685
  If the divisor is smaller than the dividend, terminate.
haftmann@53067
   686
  If not, shift the dividend to the right until termination
haftmann@53067
   687
  occurs and then reiterate single division steps in the
haftmann@53067
   688
  opposite direction.
wenzelm@60758
   689
\<close>
haftmann@53067
   690
haftmann@60867
   691
lemma divmod_divmod_step:
haftmann@53067
   692
  "divmod m n = (if m < n then (0, numeral m)
haftmann@53067
   693
    else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067
   694
proof (cases "m < n")
haftmann@53067
   695
  case True then have "numeral m < numeral n" by simp
haftmann@53067
   696
  then show ?thesis
haftmann@60867
   697
    by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
haftmann@53067
   698
next
haftmann@53067
   699
  case False
haftmann@53067
   700
  have "divmod m n =
haftmann@53067
   701
    divmod_step n (numeral m div (2 * numeral n),
haftmann@53067
   702
      numeral m mod (2 * numeral n))"
haftmann@53067
   703
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067
   704
    case True
haftmann@60867
   705
    with divmod_step_eq
haftmann@53067
   706
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   707
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@60867
   708
        by simp
haftmann@53067
   709
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067
   710
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067
   711
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067
   712
      by simp_all
haftmann@53067
   713
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   714
  next
haftmann@53067
   715
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067
   716
      by (simp add: not_le)
haftmann@60867
   717
    with divmod_step_eq
haftmann@53067
   718
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   719
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@60867
   720
        by auto
haftmann@53067
   721
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067
   722
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067
   723
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067
   724
      by (simp_all only: zero_less_numeral)
haftmann@53067
   725
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   726
  qed
haftmann@53067
   727
  then have "divmod m n =
haftmann@53067
   728
    divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067
   729
      numeral m mod numeral (Num.Bit0 n))"
lp15@60562
   730
    by (simp only: numeral.simps distrib mult_1)
haftmann@53067
   731
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067
   732
    by (simp add: divmod_def)
haftmann@53067
   733
  with False show ?thesis by simp
haftmann@53067
   734
qed
haftmann@53067
   735
wenzelm@61799
   736
text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
haftmann@60867
   737
haftmann@61275
   738
lemma divmod_trivial [simp]:
haftmann@60867
   739
  "divmod Num.One Num.One = (numeral Num.One, 0)"
haftmann@60867
   740
  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
haftmann@60867
   741
  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
haftmann@60867
   742
  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
haftmann@60867
   743
  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
haftmann@60867
   744
  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
haftmann@60867
   745
haftmann@60867
   746
text \<open>Division by an even number is a right-shift\<close>
haftmann@58953
   747
haftmann@61275
   748
lemma divmod_cancel [simp]:
haftmann@53069
   749
  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
haftmann@53069
   750
  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
haftmann@53069
   751
proof -
haftmann@53069
   752
  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069
   753
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069
   754
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069
   755
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069
   756
  then show ?P and ?Q
haftmann@60867
   757
    by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@60867
   758
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
haftmann@60867
   759
      add.commute del: numeral_times_numeral)
haftmann@58953
   760
qed
haftmann@58953
   761
haftmann@60867
   762
text \<open>The really hard work\<close>
haftmann@60867
   763
haftmann@61275
   764
lemma divmod_steps [simp]:
haftmann@60867
   765
  "divmod (num.Bit0 m) (num.Bit1 n) =
haftmann@60867
   766
      (if m \<le> n then (0, numeral (num.Bit0 m))
haftmann@60867
   767
       else divmod_step (num.Bit1 n)
haftmann@60867
   768
             (divmod (num.Bit0 m)
haftmann@60867
   769
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   770
  "divmod (num.Bit1 m) (num.Bit1 n) =
haftmann@60867
   771
      (if m < n then (0, numeral (num.Bit1 m))
haftmann@60867
   772
       else divmod_step (num.Bit1 n)
haftmann@60867
   773
             (divmod (num.Bit1 m)
haftmann@60867
   774
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   775
  by (simp_all add: divmod_divmod_step)
haftmann@60867
   776
haftmann@61275
   777
lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
haftmann@61275
   778
wenzelm@60758
   779
text \<open>Special case: divisibility\<close>
haftmann@58953
   780
haftmann@58953
   781
definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
haftmann@58953
   782
where
haftmann@58953
   783
  "divides_aux qr \<longleftrightarrow> snd qr = 0"
haftmann@58953
   784
haftmann@58953
   785
lemma divides_aux_eq [simp]:
haftmann@58953
   786
  "divides_aux (q, r) \<longleftrightarrow> r = 0"
haftmann@58953
   787
  by (simp add: divides_aux_def)
haftmann@58953
   788
haftmann@58953
   789
lemma dvd_numeral_simp [simp]:
haftmann@58953
   790
  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
haftmann@58953
   791
  by (simp add: divmod_def mod_eq_0_iff_dvd)
haftmann@53069
   792
haftmann@60867
   793
text \<open>Generic computation of quotient and remainder\<close>  
haftmann@60867
   794
haftmann@60867
   795
lemma numeral_div_numeral [simp]: 
haftmann@60867
   796
  "numeral k div numeral l = fst (divmod k l)"
haftmann@60867
   797
  by (simp add: fst_divmod)
haftmann@60867
   798
haftmann@60867
   799
lemma numeral_mod_numeral [simp]: 
haftmann@60867
   800
  "numeral k mod numeral l = snd (divmod k l)"
haftmann@60867
   801
  by (simp add: snd_divmod)
haftmann@60867
   802
haftmann@60867
   803
lemma one_div_numeral [simp]:
haftmann@60867
   804
  "1 div numeral n = fst (divmod num.One n)"
haftmann@60867
   805
  by (simp add: fst_divmod)
haftmann@60867
   806
haftmann@60867
   807
lemma one_mod_numeral [simp]:
haftmann@60867
   808
  "1 mod numeral n = snd (divmod num.One n)"
haftmann@60867
   809
  by (simp add: snd_divmod)
haftmann@60867
   810
  
haftmann@53067
   811
end
haftmann@53067
   812
lp15@60562
   813
wenzelm@60758
   814
subsection \<open>Division on @{typ nat}\<close>
wenzelm@60758
   815
haftmann@61433
   816
context
haftmann@61433
   817
begin
haftmann@61433
   818
wenzelm@60758
   819
text \<open>
haftmann@60429
   820
  We define @{const divide} and @{const mod} on @{typ nat} by means
haftmann@26100
   821
  of a characteristic relation with two input arguments
wenzelm@61076
   822
  @{term "m::nat"}, @{term "n::nat"} and two output arguments
wenzelm@61076
   823
  @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
wenzelm@60758
   824
\<close>
haftmann@26100
   825
haftmann@33340
   826
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   827
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   828
    m = fst qr * n + snd qr \<and>
haftmann@30923
   829
      (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
   830
wenzelm@60758
   831
text \<open>@{const divmod_nat_rel} is total:\<close>
haftmann@26100
   832
haftmann@61433
   833
qualified lemma divmod_nat_rel_ex:
haftmann@33340
   834
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   835
proof (cases "n = 0")
haftmann@30923
   836
  case True  with that show thesis
haftmann@33340
   837
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   838
next
haftmann@26100
   839
  case False
haftmann@26100
   840
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   841
  proof (induct m)
wenzelm@60758
   842
    case 0 with \<open>n \<noteq> 0\<close>
wenzelm@61076
   843
    have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   844
    then show ?case by blast
haftmann@26100
   845
  next
haftmann@26100
   846
    case (Suc m) then obtain q' r'
haftmann@26100
   847
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   848
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   849
      case True
haftmann@26100
   850
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   851
      with True show ?thesis by blast
haftmann@26100
   852
    next
haftmann@26100
   853
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   854
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   855
      ultimately have "n = Suc r'" by auto
haftmann@26100
   856
      with m have "Suc m = Suc q' * n + 0" by simp
wenzelm@60758
   857
      with \<open>n \<noteq> 0\<close> show ?thesis by blast
haftmann@26100
   858
    qed
haftmann@26100
   859
  qed
haftmann@26100
   860
  with that show thesis
wenzelm@60758
   861
    using \<open>n \<noteq> 0\<close> by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   862
qed
haftmann@26100
   863
wenzelm@60758
   864
text \<open>@{const divmod_nat_rel} is injective:\<close>
haftmann@26100
   865
haftmann@61433
   866
qualified lemma divmod_nat_rel_unique:
haftmann@33340
   867
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   868
    and "divmod_nat_rel m n qr'"
haftmann@30923
   869
  shows "qr = qr'"
haftmann@26100
   870
proof (cases "n = 0")
haftmann@26100
   871
  case True with assms show ?thesis
haftmann@30923
   872
    by (cases qr, cases qr')
haftmann@33340
   873
      (simp add: divmod_nat_rel_def)
haftmann@26100
   874
next
haftmann@26100
   875
  case False
wenzelm@61076
   876
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q::nat)"
haftmann@26100
   877
  apply (rule leI)
haftmann@26100
   878
  apply (subst less_iff_Suc_add)
haftmann@26100
   879
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   880
  done
wenzelm@60758
   881
  from \<open>n \<noteq> 0\<close> assms have *: "fst qr = fst qr'"
haftmann@33340
   882
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
wenzelm@53374
   883
  with assms have "snd qr = snd qr'"
haftmann@33340
   884
    by (simp add: divmod_nat_rel_def)
wenzelm@53374
   885
  with * show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   886
qed
haftmann@26100
   887
wenzelm@60758
   888
text \<open>
haftmann@26100
   889
  We instantiate divisibility on the natural numbers by
haftmann@33340
   890
  means of @{const divmod_nat_rel}:
wenzelm@60758
   891
\<close>
haftmann@25942
   892
haftmann@61433
   893
qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767
   894
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   895
haftmann@61433
   896
qualified lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   897
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   898
proof -
haftmann@33340
   899
  from divmod_nat_rel_ex
haftmann@33340
   900
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   901
  then show ?thesis
haftmann@33340
   902
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   903
qed
haftmann@30923
   904
haftmann@61433
   905
qualified lemma divmod_nat_unique:
lp15@60562
   906
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   907
  shows "divmod_nat m n = qr"
haftmann@33340
   908
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   909
haftmann@61433
   910
qualified lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
haftmann@61433
   911
  by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
haftmann@61433
   912
haftmann@61433
   913
qualified lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
haftmann@61433
   914
  by (simp add: Divides.divmod_nat_unique divmod_nat_rel_def)
haftmann@61433
   915
haftmann@61433
   916
qualified lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
haftmann@61433
   917
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@61433
   918
haftmann@61433
   919
qualified lemma divmod_nat_step:
haftmann@61433
   920
  assumes "0 < n" and "n \<le> m"
haftmann@61433
   921
  shows "divmod_nat m n = apfst Suc (divmod_nat (m - n) n)"
haftmann@61433
   922
proof (rule divmod_nat_unique)
haftmann@61433
   923
  have "divmod_nat_rel (m - n) n (divmod_nat (m - n) n)"
haftmann@61433
   924
    by (fact divmod_nat_rel_divmod_nat)
haftmann@61433
   925
  then show "divmod_nat_rel m n (apfst Suc (divmod_nat (m - n) n))"
haftmann@61433
   926
    unfolding divmod_nat_rel_def using assms by auto
haftmann@61433
   927
qed
haftmann@61433
   928
haftmann@61433
   929
end
haftmann@61433
   930
  
haftmann@60429
   931
instantiation nat :: semiring_div
haftmann@60352
   932
begin
haftmann@60352
   933
haftmann@60352
   934
definition divide_nat where
haftmann@61433
   935
  div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
haftmann@60352
   936
haftmann@60352
   937
definition mod_nat where
haftmann@61433
   938
  "m mod n = snd (Divides.divmod_nat m n)"
huffman@46551
   939
huffman@46551
   940
lemma fst_divmod_nat [simp]:
haftmann@61433
   941
  "fst (Divides.divmod_nat m n) = m div n"
huffman@46551
   942
  by (simp add: div_nat_def)
huffman@46551
   943
huffman@46551
   944
lemma snd_divmod_nat [simp]:
haftmann@61433
   945
  "snd (Divides.divmod_nat m n) = m mod n"
huffman@46551
   946
  by (simp add: mod_nat_def)
huffman@46551
   947
haftmann@33340
   948
lemma divmod_nat_div_mod:
haftmann@61433
   949
  "Divides.divmod_nat m n = (m div n, m mod n)"
huffman@46551
   950
  by (simp add: prod_eq_iff)
haftmann@26100
   951
huffman@47135
   952
lemma div_nat_unique:
lp15@60562
   953
  assumes "divmod_nat_rel m n (q, r)"
haftmann@26100
   954
  shows "m div n = q"
haftmann@61433
   955
  using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   956
huffman@47135
   957
lemma mod_nat_unique:
lp15@60562
   958
  assumes "divmod_nat_rel m n (q, r)"
haftmann@26100
   959
  shows "m mod n = r"
haftmann@61433
   960
  using assms by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   961
haftmann@33340
   962
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
haftmann@61433
   963
  using Divides.divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
haftmann@25942
   964
wenzelm@60758
   965
text \<open>The ''recursion'' equations for @{const divide} and @{const mod}\<close>
haftmann@26100
   966
haftmann@26100
   967
lemma div_less [simp]:
haftmann@26100
   968
  fixes m n :: nat
haftmann@26100
   969
  assumes "m < n"
haftmann@26100
   970
  shows "m div n = 0"
haftmann@61433
   971
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   972
haftmann@26100
   973
lemma le_div_geq:
haftmann@26100
   974
  fixes m n :: nat
haftmann@26100
   975
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   976
  shows "m div n = Suc ((m - n) div n)"
haftmann@61433
   977
  using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   978
haftmann@26100
   979
lemma mod_less [simp]:
haftmann@26100
   980
  fixes m n :: nat
haftmann@26100
   981
  assumes "m < n"
haftmann@26100
   982
  shows "m mod n = m"
haftmann@61433
   983
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   984
haftmann@26100
   985
lemma le_mod_geq:
haftmann@26100
   986
  fixes m n :: nat
haftmann@26100
   987
  assumes "n \<le> m"
haftmann@26100
   988
  shows "m mod n = (m - n) mod n"
haftmann@61433
   989
  using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   990
huffman@47136
   991
instance proof
huffman@47136
   992
  fix m n :: nat
huffman@47136
   993
  show "m div n * n + m mod n = m"
huffman@47136
   994
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136
   995
next
huffman@47136
   996
  fix m n q :: nat
huffman@47136
   997
  assume "n \<noteq> 0"
huffman@47136
   998
  then show "(q + m * n) div n = m + q div n"
huffman@47136
   999
    by (induct m) (simp_all add: le_div_geq)
huffman@47136
  1000
next
huffman@47136
  1001
  fix m n q :: nat
huffman@47136
  1002
  assume "m \<noteq> 0"
huffman@47136
  1003
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136
  1004
    unfolding divmod_nat_rel_def
huffman@47136
  1005
    by (auto split: split_if_asm, simp_all add: algebra_simps)
huffman@47136
  1006
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136
  1007
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136
  1008
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136
  1009
next
huffman@47136
  1010
  fix n :: nat show "n div 0 = 0"
haftmann@61433
  1011
    by (simp add: div_nat_def Divides.divmod_nat_zero)
huffman@47136
  1012
next
huffman@47136
  1013
  fix n :: nat show "0 div n = 0"
haftmann@61433
  1014
    by (simp add: div_nat_def Divides.divmod_nat_zero_left)
haftmann@25942
  1015
qed
haftmann@26100
  1016
haftmann@25942
  1017
end
paulson@14267
  1018
haftmann@60685
  1019
instantiation nat :: normalization_semidom
haftmann@60685
  1020
begin
haftmann@60685
  1021
haftmann@60685
  1022
definition normalize_nat
haftmann@60685
  1023
  where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
haftmann@60685
  1024
haftmann@60685
  1025
definition unit_factor_nat
haftmann@60685
  1026
  where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
haftmann@60685
  1027
haftmann@60685
  1028
lemma unit_factor_simps [simp]:
haftmann@60685
  1029
  "unit_factor 0 = (0::nat)"
haftmann@60685
  1030
  "unit_factor (Suc n) = 1"
haftmann@60685
  1031
  by (simp_all add: unit_factor_nat_def)
haftmann@60685
  1032
haftmann@60685
  1033
instance
haftmann@60685
  1034
  by standard (simp_all add: unit_factor_nat_def)
haftmann@60685
  1035
  
haftmann@60685
  1036
end
haftmann@60685
  1037
haftmann@61433
  1038
lemma divmod_nat_if [code]:
haftmann@61433
  1039
  "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@61433
  1040
    let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
blanchet@55414
  1041
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
haftmann@33361
  1042
wenzelm@60758
  1043
text \<open>Simproc for cancelling @{const divide} and @{const mod}\<close>
haftmann@25942
  1044
wenzelm@51299
  1045
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299
  1046
wenzelm@60758
  1047
ML \<open>
wenzelm@43594
  1048
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
  1049
(
haftmann@60352
  1050
  val div_name = @{const_name divide};
haftmann@30934
  1051
  val mod_name = @{const_name mod};
haftmann@30934
  1052
  val mk_binop = HOLogic.mk_binop;
huffman@48561
  1053
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
  1054
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
  1055
  fun mk_sum [] = HOLogic.zero
huffman@48561
  1056
    | mk_sum [t] = t
huffman@48561
  1057
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
  1058
  fun dest_sum tm =
huffman@48561
  1059
    if HOLogic.is_zero tm then []
huffman@48561
  1060
    else
huffman@48561
  1061
      (case try HOLogic.dest_Suc tm of
huffman@48561
  1062
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
  1063
      | NONE =>
huffman@48561
  1064
          (case try dest_plus tm of
huffman@48561
  1065
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
  1066
          | NONE => [tm]));
haftmann@25942
  1067
haftmann@30934
  1068
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
  1069
haftmann@30934
  1070
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@57514
  1071
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms ac_simps}))
wenzelm@41550
  1072
)
wenzelm@60758
  1073
\<close>
wenzelm@60758
  1074
wenzelm@60758
  1075
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = \<open>K Cancel_Div_Mod_Nat.proc\<close>
wenzelm@60758
  1076
wenzelm@60758
  1077
wenzelm@60758
  1078
subsubsection \<open>Quotient\<close>
haftmann@26100
  1079
haftmann@26100
  1080
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
  1081
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
  1082
haftmann@26100
  1083
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
  1084
by (simp add: div_geq)
haftmann@26100
  1085
haftmann@26100
  1086
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
  1087
by simp
haftmann@26100
  1088
haftmann@26100
  1089
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
  1090
by simp
haftmann@26100
  1091
haftmann@53066
  1092
lemma div_positive:
haftmann@53066
  1093
  fixes m n :: nat
haftmann@53066
  1094
  assumes "n > 0"
haftmann@53066
  1095
  assumes "m \<ge> n"
haftmann@53066
  1096
  shows "m div n > 0"
haftmann@53066
  1097
proof -
wenzelm@60758
  1098
  from \<open>m \<ge> n\<close> obtain q where "m = n + q"
haftmann@53066
  1099
    by (auto simp add: le_iff_add)
wenzelm@60758
  1100
  with \<open>n > 0\<close> show ?thesis by simp
haftmann@53066
  1101
qed
haftmann@53066
  1102
hoelzl@59000
  1103
lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
hoelzl@59000
  1104
  by (metis div_less div_positive div_by_0 gr0I less_numeral_extra(3) not_less)
haftmann@25942
  1105
wenzelm@60758
  1106
subsubsection \<open>Remainder\<close>
haftmann@25942
  1107
haftmann@26100
  1108
lemma mod_less_divisor [simp]:
haftmann@26100
  1109
  fixes m n :: nat
haftmann@26100
  1110
  assumes "n > 0"
haftmann@26100
  1111
  shows "m mod n < (n::nat)"
haftmann@33340
  1112
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
  1113
haftmann@51173
  1114
lemma mod_Suc_le_divisor [simp]:
haftmann@51173
  1115
  "m mod Suc n \<le> n"
haftmann@51173
  1116
  using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173
  1117
haftmann@26100
  1118
lemma mod_less_eq_dividend [simp]:
haftmann@26100
  1119
  fixes m n :: nat
haftmann@26100
  1120
  shows "m mod n \<le> m"
haftmann@26100
  1121
proof (rule add_leD2)
haftmann@26100
  1122
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
  1123
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
  1124
qed
haftmann@26100
  1125
wenzelm@61076
  1126
lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
  1127
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
  1128
wenzelm@61076
  1129
lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
  1130
by (simp add: le_mod_geq)
haftmann@26100
  1131
paulson@14267
  1132
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
  1133
by (induct m) (simp_all add: mod_geq)
paulson@14267
  1134
paulson@14267
  1135
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
  1136
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
huffman@47138
  1137
  using mod_div_equality2 [of n m] by arith
paulson@14267
  1138
nipkow@15439
  1139
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
  1140
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
  1141
  apply simp
wenzelm@22718
  1142
  done
paulson@14267
  1143
wenzelm@60758
  1144
subsubsection \<open>Quotient and Remainder\<close>
paulson@14267
  1145
haftmann@33340
  1146
lemma divmod_nat_rel_mult1_eq:
bulwahn@46552
  1147
  "divmod_nat_rel b c (q, r)
haftmann@33340
  1148
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
  1149
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1150
haftmann@30923
  1151
lemma div_mult1_eq:
haftmann@30923
  1152
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135
  1153
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1154
haftmann@33340
  1155
lemma divmod_nat_rel_add1_eq:
bulwahn@46552
  1156
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340
  1157
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
  1158
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1159
paulson@14267
  1160
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
  1161
lemma div_add1_eq:
nipkow@25134
  1162
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135
  1163
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1164
haftmann@33340
  1165
lemma divmod_nat_rel_mult2_eq:
haftmann@60352
  1166
  assumes "divmod_nat_rel a b (q, r)"
haftmann@60352
  1167
  shows "divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@60352
  1168
proof -
lp15@60562
  1169
  { assume "r < b" and "0 < c"
haftmann@60352
  1170
    then have "b * (q mod c) + r < b * c"
haftmann@60352
  1171
      apply (cut_tac m = q and n = c in mod_less_divisor)
haftmann@60352
  1172
      apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
haftmann@60352
  1173
      apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
haftmann@60352
  1174
      apply (simp add: add_mult_distrib2)
haftmann@60352
  1175
      done
haftmann@60352
  1176
    then have "r + b * (q mod c) < b * c"
haftmann@60352
  1177
      by (simp add: ac_simps)
haftmann@60352
  1178
  } with assms show ?thesis
haftmann@60352
  1179
    by (auto simp add: divmod_nat_rel_def algebra_simps add_mult_distrib2 [symmetric])
haftmann@60352
  1180
qed
lp15@60562
  1181
blanchet@55085
  1182
lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
huffman@47135
  1183
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267
  1184
blanchet@55085
  1185
lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
haftmann@57512
  1186
by (auto simp add: mult.commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267
  1187
haftmann@61275
  1188
instantiation nat :: semiring_numeral_div
haftmann@61275
  1189
begin
haftmann@61275
  1190
haftmann@61275
  1191
definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
haftmann@61275
  1192
where
haftmann@61275
  1193
  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
  1194
haftmann@61275
  1195
definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
haftmann@61275
  1196
where
haftmann@61275
  1197
  "divmod_step_nat l qr = (let (q, r) = qr
haftmann@61275
  1198
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
  1199
    else (2 * q, r))"
haftmann@61275
  1200
haftmann@61275
  1201
instance
haftmann@61275
  1202
  by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
haftmann@61275
  1203
haftmann@61275
  1204
end
haftmann@61275
  1205
haftmann@61275
  1206
declare divmod_algorithm_code [where ?'a = nat, code]
haftmann@61275
  1207
  
paulson@14267
  1208
wenzelm@60758
  1209
subsubsection \<open>Further Facts about Quotient and Remainder\<close>
paulson@14267
  1210
haftmann@58786
  1211
lemma div_1 [simp]:
haftmann@58786
  1212
  "m div Suc 0 = m"
haftmann@58786
  1213
  using div_by_1 [of m] by simp
paulson@14267
  1214
paulson@14267
  1215
(* Monotonicity of div in first argument *)
haftmann@30923
  1216
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
  1217
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
  1218
apply (case_tac "k=0", simp)
paulson@15251
  1219
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
  1220
apply (case_tac "n<k")
paulson@14267
  1221
(* 1  case n<k *)
paulson@14267
  1222
apply simp
paulson@14267
  1223
(* 2  case n >= k *)
paulson@14267
  1224
apply (case_tac "m<k")
paulson@14267
  1225
(* 2.1  case m<k *)
paulson@14267
  1226
apply simp
paulson@14267
  1227
(* 2.2  case m>=k *)
nipkow@15439
  1228
apply (simp add: div_geq diff_le_mono)
paulson@14267
  1229
done
paulson@14267
  1230
paulson@14267
  1231
(* Antimonotonicity of div in second argument *)
paulson@14267
  1232
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
  1233
apply (subgoal_tac "0<n")
wenzelm@22718
  1234
 prefer 2 apply simp
paulson@15251
  1235
apply (induct_tac k rule: nat_less_induct)
paulson@14267
  1236
apply (rename_tac "k")
paulson@14267
  1237
apply (case_tac "k<n", simp)
paulson@14267
  1238
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
  1239
 prefer 2 apply simp
paulson@14267
  1240
apply (simp add: div_geq)
paulson@15251
  1241
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
  1242
 prefer 2
paulson@14267
  1243
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
  1244
apply (rule le_trans, simp)
nipkow@15439
  1245
apply (simp)
paulson@14267
  1246
done
paulson@14267
  1247
paulson@14267
  1248
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
  1249
apply (case_tac "n=0", simp)
paulson@14267
  1250
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
  1251
apply (rule div_le_mono2)
paulson@14267
  1252
apply (simp_all (no_asm_simp))
paulson@14267
  1253
done
paulson@14267
  1254
wenzelm@22718
  1255
(* Similar for "less than" *)
huffman@47138
  1256
lemma div_less_dividend [simp]:
huffman@47138
  1257
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
  1258
apply (induct m rule: nat_less_induct)
paulson@14267
  1259
apply (rename_tac "m")
paulson@14267
  1260
apply (case_tac "m<n", simp)
paulson@14267
  1261
apply (subgoal_tac "0<n")
wenzelm@22718
  1262
 prefer 2 apply simp
paulson@14267
  1263
apply (simp add: div_geq)
paulson@14267
  1264
apply (case_tac "n<m")
paulson@15251
  1265
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
  1266
  apply (rule impI less_trans_Suc)+
paulson@14267
  1267
apply assumption
nipkow@15439
  1268
  apply (simp_all)
paulson@14267
  1269
done
paulson@14267
  1270
wenzelm@60758
  1271
text\<open>A fact for the mutilated chess board\<close>
paulson@14267
  1272
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
  1273
apply (case_tac "n=0", simp)
paulson@15251
  1274
apply (induct "m" rule: nat_less_induct)
paulson@14267
  1275
apply (case_tac "Suc (na) <n")
paulson@14267
  1276
(* case Suc(na) < n *)
paulson@14267
  1277
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
  1278
(* case n \<le> Suc(na) *)
paulson@16796
  1279
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
  1280
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
  1281
done
paulson@14267
  1282
paulson@14267
  1283
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
  1284
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
  1285
wenzelm@22718
  1286
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
  1287
paulson@14267
  1288
(*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@57514
  1289
lemma mod_eqD:
haftmann@57514
  1290
  fixes m d r q :: nat
haftmann@57514
  1291
  assumes "m mod d = r"
haftmann@57514
  1292
  shows "\<exists>q. m = r + q * d"
haftmann@57514
  1293
proof -
haftmann@57514
  1294
  from mod_div_equality obtain q where "q * d + m mod d = m" by blast
haftmann@57514
  1295
  with assms have "m = r + q * d" by simp
haftmann@57514
  1296
  then show ?thesis ..
haftmann@57514
  1297
qed
paulson@14267
  1298
nipkow@13152
  1299
lemma split_div:
nipkow@13189
  1300
 "P(n div k :: nat) =
nipkow@13189
  1301
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
  1302
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1303
proof
nipkow@13189
  1304
  assume P: ?P
nipkow@13189
  1305
  show ?Q
nipkow@13189
  1306
  proof (cases)
nipkow@13189
  1307
    assume "k = 0"
haftmann@27651
  1308
    with P show ?Q by simp
nipkow@13189
  1309
  next
nipkow@13189
  1310
    assume not0: "k \<noteq> 0"
nipkow@13189
  1311
    thus ?Q
nipkow@13189
  1312
    proof (simp, intro allI impI)
nipkow@13189
  1313
      fix i j
nipkow@13189
  1314
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
  1315
      show "P i"
nipkow@13189
  1316
      proof (cases)
wenzelm@22718
  1317
        assume "i = 0"
wenzelm@22718
  1318
        with n j P show "P i" by simp
nipkow@13189
  1319
      next
wenzelm@22718
  1320
        assume "i \<noteq> 0"
haftmann@57514
  1321
        with not0 n j P show "P i" by(simp add:ac_simps)
nipkow@13189
  1322
      qed
nipkow@13189
  1323
    qed
nipkow@13189
  1324
  qed
nipkow@13189
  1325
next
nipkow@13189
  1326
  assume Q: ?Q
nipkow@13189
  1327
  show ?P
nipkow@13189
  1328
  proof (cases)
nipkow@13189
  1329
    assume "k = 0"
haftmann@27651
  1330
    with Q show ?P by simp
nipkow@13189
  1331
  next
nipkow@13189
  1332
    assume not0: "k \<noteq> 0"
nipkow@13189
  1333
    with Q have R: ?R by simp
nipkow@13189
  1334
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1335
    show ?P by simp
nipkow@13189
  1336
  qed
nipkow@13189
  1337
qed
nipkow@13189
  1338
berghofe@13882
  1339
lemma split_div_lemma:
haftmann@26100
  1340
  assumes "0 < n"
wenzelm@61076
  1341
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m::nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
  1342
proof
haftmann@26100
  1343
  assume ?rhs
haftmann@26100
  1344
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
  1345
  then have A: "n * q \<le> m" by simp
haftmann@26100
  1346
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
  1347
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
  1348
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
  1349
  with nq have "m < n + n * q" by simp
haftmann@26100
  1350
  then have B: "m < n * Suc q" by simp
haftmann@26100
  1351
  from A B show ?lhs ..
haftmann@26100
  1352
next
haftmann@26100
  1353
  assume P: ?lhs
haftmann@33340
  1354
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@57514
  1355
    unfolding divmod_nat_rel_def by (auto simp add: ac_simps)
haftmann@61433
  1356
  then have "m div n = q"
haftmann@61433
  1357
    by (rule div_nat_unique)
haftmann@30923
  1358
  then show ?rhs by simp
haftmann@26100
  1359
qed
berghofe@13882
  1360
berghofe@13882
  1361
theorem split_div':
berghofe@13882
  1362
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
  1363
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
haftmann@61433
  1364
  apply (cases "0 < n")
berghofe@13882
  1365
  apply (simp only: add: split_div_lemma)
haftmann@27651
  1366
  apply simp_all
berghofe@13882
  1367
  done
berghofe@13882
  1368
nipkow@13189
  1369
lemma split_mod:
nipkow@13189
  1370
 "P(n mod k :: nat) =
nipkow@13189
  1371
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
  1372
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1373
proof
nipkow@13189
  1374
  assume P: ?P
nipkow@13189
  1375
  show ?Q
nipkow@13189
  1376
  proof (cases)
nipkow@13189
  1377
    assume "k = 0"
haftmann@27651
  1378
    with P show ?Q by simp
nipkow@13189
  1379
  next
nipkow@13189
  1380
    assume not0: "k \<noteq> 0"
nipkow@13189
  1381
    thus ?Q
nipkow@13189
  1382
    proof (simp, intro allI impI)
nipkow@13189
  1383
      fix i j
nipkow@13189
  1384
      assume "n = k*i + j" "j < k"
haftmann@58786
  1385
      thus "P j" using not0 P by (simp add: ac_simps)
nipkow@13189
  1386
    qed
nipkow@13189
  1387
  qed
nipkow@13189
  1388
next
nipkow@13189
  1389
  assume Q: ?Q
nipkow@13189
  1390
  show ?P
nipkow@13189
  1391
  proof (cases)
nipkow@13189
  1392
    assume "k = 0"
haftmann@27651
  1393
    with Q show ?P by simp
nipkow@13189
  1394
  next
nipkow@13189
  1395
    assume not0: "k \<noteq> 0"
nipkow@13189
  1396
    with Q have R: ?R by simp
nipkow@13189
  1397
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1398
    show ?P by simp
nipkow@13189
  1399
  qed
nipkow@13189
  1400
qed
nipkow@13189
  1401
haftmann@60868
  1402
theorem mod_div_equality' [nitpick_unfold]: "(m::nat) mod n = m - (m div n) * n"
huffman@47138
  1403
  using mod_div_equality [of m n] by arith
huffman@47138
  1404
huffman@47138
  1405
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
huffman@47138
  1406
  using mod_div_equality [of m n] by arith
huffman@47138
  1407
(* FIXME: very similar to mult_div_cancel *)
haftmann@22800
  1408
noschinl@52398
  1409
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398
  1410
  apply rule
noschinl@52398
  1411
  apply (cases "b = 0")
noschinl@52398
  1412
  apply simp_all
noschinl@52398
  1413
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398
  1414
  done
noschinl@52398
  1415
haftmann@22800
  1416
wenzelm@60758
  1417
subsubsection \<open>An ``induction'' law for modulus arithmetic.\<close>
paulson@14640
  1418
paulson@14640
  1419
lemma mod_induct_0:
paulson@14640
  1420
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1421
  and base: "P i" and i: "i<p"
paulson@14640
  1422
  shows "P 0"
paulson@14640
  1423
proof (rule ccontr)
paulson@14640
  1424
  assume contra: "\<not>(P 0)"
paulson@14640
  1425
  from i have p: "0<p" by simp
paulson@14640
  1426
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1427
  proof
paulson@14640
  1428
    fix k
paulson@14640
  1429
    show "?A k"
paulson@14640
  1430
    proof (induct k)
wenzelm@61799
  1431
      show "?A 0" by simp  \<comment> "by contradiction"
paulson@14640
  1432
    next
paulson@14640
  1433
      fix n
paulson@14640
  1434
      assume ih: "?A n"
paulson@14640
  1435
      show "?A (Suc n)"
paulson@14640
  1436
      proof (clarsimp)
wenzelm@22718
  1437
        assume y: "P (p - Suc n)"
wenzelm@22718
  1438
        have n: "Suc n < p"
wenzelm@22718
  1439
        proof (rule ccontr)
wenzelm@22718
  1440
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1441
          hence "p - Suc n = 0"
wenzelm@22718
  1442
            by simp
wenzelm@22718
  1443
          with y contra show "False"
wenzelm@22718
  1444
            by simp
wenzelm@22718
  1445
        qed
wenzelm@22718
  1446
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1447
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1448
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1449
          by blast
wenzelm@22718
  1450
        show "False"
wenzelm@22718
  1451
        proof (cases "n=0")
wenzelm@22718
  1452
          case True
wenzelm@22718
  1453
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1454
        next
wenzelm@22718
  1455
          case False
wenzelm@22718
  1456
          with p have "p-n < p" by arith
wenzelm@22718
  1457
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1458
        qed
paulson@14640
  1459
      qed
paulson@14640
  1460
    qed
paulson@14640
  1461
  qed
paulson@14640
  1462
  moreover
paulson@14640
  1463
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1464
    by (blast dest: less_imp_add_positive)
paulson@14640
  1465
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1466
  moreover
paulson@14640
  1467
  note base
paulson@14640
  1468
  ultimately
paulson@14640
  1469
  show "False" by blast
paulson@14640
  1470
qed
paulson@14640
  1471
paulson@14640
  1472
lemma mod_induct:
paulson@14640
  1473
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1474
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1475
  shows "P j"
paulson@14640
  1476
proof -
paulson@14640
  1477
  have "\<forall>j<p. P j"
paulson@14640
  1478
  proof
paulson@14640
  1479
    fix j
paulson@14640
  1480
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1481
    proof (induct j)
paulson@14640
  1482
      from step base i show "?A 0"
wenzelm@22718
  1483
        by (auto elim: mod_induct_0)
paulson@14640
  1484
    next
paulson@14640
  1485
      fix k
paulson@14640
  1486
      assume ih: "?A k"
paulson@14640
  1487
      show "?A (Suc k)"
paulson@14640
  1488
      proof
wenzelm@22718
  1489
        assume suc: "Suc k < p"
wenzelm@22718
  1490
        hence k: "k<p" by simp
wenzelm@22718
  1491
        with ih have "P k" ..
wenzelm@22718
  1492
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1493
          by blast
wenzelm@22718
  1494
        moreover
wenzelm@22718
  1495
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1496
          by simp
wenzelm@22718
  1497
        ultimately
wenzelm@22718
  1498
        show "P (Suc k)" by simp
paulson@14640
  1499
      qed
paulson@14640
  1500
    qed
paulson@14640
  1501
  qed
paulson@14640
  1502
  with j show ?thesis by blast
paulson@14640
  1503
qed
paulson@14640
  1504
haftmann@33296
  1505
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1506
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1507
huffman@47138
  1508
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1509
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1510
haftmann@33296
  1511
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1512
by (simp add: mult_2 [symmetric])
haftmann@33296
  1513
wenzelm@61076
  1514
lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1515
proof -
boehmes@35815
  1516
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1517
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1518
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1519
  then show ?thesis by auto
haftmann@33296
  1520
qed
haftmann@33296
  1521
wenzelm@60758
  1522
text\<open>These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1523
    at least three Sucs, since two and fewer are rewritten back to Suc again!
wenzelm@60758
  1524
    We already have some rules to simplify operands smaller than 3.\<close>
haftmann@33296
  1525
haftmann@33296
  1526
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1527
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1528
haftmann@33296
  1529
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1530
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1531
haftmann@33296
  1532
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1533
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1534
haftmann@33296
  1535
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1536
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1537
huffman@47108
  1538
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1539
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1540
lp15@60562
  1541
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
haftmann@33361
  1542
apply (induct "m")
haftmann@33361
  1543
apply (simp_all add: mod_Suc)
haftmann@33361
  1544
done
haftmann@33361
  1545
huffman@47108
  1546
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1547
haftmann@60868
  1548
lemma mod_greater_zero_iff_not_dvd:
haftmann@60868
  1549
  fixes m n :: nat
haftmann@60868
  1550
  shows "m mod n > 0 \<longleftrightarrow> \<not> n dvd m"
haftmann@60868
  1551
  by (simp add: dvd_eq_mod_eq_0)
haftmann@60868
  1552
huffman@47138
  1553
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1554
by (simp add: div_le_mono)
haftmann@33361
  1555
haftmann@33361
  1556
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1557
by (cases n) simp_all
haftmann@33361
  1558
boehmes@35815
  1559
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1560
proof -
boehmes@35815
  1561
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
lp15@60562
  1562
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
boehmes@35815
  1563
qed
haftmann@33361
  1564
haftmann@33361
  1565
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1566
proof -
haftmann@33361
  1567
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
lp15@60562
  1568
  also have "... = Suc m mod n" by (rule mod_mult_self3)
haftmann@33361
  1569
  finally show ?thesis .
haftmann@33361
  1570
qed
haftmann@33361
  1571
haftmann@33361
  1572
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
lp15@60562
  1573
apply (subst mod_Suc [of m])
lp15@60562
  1574
apply (subst mod_Suc [of "m mod n"], simp)
haftmann@33361
  1575
done
haftmann@33361
  1576
huffman@47108
  1577
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1578
  fixes n :: nat
huffman@47108
  1579
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
haftmann@58786
  1580
  by (fact not_mod_2_eq_0_eq_1)
lp15@60562
  1581
haftmann@58778
  1582
lemma even_Suc_div_two [simp]:
haftmann@58778
  1583
  "even n \<Longrightarrow> Suc n div 2 = n div 2"
haftmann@58778
  1584
  using even_succ_div_two [of n] by simp
lp15@60562
  1585
haftmann@58778
  1586
lemma odd_Suc_div_two [simp]:
haftmann@58778
  1587
  "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
haftmann@58778
  1588
  using odd_succ_div_two [of n] by simp
haftmann@58778
  1589
haftmann@58834
  1590
lemma odd_two_times_div_two_nat [simp]:
haftmann@60352
  1591
  assumes "odd n"
haftmann@60352
  1592
  shows "2 * (n div 2) = n - (1 :: nat)"
haftmann@60352
  1593
proof -
haftmann@60352
  1594
  from assms have "2 * (n div 2) + 1 = n"
haftmann@60352
  1595
    by (rule odd_two_times_div_two_succ)
haftmann@60352
  1596
  then have "Suc (2 * (n div 2)) - 1 = n - 1"
haftmann@60352
  1597
    by simp
haftmann@60352
  1598
  then show ?thesis
haftmann@60352
  1599
    by simp
haftmann@60352
  1600
qed
haftmann@58778
  1601
haftmann@58778
  1602
lemma parity_induct [case_names zero even odd]:
haftmann@58778
  1603
  assumes zero: "P 0"
haftmann@58778
  1604
  assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
haftmann@58778
  1605
  assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
haftmann@58778
  1606
  shows "P n"
haftmann@58778
  1607
proof (induct n rule: less_induct)
haftmann@58778
  1608
  case (less n)
haftmann@58778
  1609
  show "P n"
haftmann@58778
  1610
  proof (cases "n = 0")
haftmann@58778
  1611
    case True with zero show ?thesis by simp
haftmann@58778
  1612
  next
haftmann@58778
  1613
    case False
haftmann@58778
  1614
    with less have hyp: "P (n div 2)" by simp
haftmann@58778
  1615
    show ?thesis
haftmann@58778
  1616
    proof (cases "even n")
haftmann@58778
  1617
      case True
haftmann@58778
  1618
      with hyp even [of "n div 2"] show ?thesis
haftmann@58834
  1619
        by simp
haftmann@58778
  1620
    next
haftmann@58778
  1621
      case False
lp15@60562
  1622
      with hyp odd [of "n div 2"] show ?thesis
haftmann@58834
  1623
        by simp
haftmann@58778
  1624
    qed
haftmann@58778
  1625
  qed
haftmann@58778
  1626
qed
haftmann@58778
  1627
haftmann@60868
  1628
lemma Suc_0_div_numeral [simp]:
haftmann@60868
  1629
  fixes k l :: num
haftmann@60868
  1630
  shows "Suc 0 div numeral k = fst (divmod Num.One k)"
haftmann@60868
  1631
  by (simp_all add: fst_divmod)
haftmann@60868
  1632
haftmann@60868
  1633
lemma Suc_0_mod_numeral [simp]:
haftmann@60868
  1634
  fixes k l :: num
haftmann@60868
  1635
  shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
haftmann@60868
  1636
  by (simp_all add: snd_divmod)
haftmann@60868
  1637
haftmann@33361
  1638
wenzelm@60758
  1639
subsection \<open>Division on @{typ int}\<close>
haftmann@33361
  1640
wenzelm@61799
  1641
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" \<comment> \<open>definition of quotient and remainder\<close>
haftmann@60868
  1642
  where "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139
  1643
    (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
  1644
haftmann@33361
  1645
lemma unique_quotient_lemma:
haftmann@60868
  1646
  "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
haftmann@33361
  1647
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1648
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1649
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1650
apply (erule_tac [2] order_le_less_trans)
webertj@49962
  1651
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1652
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
  1653
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1654
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1655
done
haftmann@33361
  1656
haftmann@33361
  1657
lemma unique_quotient_lemma_neg:
haftmann@60868
  1658
  "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
haftmann@60868
  1659
  by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
haftmann@33361
  1660
haftmann@33361
  1661
lemma unique_quotient:
haftmann@60868
  1662
  "divmod_int_rel a b (q, r) \<Longrightarrow> divmod_int_rel a b (q', r') \<Longrightarrow> q = q'"
haftmann@33361
  1663
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361
  1664
apply (blast intro: order_antisym
lp15@60562
  1665
             dest: order_eq_refl [THEN unique_quotient_lemma]
haftmann@33361
  1666
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1667
done
haftmann@33361
  1668
haftmann@33361
  1669
lemma unique_remainder:
haftmann@60868
  1670
  "divmod_int_rel a b (q, r) \<Longrightarrow> divmod_int_rel a b (q', r') \<Longrightarrow> r = r'"
haftmann@33361
  1671
apply (subgoal_tac "q = q'")
haftmann@33361
  1672
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1673
apply (blast intro: unique_quotient)
haftmann@33361
  1674
done
haftmann@33361
  1675
haftmann@60868
  1676
instantiation int :: Divides.div
haftmann@60868
  1677
begin
haftmann@60868
  1678
haftmann@60868
  1679
definition divide_int
haftmann@60868
  1680
  where "k div l = (if l = 0 \<or> k = 0 then 0
haftmann@60868
  1681
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
  1682
      then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
  1683
      else
haftmann@60868
  1684
        if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
  1685
        else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
haftmann@60868
  1686
haftmann@60868
  1687
definition mod_int
haftmann@60868
  1688
  where "k mod l = (if l = 0 then k else if l dvd k then 0
haftmann@60868
  1689
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
  1690
      then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
haftmann@60868
  1691
      else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
haftmann@60868
  1692
haftmann@60868
  1693
instance ..      
haftmann@60868
  1694
haftmann@60868
  1695
end
haftmann@60868
  1696
haftmann@60868
  1697
lemma divmod_int_rel:
haftmann@60868
  1698
  "divmod_int_rel k l (k div l, k mod l)"
lp15@61649
  1699
  unfolding divmod_int_rel_def divide_int_def mod_int_def
haftmann@60868
  1700
  apply (cases k rule: int_cases3)
lp15@61649
  1701
  apply (simp add: mod_greater_zero_iff_not_dvd not_le algebra_simps)
haftmann@60868
  1702
  apply (cases l rule: int_cases3)
lp15@61649
  1703
  apply (simp add: mod_greater_zero_iff_not_dvd not_le algebra_simps)
lp15@61649
  1704
  apply (simp_all del: of_nat_add of_nat_mult add: mod_greater_zero_iff_not_dvd not_le algebra_simps int_dvd_iff of_nat_add [symmetric] of_nat_mult [symmetric])
haftmann@60868
  1705
  apply (cases l rule: int_cases3)
lp15@61649
  1706
  apply (simp_all del: of_nat_add of_nat_mult add: not_le algebra_simps int_dvd_iff of_nat_add [symmetric] of_nat_mult [symmetric])
wenzelm@41550
  1707
  done
haftmann@33361
  1708
haftmann@60868
  1709
instantiation int :: ring_div
haftmann@60868
  1710
begin
haftmann@60868
  1711
haftmann@60868
  1712
subsubsection \<open>Uniqueness and Monotonicity of Quotients and Remainders\<close>
haftmann@33361
  1713
huffman@47141
  1714
lemma divmod_int_unique:
haftmann@60868
  1715
  assumes "divmod_int_rel k l (q, r)"
haftmann@60868
  1716
  shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
haftmann@60868
  1717
  using assms divmod_int_rel [of k l]
haftmann@60868
  1718
  using unique_quotient [of k l] unique_remainder [of k l]
haftmann@60868
  1719
  by auto
haftmann@60868
  1720
  
haftmann@60429
  1721
instance
huffman@47141
  1722
proof
huffman@47141
  1723
  fix a b :: int
huffman@47141
  1724
  show "a div b * b + a mod b = a"
haftmann@60868
  1725
    using divmod_int_rel [of a b]
haftmann@57512
  1726
    unfolding divmod_int_rel_def by (simp add: mult.commute)
huffman@47141
  1727
next
huffman@47141
  1728
  fix a b c :: int
huffman@47141
  1729
  assume "b \<noteq> 0"
huffman@47141
  1730
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
haftmann@60868
  1731
    using divmod_int_rel [of a b]
huffman@47141
  1732
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141
  1733
  thus "(a + c * b) div b = c + a div b"
huffman@47141
  1734
    by (rule div_int_unique)
huffman@47141
  1735
next
huffman@47141
  1736
  fix a b c :: int
huffman@47141
  1737
  assume "c \<noteq> 0"
huffman@47141
  1738
  hence "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141
  1739
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141
  1740
    unfolding divmod_int_rel_def
huffman@47141
  1741
    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
huffman@47141
  1742
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
huffman@47141
  1743
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
huffman@47141
  1744
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
haftmann@60868
  1745
    using divmod_int_rel [of a b] .
huffman@47141
  1746
  thus "(c * a) div (c * b) = a div b"
huffman@47141
  1747
    by (rule div_int_unique)
huffman@47141
  1748
next
huffman@47141
  1749
  fix a :: int show "a div 0 = 0"
huffman@47141
  1750
    by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141
  1751
next
huffman@47141
  1752
  fix a :: int show "0 div a = 0"
huffman@47141
  1753
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141
  1754
qed
huffman@47141
  1755
haftmann@60429
  1756
end
haftmann@60429
  1757
haftmann@60517
  1758
lemma is_unit_int:
haftmann@60517
  1759
  "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
haftmann@60517
  1760
  by auto
haftmann@60517
  1761
haftmann@60685
  1762
instantiation int :: normalization_semidom
haftmann@60685
  1763
begin
haftmann@60685
  1764
haftmann@60685
  1765
definition normalize_int
haftmann@60685
  1766
  where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
haftmann@60685
  1767
haftmann@60685
  1768
definition unit_factor_int
haftmann@60685
  1769
  where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
haftmann@60685
  1770
haftmann@60685
  1771
instance
haftmann@60685
  1772
proof
haftmann@60685
  1773
  fix k :: int
haftmann@60685
  1774
  assume "k \<noteq> 0"
haftmann@60685
  1775
  then have "\<bar>sgn k\<bar> = 1"
haftmann@60685
  1776
    by (cases "0::int" k rule: linorder_cases) simp_all
haftmann@60685
  1777
  then show "is_unit (unit_factor k)"
haftmann@60685
  1778
    by simp
haftmann@60685
  1779
qed (simp_all add: sgn_times mult_sgn_abs)
haftmann@60685
  1780
  
haftmann@60685
  1781
end
haftmann@60685
  1782
  
wenzelm@60758
  1783
text\<open>Basic laws about division and remainder\<close>
haftmann@33361
  1784
haftmann@33361
  1785
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
huffman@47141
  1786
  by (fact mod_div_equality2 [symmetric])
haftmann@33361
  1787
haftmann@60868
  1788
lemma zdiv_int: "int (a div b) = int a div int b"
haftmann@60868
  1789
  by (simp add: divide_int_def)
haftmann@60868
  1790
haftmann@60868
  1791
lemma zmod_int: "int (a mod b) = int a mod int b"
haftmann@60868
  1792
  by (simp add: mod_int_def int_dvd_iff)
haftmann@60868
  1793
  
wenzelm@60758
  1794
text \<open>Tool setup\<close>
wenzelm@60758
  1795
wenzelm@60758
  1796
ML \<open>
wenzelm@43594
  1797
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550
  1798
(
haftmann@60352
  1799
  val div_name = @{const_name Rings.divide};
haftmann@33361
  1800
  val mod_name = @{const_name mod};
haftmann@33361
  1801
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1802
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1803
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1804
huffman@47165
  1805
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361
  1806
lp15@60562
  1807
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@59556
  1808
    (@{thm diff_conv_add_uminus} :: @{thms add_0_left add_0_right} @ @{thms ac_simps}))
wenzelm@41550
  1809
)
wenzelm@60758
  1810
\<close>
wenzelm@60758
  1811
wenzelm@60758
  1812
simproc_setup cancel_div_mod_int ("(k::int) + l") = \<open>K Cancel_Div_Mod_Int.proc\<close>
wenzelm@43594
  1813
huffman@47141
  1814
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
haftmann@60868
  1815
  using divmod_int_rel [of a b]
huffman@47141
  1816
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1817
wenzelm@45607
  1818
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1819
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1820
huffman@47141
  1821
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
haftmann@60868
  1822
  using divmod_int_rel [of a b]
huffman@47141
  1823
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1824
wenzelm@45607
  1825
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1826
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1827
haftmann@33361
  1828
wenzelm@60758
  1829
subsubsection \<open>General Properties of div and mod\<close>
haftmann@33361
  1830
haftmann@33361
  1831
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1832
apply (rule div_int_unique)
haftmann@33361
  1833
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1834
done
haftmann@33361
  1835
haftmann@33361
  1836
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1837
apply (rule div_int_unique)
haftmann@33361
  1838
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1839
done
haftmann@33361
  1840
haftmann@33361
  1841
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1842
apply (rule div_int_unique)
haftmann@33361
  1843
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1844
done
haftmann@33361
  1845
haftmann@33361
  1846
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1847
haftmann@33361
  1848
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1849
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1850
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1851
done
haftmann@33361
  1852
haftmann@33361
  1853
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1854
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1855
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1856
done
haftmann@33361
  1857
haftmann@33361
  1858
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1859
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361
  1860
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1861
done
haftmann@33361
  1862
wenzelm@61799
  1863
text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
wenzelm@60758
  1864
wenzelm@60758
  1865
wenzelm@60758
  1866
subsubsection \<open>Laws for div and mod with Unary Minus\<close>
haftmann@33361
  1867
haftmann@33361
  1868
lemma zminus1_lemma:
huffman@47139
  1869
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
lp15@60562
  1870
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,
haftmann@33361
  1871
                          if r=0 then 0 else b-r)"
haftmann@33361
  1872
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1873
haftmann@33361
  1874
haftmann@33361
  1875
lemma zdiv_zminus1_eq_if:
lp15@60562
  1876
     "b \<noteq> (0::int)
lp15@60562
  1877
      ==> (-a) div b =
haftmann@33361
  1878
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@60868
  1879
by (blast intro: divmod_int_rel [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1880
haftmann@33361
  1881
lemma zmod_zminus1_eq_if:
haftmann@33361
  1882
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1883
apply (case_tac "b = 0", simp)
haftmann@60868
  1884
apply (blast intro: divmod_int_rel [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1885
done
haftmann@33361
  1886
haftmann@33361
  1887
lemma zmod_zminus1_not_zero:
haftmann@33361
  1888
  fixes k l :: int
haftmann@33361
  1889
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1890
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1891
haftmann@33361
  1892
lemma zdiv_zminus2_eq_if:
lp15@60562
  1893
     "b \<noteq> (0::int)
lp15@60562
  1894
      ==> a div (-b) =
haftmann@33361
  1895
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1896
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1897
haftmann@33361
  1898
lemma zmod_zminus2_eq_if:
haftmann@33361
  1899
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1900
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1901
haftmann@33361
  1902
lemma zmod_zminus2_not_zero:
haftmann@33361
  1903
  fixes k l :: int
haftmann@33361
  1904
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
lp15@60562
  1905
  unfolding zmod_zminus2_eq_if by auto
haftmann@33361
  1906
haftmann@33361
  1907
wenzelm@60758
  1908
subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
haftmann@33361
  1909
haftmann@33361
  1910
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361
  1911
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1912
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1913
apply (rule unique_quotient_lemma)
haftmann@33361
  1914
apply (erule subst)
haftmann@33361
  1915
apply (erule subst, simp_all)
haftmann@33361
  1916
done
haftmann@33361
  1917
haftmann@33361
  1918
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361
  1919
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1920
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1921
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1922
apply (erule subst)
haftmann@33361
  1923
apply (erule subst, simp_all)
haftmann@33361
  1924
done
haftmann@33361
  1925
haftmann@33361
  1926
wenzelm@60758
  1927
subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
haftmann@33361
  1928
haftmann@33361
  1929
lemma q_pos_lemma:
haftmann@33361
  1930
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1931
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1932
 apply (simp add: zero_less_mult_iff)
webertj@49962
  1933
apply (simp add: distrib_left)
haftmann@33361
  1934
done
haftmann@33361
  1935
haftmann@33361
  1936
lemma zdiv_mono2_lemma:
lp15@60562
  1937
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
lp15@60562
  1938
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
haftmann@33361
  1939
      ==> q \<le> (q'::int)"
lp15@60562
  1940
apply (frule q_pos_lemma, assumption+)
haftmann@33361
  1941
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  1942
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1943
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  1944
 prefer 2 apply simp
webertj@49962
  1945
apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512
  1946
apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361
  1947
apply (rule mult_right_mono, auto)
haftmann@33361
  1948
done
haftmann@33361
  1949
haftmann@33361
  1950
lemma zdiv_mono2:
haftmann@33361
  1951
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  1952
apply (subgoal_tac "b \<noteq> 0")
haftmann@33361
  1953
 prefer 2 apply arith
haftmann@33361
  1954
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1955
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1956
apply (rule zdiv_mono2_lemma)
haftmann@33361
  1957
apply (erule subst)
haftmann@33361
  1958
apply (erule subst, simp_all)
haftmann@33361
  1959
done
haftmann@33361
  1960
haftmann@33361
  1961
lemma q_neg_lemma:
haftmann@33361
  1962
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  1963
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  1964
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  1965
done
haftmann@33361
  1966
haftmann@33361
  1967
lemma zdiv_mono2_neg_lemma:
lp15@60562
  1968
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
lp15@60562
  1969
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
haftmann@33361
  1970
      ==> q' \<le> (q::int)"
lp15@60562
  1971
apply (frule q_neg_lemma, assumption+)
haftmann@33361
  1972
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  1973
 apply (simp add: mult_less_cancel_left)
webertj@49962
  1974
apply (simp add: distrib_left)
haftmann@33361
  1975
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  1976
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  1977
done
haftmann@33361
  1978
haftmann@33361
  1979
lemma zdiv_mono2_neg:
haftmann@33361
  1980
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361
  1981
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1982
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1983
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  1984
apply (erule subst)
haftmann@33361
  1985
apply (erule subst, simp_all)
haftmann@33361
  1986
done
haftmann@33361
  1987
haftmann@33361
  1988
wenzelm@60758
  1989
subsubsection \<open>More Algebraic Laws for div and mod\<close>
wenzelm@60758
  1990
wenzelm@60758
  1991
text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
haftmann@33361
  1992
haftmann@33361
  1993
lemma zmult1_lemma:
lp15@60562
  1994
     "[| divmod_int_rel b c (q, r) |]
haftmann@33361
  1995
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@57514
  1996
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left ac_simps)
haftmann@33361
  1997
haftmann@33361
  1998
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  1999
apply (case_tac "c = 0", simp)
haftmann@60868
  2000
apply (blast intro: divmod_int_rel [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  2001
done
haftmann@33361
  2002
wenzelm@60758
  2003
text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
haftmann@33361
  2004
haftmann@33361
  2005
lemma zadd1_lemma:
lp15@60562
  2006
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]
haftmann@33361
  2007
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
webertj@49962
  2008
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
haftmann@33361
  2009
haftmann@33361
  2010
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  2011
lemma zdiv_zadd1_eq:
haftmann@33361
  2012
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  2013
apply (case_tac "c = 0", simp)
haftmann@60868
  2014
apply (blast intro: zadd1_lemma [OF divmod_int_rel divmod_int_rel] div_int_unique)
haftmann@33361
  2015
done
haftmann@33361
  2016
haftmann@33361
  2017
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  2018
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  2019
haftmann@33361
  2020
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  2021
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  2022
haftmann@60868
  2023
lemma zmod_zdiv_equality' [nitpick_unfold]:
wenzelm@61076
  2024
  "(m::int) mod n = m - (m div n) * n"
huffman@47141
  2025
  using mod_div_equality [of m n] by arith
huffman@47108
  2026
haftmann@33361
  2027
wenzelm@60758
  2028
subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
haftmann@33361
  2029
haftmann@33361
  2030
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  2031
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  2032
  to cause particular problems.*)
haftmann@33361
  2033
wenzelm@60758
  2034
text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
haftmann@33361
  2035
blanchet@55085
  2036
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361
  2037
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  2038
 apply (simp add: algebra_simps)
haftmann@33361
  2039
apply (rule order_le_less_trans)
haftmann@33361
  2040
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  2041
 apply (rule mult_left_mono_neg)
huffman@35216
  2042
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  2043
 apply (simp)
haftmann@33361
  2044
apply (simp)
haftmann@33361
  2045
done
haftmann@33361
  2046
haftmann@33361
  2047
lemma zmult2_lemma_aux2:
haftmann@33361
  2048
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  2049
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  2050
 apply arith
haftmann@33361
  2051
apply (simp add: mult_le_0_iff)
haftmann@33361
  2052
done
haftmann@33361
  2053
haftmann@33361
  2054
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  2055
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  2056
apply arith
haftmann@33361
  2057
apply (simp add: zero_le_mult_iff)
haftmann@33361
  2058
done
haftmann@33361
  2059
haftmann@33361
  2060
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  2061
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  2062
 apply (simp add: right_diff_distrib)
haftmann@33361
  2063
apply (rule order_less_le_trans)
haftmann@33361
  2064
 apply (erule mult_strict_right_mono)
haftmann@33361
  2065
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  2066
  apply simp
huffman@35216
  2067
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  2068
apply simp
haftmann@33361
  2069
done
haftmann@33361
  2070
lp15@60562
  2071
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]
haftmann@33361
  2072
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@57514
  2073
by (auto simp add: mult.assoc divmod_int_rel_def linorder_neq_iff
lp15@60562
  2074
                   zero_less_mult_iff distrib_left [symmetric]
huffman@47139
  2075
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
haftmann@33361
  2076
haftmann@53068
  2077
lemma zdiv_zmult2_eq:
haftmann@53068
  2078
  fixes a b c :: int
haftmann@53068
  2079
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
  2080
apply (case_tac "b = 0", simp)
haftmann@60868
  2081
apply (force simp add: le_less divmod_int_rel [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  2082
done
haftmann@33361
  2083
haftmann@33361
  2084
lemma zmod_zmult2_eq:
haftmann@53068
  2085
  fixes a b c :: int
haftmann@53068
  2086
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
  2087
apply (case_tac "b = 0", simp)
haftmann@60868
  2088
apply (force simp add: le_less divmod_int_rel [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  2089
done
haftmann@33361
  2090
huffman@47108
  2091
lemma div_pos_geq:
huffman@47108
  2092
  fixes k l :: int
huffman@47108
  2093
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2094
  shows "k div l = (k - l) div l + 1"
huffman@47108
  2095
proof -
huffman@47108
  2096
  have "k = (k - l) + l" by simp
huffman@47108
  2097
  then obtain j where k: "k = j + l" ..
huffman@47108
  2098
  with assms show ?thesis by simp
huffman@47108
  2099
qed
huffman@47108
  2100
huffman@47108
  2101
lemma mod_pos_geq:
huffman@47108
  2102
  fixes k l :: int
huffman@47108
  2103
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2104
  shows "k mod l = (k - l) mod l"
huffman@47108
  2105
proof -
huffman@47108
  2106
  have "k = (k - l) + l" by simp
huffman@47108
  2107
  then obtain j where k: "k = j + l" ..
huffman@47108
  2108
  with assms show ?thesis by simp
huffman@47108
  2109
qed
huffman@47108
  2110
haftmann@33361
  2111
wenzelm@60758
  2112
subsubsection \<open>Splitting Rules for div and mod\<close>
wenzelm@60758
  2113
wenzelm@60758
  2114
text\<open>The proofs of the two lemmas below are essentially identical\<close>
haftmann@33361
  2115
haftmann@33361
  2116
lemma split_pos_lemma:
lp15@60562
  2117
 "0<k ==>
haftmann@33361
  2118
    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
  2119
apply (rule iffI, clarify)
lp15@60562
  2120
 apply (erule_tac P="P x y" for x y in rev_mp)
lp15@60562
  2121
 apply (subst mod_add_eq)
lp15@60562
  2122
 apply (subst zdiv_zadd1_eq)
lp15@60562
  2123
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
wenzelm@60758
  2124
txt\<open>converse direction\<close>
lp15@60562
  2125
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  2126
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2127
done
haftmann@33361
  2128
haftmann@33361
  2129
lemma split_neg_lemma:
haftmann@33361
  2130
 "k<0 ==>
haftmann@33361
  2131
    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
  2132
apply (rule iffI, clarify)
lp15@60562
  2133
 apply (erule_tac P="P x y" for x y in rev_mp)
lp15@60562
  2134
 apply (subst mod_add_eq)
lp15@60562
  2135
 apply (subst zdiv_zadd1_eq)
lp15@60562
  2136
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
wenzelm@60758
  2137
txt\<open>converse direction\<close>
lp15@60562
  2138
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  2139
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2140
done
haftmann@33361
  2141
haftmann@33361
  2142
lemma split_zdiv:
haftmann@33361
  2143
 "P(n div k :: int) =
lp15@60562
  2144
  ((k = 0 --> P 0) &
lp15@60562
  2145
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
haftmann@33361
  2146
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  2147
apply (case_tac "k=0", simp)
haftmann@33361
  2148
apply (simp only: linorder_neq_iff)
lp15@60562
  2149
apply (erule disjE)
lp15@60562
  2150
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
haftmann@33361
  2151
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  2152
done
haftmann@33361
  2153
haftmann@33361
  2154
lemma split_zmod:
haftmann@33361
  2155
 "P(n mod k :: int) =
lp15@60562
  2156
  ((k = 0 --> P n) &
lp15@60562
  2157
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
haftmann@33361
  2158
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  2159
apply (case_tac "k=0", simp)
haftmann@33361
  2160
apply (simp only: linorder_neq_iff)
lp15@60562
  2161
apply (erule disjE)
lp15@60562
  2162
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
haftmann@33361
  2163
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  2164
done
haftmann@33361
  2165
wenzelm@60758
  2166
text \<open>Enable (lin)arith to deal with @{const divide} and @{const mod}
webertj@33730
  2167
  when these are applied to some constant that is of the form
wenzelm@60758
  2168
  @{term "numeral k"}:\<close>
huffman@47108
  2169
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  2170
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  2171
haftmann@33361
  2172
wenzelm@61799
  2173
subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
huffman@47166
  2174
huffman@47166
  2175
lemma pos_divmod_int_rel_mult_2:
huffman@47166
  2176
  assumes "0 \<le> b"
huffman@47166
  2177
  assumes "divmod_int_rel a b (q, r)"
huffman@47166
  2178
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166
  2179
  using assms unfolding divmod_int_rel_def by auto
huffman@47166
  2180
wenzelm@60758
  2181
declaration \<open>K (Lin_Arith.add_simps @{thms uminus_numeral_One})\<close>
haftmann@54489
  2182
huffman@47166
  2183
lemma neg_divmod_int_rel_mult_2:
huffman@47166
  2184
  assumes "b \<le> 0"
huffman@47166
  2185
  assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166
  2186
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166
  2187
  using assms unfolding divmod_int_rel_def by auto
haftmann@33361
  2188
wenzelm@60758
  2189
text\<open>computing div by shifting\<close>
haftmann@33361
  2190
haftmann@33361
  2191
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
haftmann@60868
  2192
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel]
huffman@47166
  2193
  by (rule div_int_unique)
haftmann@33361
  2194
lp15@60562
  2195
lemma neg_zdiv_mult_2:
boehmes@35815
  2196
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
haftmann@60868
  2197
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel]
huffman@47166
  2198
  by (rule div_int_unique)
haftmann@33361
  2199
huffman@47108
  2200
(* FIXME: add rules for negative numerals *)
huffman@47108
  2201
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  2202
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  2203
    numeral v div (numeral w :: int)"
huffman@47108
  2204
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  2205
  by (rule div_mult_mult1, simp)
huffman@47108
  2206
huffman@47108
  2207
lemma zdiv_numeral_Bit1 [simp]:
lp15@60562
  2208
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
huffman@47108
  2209
    (numeral v div (numeral w :: int))"
huffman@47108
  2210
  unfolding numeral.simps
haftmann@57512
  2211
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  2212
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  2213
haftmann@33361
  2214
lemma pos_zmod_mult_2:
haftmann@33361
  2215
  fixes a b :: int
haftmann@33361
  2216
  assumes "0 \<le> a"
haftmann@33361
  2217
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
haftmann@60868
  2218
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel]
huffman@47166
  2219
  by (rule mod_int_unique)
haftmann@33361
  2220
haftmann@33361
  2221
lemma neg_zmod_mult_2:
haftmann@33361
  2222
  fixes a b :: int
haftmann@33361
  2223
  assumes "a \<le> 0"
haftmann@33361
  2224
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
haftmann@60868
  2225
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel]
huffman@47166
  2226
  by (rule mod_int_unique)
haftmann@33361
  2227
huffman@47108
  2228
(* FIXME: add rules for negative numerals *)
huffman@47108
  2229
lemma zmod_numeral_Bit0 [simp]:
lp15@60562
  2230
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2231
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  2232
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  2233
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  2234
huffman@47108
  2235
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  2236
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2237
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  2238
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512
  2239
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  2240
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  2241
nipkow@39489
  2242
lemma zdiv_eq_0_iff:
nipkow@39489
  2243
 "(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
  2244
proof
nipkow@39489
  2245
  assume ?L
nipkow@39489
  2246
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
wenzelm@60758
  2247
  with \<open>?L\<close> show ?R by blast
nipkow@39489
  2248
next
nipkow@39489
  2249
  assume ?R thus ?L
nipkow@39489
  2250
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  2251
qed
nipkow@39489
  2252
nipkow@39489
  2253
wenzelm@60758
  2254
subsubsection \<open>Quotients of Signs\<close>
haftmann@33361
  2255
haftmann@60868
  2256
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@60868
  2257
by (simp add: divide_int_def)
haftmann@60868
  2258
haftmann@60868
  2259
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@60868
  2260
by (simp add: mod_int_def)
haftmann@60868
  2261
haftmann@33361
  2262
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2263
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2264
apply (rule order_trans)
haftmann@33361
  2265
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2266
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2267
done
haftmann@33361
  2268
haftmann@33361
  2269
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2270
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2271
haftmann@33361
  2272
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2273
by (drule zdiv_mono1, auto)
haftmann@33361
  2274
wenzelm@61799
  2275
text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
wenzelm@61799
  2276
conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
wenzelm@60758
  2277
They should all be simp rules unless that causes too much search.\<close>
nipkow@33804
  2278
haftmann@33361
  2279
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2280
apply auto
haftmann@33361
  2281
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2282
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2283
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2284
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2285
done
haftmann@33361
  2286
haftmann@60868
  2287
lemma pos_imp_zdiv_pos_iff:
haftmann@60868
  2288
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
haftmann@60868
  2289
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
haftmann@60868
  2290
by arith
haftmann@60868
  2291
haftmann@33361
  2292
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2293
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2294
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2295
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2296
done
haftmann@33361
  2297
haftmann@33361
  2298
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2299
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2300
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2301
haftmann@33361
  2302
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2303
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2304
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2305
nipkow@33804
  2306
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2307
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2308
apply rule
nipkow@33804
  2309
 apply rule
nipkow@33804
  2310
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2311
 apply(cases "b=0")apply simp
nipkow@33804
  2312
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2313
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2314
done
nipkow@33804
  2315
nipkow@39489
  2316
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2317
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2318
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2319
done
nipkow@39489
  2320
haftmann@60868
  2321
lemma zmult_div_cancel:
haftmann@60868
  2322
  "(n::int) * (m div n) = m - (m mod n)"
haftmann@60868
  2323
  using zmod_zdiv_equality [where a="m" and b="n"]
huffman@47142
  2324
  by (simp add: algebra_simps) (* FIXME: generalize *)
haftmann@33361
  2325
haftmann@60868
  2326
haftmann@60868
  2327
subsubsection \<open>Computation of Division and Remainder\<close>
haftmann@60868
  2328
haftmann@61275
  2329
instantiation int :: semiring_numeral_div
haftmann@61275
  2330
begin
haftmann@61275
  2331
haftmann@61275
  2332
definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
haftmann@61275
  2333
where
haftmann@61275
  2334
  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
  2335
haftmann@61275
  2336
definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
haftmann@61275
  2337
where
haftmann@61275
  2338
  "divmod_step_int l qr = (let (q, r) = qr
haftmann@61275
  2339
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
  2340
    else (2 * q, r))"
haftmann@61275
  2341
haftmann@61275
  2342
instance
haftmann@61275
  2343
  by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
haftmann@61275
  2344
    pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
haftmann@61275
  2345
haftmann@61275
  2346
end
haftmann@61275
  2347
haftmann@61275
  2348
declare divmod_algorithm_code [where ?'a = int, code]
lp15@60562
  2349
haftmann@60930
  2350
context
haftmann@60930
  2351
begin
haftmann@60930
  2352
  
haftmann@60930
  2353
qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
haftmann@60868
  2354
where
haftmann@60868
  2355
  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
haftmann@60868
  2356
haftmann@60930
  2357
qualified lemma adjust_div_eq [simp, code]:
haftmann@60868
  2358
  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
haftmann@60868
  2359
  by (simp add: adjust_div_def)
haftmann@60868
  2360
haftmann@60930
  2361
qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  2362
where
haftmann@60868
  2363
  [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
haftmann@60868
  2364
haftmann@60868
  2365
lemma minus_numeral_div_numeral [simp]:
haftmann@60868
  2366
  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  2367
proof -
haftmann@60868
  2368
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  2369
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  2370
  then show ?thesis
haftmann@60868
  2371
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  2372
qed
haftmann@60868
  2373
haftmann@60868
  2374
lemma minus_numeral_mod_numeral [simp]:
haftmann@60868
  2375
  "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  2376
proof -
haftmann@60868
  2377
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@60868
  2378
    using that by (simp only: snd_divmod mod_int_def) auto
haftmann@60868
  2379
  then show ?thesis
haftmann@60868
  2380
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def mod_int_def)
haftmann@60868
  2381
qed
haftmann@60868
  2382
haftmann@60868
  2383
lemma numeral_div_minus_numeral [simp]:
haftmann@60868
  2384
  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  2385
proof -
haftmann@60868
  2386
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  2387
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  2388
  then show ?thesis
haftmann@60868
  2389
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  2390
qed
haftmann@60868
  2391
  
haftmann@60868
  2392
lemma numeral_mod_minus_numeral [simp]:
haftmann@60868
  2393
  "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  2394
proof -
haftmann@60868
  2395
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@60868
  2396
    using that by (simp only: snd_divmod mod_int_def) auto
haftmann@60868
  2397
  then show ?thesis
haftmann@60868
  2398
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def mod_int_def)
haftmann@60868
  2399
qed
haftmann@60868
  2400
haftmann@60868
  2401
lemma minus_one_div_numeral [simp]:
haftmann@60868
  2402
  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  2403
  using minus_numeral_div_numeral [of Num.One n] by simp  
haftmann@60868
  2404
haftmann@60868
  2405
lemma minus_one_mod_numeral [simp]:
haftmann@60868
  2406
  "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  2407
  using minus_numeral_mod_numeral [of Num.One n] by simp
haftmann@60868
  2408
haftmann@60868
  2409
lemma one_div_minus_numeral [simp]:
haftmann@60868
  2410
  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  2411
  using numeral_div_minus_numeral [of Num.One n] by simp
haftmann@60868
  2412
  
haftmann@60868
  2413
lemma one_mod_minus_numeral [simp]:
haftmann@60868
  2414
  "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  2415
  using numeral_mod_minus_numeral [of Num.One n] by simp
haftmann@60868
  2416
haftmann@60930
  2417
end
haftmann@60930
  2418
haftmann@60868
  2419
haftmann@60868
  2420
subsubsection \<open>Further properties\<close>
haftmann@60868
  2421
haftmann@60868
  2422
text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
haftmann@60868
  2423
haftmann@60868
  2424
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  2425
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
haftmann@60868
  2426
haftmann@60868
  2427
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  2428
  by (rule div_int_unique [of a b q r],
haftmann@60868
  2429
    simp add: divmod_int_rel_def)
haftmann@60868
  2430
haftmann@60868
  2431
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  2432
  by (rule mod_int_unique [of a b q r],
haftmann@60868
  2433
    simp add: divmod_int_rel_def)
haftmann@60868
  2434
haftmann@60868
  2435
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  2436
  by (rule mod_int_unique [of a b q r],
haftmann@60868
  2437
    simp add: divmod_int_rel_def)
haftmann@33361
  2438
wenzelm@61944
  2439
lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
haftmann@33361
  2440
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2441
wenzelm@60758
  2442
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  2443
lemma int_power_div_base:
haftmann@33361
  2444
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2445
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2446
 apply (erule ssubst)
haftmann@33361
  2447
 apply (simp only: power_add)
haftmann@33361
  2448
 apply simp_all
haftmann@33361
  2449
done
haftmann@33361
  2450
wenzelm@60758
  2451
text \<open>by Brian Huffman\<close>
haftmann@33361
  2452
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2453
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2454
haftmann@33361
  2455
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2456
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2457
haftmann@33361
  2458
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2459
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2460
haftmann@33361
  2461
lemmas zmod_simps =
haftmann@33361
  2462
  mod_add_left_eq  [symmetric]
haftmann@33361
  2463
  mod_add_right_eq [symmetric]
huffman@47142
  2464
  mod_mult_right_eq[symmetric]
haftmann@33361
  2465
  mod_mult_left_eq [symmetric]
huffman@47164
  2466
  power_mod
haftmann@33361
  2467
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2468
wenzelm@61799
  2469
text \<open>Distributive laws for function \<open>nat\<close>.\<close>
haftmann@33361
  2470
haftmann@33361
  2471
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2472
apply (rule linorder_cases [of y 0])
haftmann@33361
  2473
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2474
apply simp
haftmann@33361
  2475
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2476
done
haftmann@33361
  2477
haftmann@33361
  2478
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2479
lemma nat_mod_distrib:
haftmann@33361
  2480
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2481
apply (case_tac "y = 0", simp)
haftmann@33361
  2482
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2483
done
haftmann@33361
  2484
wenzelm@60758
  2485
text  \<open>transfer setup\<close>
haftmann@33361
  2486
haftmann@33361
  2487
lemma transfer_nat_int_functions:
haftmann@33361
  2488
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2489
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2490
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2491
haftmann@33361
  2492
lemma transfer_nat_int_function_closures:
haftmann@33361
  2493
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2494
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2495
  apply (cases "y = 0")
haftmann@33361
  2496
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2497
  apply (cases "y = 0")
haftmann@33361
  2498
  apply auto
haftmann@33361
  2499
done
haftmann@33361
  2500
haftmann@35644
  2501
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2502
  transfer_nat_int_functions
haftmann@33361
  2503
  transfer_nat_int_function_closures
haftmann@33361
  2504
]
haftmann@33361
  2505
haftmann@33361
  2506
lemma transfer_int_nat_functions:
haftmann@33361
  2507
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2508
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2509
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2510
haftmann@33361
  2511
lemma transfer_int_nat_function_closures:
haftmann@33361
  2512
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  2513
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  2514
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  2515
haftmann@35644
  2516
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  2517
  transfer_int_nat_functions
haftmann@33361
  2518
  transfer_int_nat_function_closures
haftmann@33361
  2519
]
haftmann@33361
  2520
wenzelm@60758
  2521
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  2522
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  2523
apply (subgoal_tac "nat x div nat k < nat x")