src/HOL/Divides.thy
author haftmann
Thu Oct 31 11:44:20 2013 +0100 (2013-10-31)
changeset 54221 56587960e444
parent 53374 a14d2a854c02
child 54226 e3df2a4e02fc
permissions -rw-r--r--
more lemmas on division
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(*  Title:      HOL/Divides.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div and mod *}
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theory Divides
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imports Nat_Transfer
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begin
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subsection {* Syntactic division operations *}
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class div = dvd +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract division in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0 [simp]: "a div 0 = 0"
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    and div_0 [simp]: "0 div a = 0"
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    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
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begin
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text {* @{const div} and @{const mod} *}
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma mod_div_equality': "a mod b + a div b * b = a"
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  using mod_div_equality [of a b]
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  by (simp only: add_ac)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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lemma mod_by_0 [simp]: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mod_0 [simp]: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp
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lemma div_mult_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b * c) div b = c + a div b"
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  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
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lemma div_mult_self3 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(c * b + a) div b = c + a div b"
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  using assms by (simp add: add.commute)
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lemma div_mult_self4 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b * c + a) div b = c + a div b"
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  using assms by (simp add: add.commute)
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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proof (cases "b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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    by (simp add: mod_div_equality)
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  also from False div_mult_self1 [of b a c] have
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    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
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      by (simp add: algebra_simps)
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  finally have "a = a div b * b + (a + c * b) mod b"
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    by (simp add: add_commute [of a] add_assoc distrib_right)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: 
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  "(a + b * c) mod b = a mod b"
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  by (simp add: mult_commute [of b])
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lemma mod_mult_self3 [simp]:
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  "(c * b + a) mod b = a mod b"
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  by (simp add: add.commute)
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lemma mod_mult_self4 [simp]:
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  "(b * c + a) mod b = a mod b"
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  by (simp add: add.commute)
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  using div_mult_self2 [of b 0 a] by simp
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  using div_mult_self1 [of b 0 a] by simp
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
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  using mod_mult_self2 [of 0 b a] by simp
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma div_by_1 [simp]: "a div 1 = a"
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  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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lemma mod_by_1 [simp]: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self [simp]: "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using div_mult_self2_is_id [of _ 1] by simp
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lemma div_add_self1 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b + a) div b = a div b + 1"
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  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
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lemma div_add_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b) div b = a div b + 1"
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  using assms div_add_self1 [of b a] by (simp add: add_commute)
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lemma mod_add_self1 [simp]:
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  "(b + a) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
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lemma mod_add_self2 [simp]:
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  "(a + b) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by simp
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lemma mod_div_decomp:
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  fixes a b
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  obtains q r where "q = a div b" and "r = a mod b"
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    and "a = q * b + r"
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proof -
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  from mod_div_equality have "a = a div b * b + a mod b" by simp
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  moreover have "a div b = a div b" ..
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  moreover have "a mod b = a mod b" ..
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  note that ultimately show thesis by blast
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qed
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lemma dvd_eq_mod_eq_0 [code]: "a dvd b \<longleftrightarrow> b mod a = 0"
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proof
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  assume "b mod a = 0"
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  with mod_div_equality [of b a] have "b div a * a = b" by simp
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  then have "b = a * (b div a)" unfolding mult_commute ..
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  then have "\<exists>c. b = a * c" ..
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  then show "a dvd b" unfolding dvd_def .
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next
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  assume "a dvd b"
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  then have "\<exists>c. b = a * c" unfolding dvd_def .
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  then obtain c where "b = a * c" ..
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  then have "b mod a = a * c mod a" by simp
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  then have "b mod a = c * a mod a" by (simp add: mult_commute)
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  then show "b mod a = 0" by simp
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qed
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lemma mod_div_trivial [simp]: "a mod b div b = 0"
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proof (cases "b = 0")
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  assume "b = 0"
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  thus ?thesis by simp
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next
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  assume "b \<noteq> 0"
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  hence "a div b + a mod b div b = (a mod b + a div b * b) div b"
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    by (rule div_mult_self1 [symmetric])
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  also have "\<dots> = a div b"
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    by (simp only: mod_div_equality')
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  also have "\<dots> = a div b + 0"
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    by simp
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  finally show ?thesis
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    by (rule add_left_imp_eq)
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qed
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lemma mod_mod_trivial [simp]: "a mod b mod b = a mod b"
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proof -
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  have "a mod b mod b = (a mod b + a div b * b) mod b"
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    by (simp only: mod_mult_self1)
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  also have "\<dots> = a mod b"
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    by (simp only: mod_div_equality')
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  finally show ?thesis .
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qed
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lemma dvd_imp_mod_0: "a dvd b \<Longrightarrow> b mod a = 0"
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by (rule dvd_eq_mod_eq_0[THEN iffD1])
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lemma dvd_div_mult_self: "a dvd b \<Longrightarrow> (b div a) * a = b"
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by (subst (2) mod_div_equality [of b a, symmetric]) (simp add:dvd_imp_mod_0)
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lemma dvd_mult_div_cancel: "a dvd b \<Longrightarrow> a * (b div a) = b"
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by (drule dvd_div_mult_self) (simp add: mult_commute)
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lemma dvd_div_mult: "a dvd b \<Longrightarrow> (b div a) * c = b * c div a"
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apply (cases "a = 0")
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 apply simp
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apply (auto simp: dvd_def mult_assoc)
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done
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lemma div_dvd_div[simp]:
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  "a dvd b \<Longrightarrow> a dvd c \<Longrightarrow> (b div a dvd c div a) = (b dvd c)"
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apply (cases "a = 0")
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 apply simp
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apply (unfold dvd_def)
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apply auto
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 apply(blast intro:mult_assoc[symmetric])
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apply(fastforce simp add: mult_assoc)
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done
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lemma dvd_mod_imp_dvd: "[| k dvd m mod n;  k dvd n |] ==> k dvd m"
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  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
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   apply (simp add: mod_div_equality)
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  apply (simp only: dvd_add dvd_mult)
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  done
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text {* Addition respects modular equivalence. *}
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lemma mod_add_left_eq: "(a + b) mod c = (a mod c + b) mod c"
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proof -
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  have "(a + b) mod c = (a div c * c + a mod c + b) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a mod c + b + a div c * c) mod c"
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    by (simp only: add_ac)
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  also have "\<dots> = (a mod c + b) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_add_right_eq: "(a + b) mod c = (a + b mod c) mod c"
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proof -
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  have "(a + b) mod c = (a + (b div c * c + b mod c)) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a + b mod c + b div c * c) mod c"
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    by (simp only: add_ac)
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  also have "\<dots> = (a + b mod c) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_add_eq: "(a + b) mod c = (a mod c + b mod c) mod c"
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by (rule trans [OF mod_add_left_eq mod_add_right_eq])
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lemma mod_add_cong:
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  assumes "a mod c = a' mod c"
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  assumes "b mod c = b' mod c"
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  shows "(a + b) mod c = (a' + b') mod c"
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proof -
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  have "(a mod c + b mod c) mod c = (a' mod c + b' mod c) mod c"
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    unfolding assms ..
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  thus ?thesis
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    by (simp only: mod_add_eq [symmetric])
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qed
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lemma div_add [simp]: "z dvd x \<Longrightarrow> z dvd y
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  \<Longrightarrow> (x + y) div z = x div z + y div z"
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by (cases "z = 0", simp, unfold dvd_def, auto simp add: algebra_simps)
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text {* Multiplication respects modular equivalence. *}
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lemma mod_mult_left_eq: "(a * b) mod c = ((a mod c) * b) mod c"
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proof -
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  have "(a * b) mod c = ((a div c * c + a mod c) * b) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a mod c * b + a div c * b * c) mod c"
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    by (simp only: algebra_simps)
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  also have "\<dots> = (a mod c * b) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_mult_right_eq: "(a * b) mod c = (a * (b mod c)) mod c"
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proof -
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  have "(a * b) mod c = (a * (b div c * c + b mod c)) mod c"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (a * (b mod c) + a * (b div c) * c) mod c"
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    by (simp only: algebra_simps)
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  also have "\<dots> = (a * (b mod c)) mod c"
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    by (rule mod_mult_self1)
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  finally show ?thesis .
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qed
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lemma mod_mult_eq: "(a * b) mod c = ((a mod c) * (b mod c)) mod c"
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by (rule trans [OF mod_mult_left_eq mod_mult_right_eq])
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lemma mod_mult_cong:
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  assumes "a mod c = a' mod c"
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  assumes "b mod c = b' mod c"
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  shows "(a * b) mod c = (a' * b') mod c"
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proof -
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  have "(a mod c * (b mod c)) mod c = (a' mod c * (b' mod c)) mod c"
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    unfolding assms ..
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  thus ?thesis
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    by (simp only: mod_mult_eq [symmetric])
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qed
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text {* Exponentiation respects modular equivalence. *}
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lemma power_mod: "(a mod b)^n mod b = a^n mod b"
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apply (induct n, simp_all)
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apply (rule mod_mult_right_eq [THEN trans])
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apply (simp (no_asm_simp))
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apply (rule mod_mult_eq [symmetric])
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done
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lemma mod_mod_cancel:
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  assumes "c dvd b"
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  shows "a mod b mod c = a mod c"
huffman@29404
   316
proof -
huffman@29404
   317
  from `c dvd b` obtain k where "b = c * k"
huffman@29404
   318
    by (rule dvdE)
huffman@29404
   319
  have "a mod b mod c = a mod (c * k) mod c"
huffman@29404
   320
    by (simp only: `b = c * k`)
huffman@29404
   321
  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
huffman@29404
   322
    by (simp only: mod_mult_self1)
huffman@29404
   323
  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
huffman@29404
   324
    by (simp only: add_ac mult_ac)
huffman@29404
   325
  also have "\<dots> = a mod c"
huffman@29404
   326
    by (simp only: mod_div_equality)
huffman@29404
   327
  finally show ?thesis .
huffman@29404
   328
qed
huffman@29404
   329
haftmann@30930
   330
lemma div_mult_div_if_dvd:
haftmann@30930
   331
  "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
haftmann@30930
   332
  apply (cases "y = 0", simp)
haftmann@30930
   333
  apply (cases "z = 0", simp)
haftmann@30930
   334
  apply (auto elim!: dvdE simp add: algebra_simps)
nipkow@30476
   335
  apply (subst mult_assoc [symmetric])
nipkow@30476
   336
  apply (simp add: no_zero_divisors)
haftmann@30930
   337
  done
haftmann@30930
   338
haftmann@35367
   339
lemma div_mult_swap:
haftmann@35367
   340
  assumes "c dvd b"
haftmann@35367
   341
  shows "a * (b div c) = (a * b) div c"
haftmann@35367
   342
proof -
haftmann@35367
   343
  from assms have "b div c * (a div 1) = b * a div (c * 1)"
haftmann@35367
   344
    by (simp only: div_mult_div_if_dvd one_dvd)
haftmann@35367
   345
  then show ?thesis by (simp add: mult_commute)
haftmann@35367
   346
qed
haftmann@35367
   347
   
haftmann@30930
   348
lemma div_mult_mult2 [simp]:
haftmann@30930
   349
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@30930
   350
  by (drule div_mult_mult1) (simp add: mult_commute)
haftmann@30930
   351
haftmann@30930
   352
lemma div_mult_mult1_if [simp]:
haftmann@30930
   353
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   354
  by simp_all
nipkow@30476
   355
haftmann@30930
   356
lemma mod_mult_mult1:
haftmann@30930
   357
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   358
proof (cases "c = 0")
haftmann@30930
   359
  case True then show ?thesis by simp
haftmann@30930
   360
next
haftmann@30930
   361
  case False
haftmann@30930
   362
  from mod_div_equality
haftmann@30930
   363
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   364
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   365
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@30930
   366
  with mod_div_equality show ?thesis by simp 
haftmann@30930
   367
qed
haftmann@30930
   368
  
haftmann@30930
   369
lemma mod_mult_mult2:
haftmann@30930
   370
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@30930
   371
  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
haftmann@30930
   372
huffman@47159
   373
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159
   374
  by (fact mod_mult_mult2 [symmetric])
huffman@47159
   375
huffman@47159
   376
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159
   377
  by (fact mod_mult_mult1 [symmetric])
huffman@47159
   378
huffman@31662
   379
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   380
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   381
huffman@31662
   382
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   383
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   384
haftmann@31009
   385
lemma div_power:
huffman@31661
   386
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
nipkow@30476
   387
apply (induct n)
nipkow@30476
   388
 apply simp
nipkow@30476
   389
apply(simp add: div_mult_div_if_dvd dvd_power_same)
nipkow@30476
   390
done
nipkow@30476
   391
haftmann@35367
   392
lemma dvd_div_eq_mult:
haftmann@35367
   393
  assumes "a \<noteq> 0" and "a dvd b"  
haftmann@35367
   394
  shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@35367
   395
proof
haftmann@35367
   396
  assume "b = c * a"
haftmann@35367
   397
  then show "b div a = c" by (simp add: assms)
haftmann@35367
   398
next
haftmann@35367
   399
  assume "b div a = c"
haftmann@35367
   400
  then have "b div a * a = c * a" by simp
haftmann@35367
   401
  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
haftmann@35367
   402
  ultimately show "b = c * a" by simp
haftmann@35367
   403
qed
haftmann@35367
   404
   
haftmann@35367
   405
lemma dvd_div_div_eq_mult:
haftmann@35367
   406
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@35367
   407
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
haftmann@35367
   408
  using assms by (auto simp add: mult_commute [of _ a] dvd_div_mult_self dvd_div_eq_mult div_mult_swap intro: sym)
haftmann@35367
   409
huffman@31661
   410
end
huffman@31661
   411
haftmann@35673
   412
class ring_div = semiring_div + comm_ring_1
huffman@29405
   413
begin
huffman@29405
   414
haftmann@36634
   415
subclass ring_1_no_zero_divisors ..
haftmann@36634
   416
huffman@29405
   417
text {* Negation respects modular equivalence. *}
huffman@29405
   418
huffman@29405
   419
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   420
proof -
huffman@29405
   421
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405
   422
    by (simp only: mod_div_equality)
huffman@29405
   423
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
huffman@29405
   424
    by (simp only: minus_add_distrib minus_mult_left add_ac)
huffman@29405
   425
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   426
    by (rule mod_mult_self1)
huffman@29405
   427
  finally show ?thesis .
huffman@29405
   428
qed
huffman@29405
   429
huffman@29405
   430
lemma mod_minus_cong:
huffman@29405
   431
  assumes "a mod b = a' mod b"
huffman@29405
   432
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   433
proof -
huffman@29405
   434
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   435
    unfolding assms ..
huffman@29405
   436
  thus ?thesis
huffman@29405
   437
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   438
qed
huffman@29405
   439
huffman@29405
   440
text {* Subtraction respects modular equivalence. *}
huffman@29405
   441
huffman@29405
   442
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
huffman@29405
   443
  unfolding diff_minus
huffman@29405
   444
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   445
huffman@29405
   446
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
huffman@29405
   447
  unfolding diff_minus
huffman@29405
   448
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   449
huffman@29405
   450
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
huffman@29405
   451
  unfolding diff_minus
huffman@29405
   452
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   453
huffman@29405
   454
lemma mod_diff_cong:
huffman@29405
   455
  assumes "a mod c = a' mod c"
huffman@29405
   456
  assumes "b mod c = b' mod c"
huffman@29405
   457
  shows "(a - b) mod c = (a' - b') mod c"
huffman@29405
   458
  unfolding diff_minus using assms
huffman@29405
   459
  by (intro mod_add_cong mod_minus_cong)
huffman@29405
   460
nipkow@30180
   461
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   462
apply (case_tac "y = 0") apply simp
nipkow@30180
   463
apply (auto simp add: dvd_def)
nipkow@30180
   464
apply (subgoal_tac "-(y * k) = y * - k")
nipkow@30180
   465
 apply (erule ssubst)
nipkow@30180
   466
 apply (erule div_mult_self1_is_id)
nipkow@30180
   467
apply simp
nipkow@30180
   468
done
nipkow@30180
   469
nipkow@30180
   470
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   471
apply (case_tac "y = 0") apply simp
nipkow@30180
   472
apply (auto simp add: dvd_def)
nipkow@30180
   473
apply (subgoal_tac "y * k = -y * -k")
nipkow@30180
   474
 apply (erule ssubst)
nipkow@30180
   475
 apply (rule div_mult_self1_is_id)
nipkow@30180
   476
 apply simp
nipkow@30180
   477
apply simp
nipkow@30180
   478
done
nipkow@30180
   479
huffman@47159
   480
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159
   481
  using div_mult_mult1 [of "- 1" a b]
huffman@47159
   482
  unfolding neg_equal_0_iff_equal by simp
huffman@47159
   483
huffman@47159
   484
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159
   485
  using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159
   486
huffman@47159
   487
lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159
   488
  using div_minus_minus [of "-a" b] by simp
huffman@47159
   489
huffman@47159
   490
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159
   491
  using mod_minus_minus [of "-a" b] by simp
huffman@47159
   492
huffman@47160
   493
lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160
   494
  using div_minus_right [of a 1] by simp
huffman@47160
   495
huffman@47160
   496
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160
   497
  using mod_minus_right [of a 1] by simp
huffman@47160
   498
haftmann@54221
   499
lemma minus_mod_self2 [simp]: 
haftmann@54221
   500
  "(a - b) mod b = a mod b"
haftmann@54221
   501
  by (simp add: mod_diff_right_eq)
haftmann@54221
   502
haftmann@54221
   503
lemma minus_mod_self1 [simp]: 
haftmann@54221
   504
  "(b - a) mod b = - a mod b"
haftmann@54221
   505
proof -
haftmann@54221
   506
  have "b - a = - a + b" by (simp add: diff_minus add.commute)
haftmann@54221
   507
  then show ?thesis by simp
haftmann@54221
   508
qed
haftmann@54221
   509
huffman@29405
   510
end
huffman@29405
   511
haftmann@25942
   512
haftmann@53067
   513
subsection {* Generic numeral division with a pragmatic type class *}
haftmann@53067
   514
haftmann@53067
   515
text {*
haftmann@53067
   516
  The following type class contains everything necessary to formulate
haftmann@53067
   517
  a division algorithm in ring structures with numerals, restricted
haftmann@53067
   518
  to its positive segments.  This is its primary motiviation, and it
haftmann@53067
   519
  could surely be formulated using a more fine-grained, more algebraic
haftmann@53067
   520
  and less technical class hierarchy.
haftmann@53067
   521
*}
haftmann@53067
   522
haftmann@53067
   523
haftmann@53067
   524
class semiring_numeral_div = linordered_semidom + minus + semiring_div +
haftmann@53067
   525
  assumes diff_invert_add1: "a + b = c \<Longrightarrow> a = c - b"
haftmann@53067
   526
    and le_add_diff_inverse2: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@53067
   527
  assumes mult_div_cancel: "b * (a div b) = a - a mod b"
haftmann@53067
   528
    and div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067
   529
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067
   530
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067
   531
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067
   532
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067
   533
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067
   534
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067
   535
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067
   536
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@53067
   537
begin
haftmann@53067
   538
haftmann@53067
   539
lemma diff_zero [simp]:
haftmann@53067
   540
  "a - 0 = a"
haftmann@53067
   541
  by (rule diff_invert_add1 [symmetric]) simp
haftmann@53067
   542
haftmann@53067
   543
lemma parity:
haftmann@53067
   544
  "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@53067
   545
proof (rule ccontr)
haftmann@53067
   546
  assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@53067
   547
  then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@53067
   548
  have "0 < 2" by simp
haftmann@53067
   549
  with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
haftmann@53067
   550
  with `a mod 2 \<noteq> 0` have "0 < a mod 2" by simp
haftmann@53067
   551
  with discrete have "1 \<le> a mod 2" by simp
haftmann@53067
   552
  with `a mod 2 \<noteq> 1` have "1 < a mod 2" by simp
haftmann@53067
   553
  with discrete have "2 \<le> a mod 2" by simp
haftmann@53067
   554
  with `a mod 2 < 2` show False by simp
haftmann@53067
   555
qed
haftmann@53067
   556
haftmann@53067
   557
lemma divmod_digit_1:
haftmann@53067
   558
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067
   559
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067
   560
    and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067
   561
proof -
haftmann@53067
   562
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067
   563
    by (auto intro: trans)
haftmann@53067
   564
  with `0 < b` have "0 < a div b" by (auto intro: div_positive)
haftmann@53067
   565
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
haftmann@53067
   566
  with `0 < b` have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
haftmann@53067
   567
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   568
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   569
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   570
  from assms w_exhaust have "w = 1"
haftmann@53067
   571
    by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067
   572
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067
   573
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   574
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067
   575
  with `w = 1` have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067
   576
  then show ?P and ?Q
haftmann@53067
   577
    by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)
haftmann@53067
   578
qed
haftmann@53067
   579
haftmann@53067
   580
lemma divmod_digit_0:
haftmann@53067
   581
  assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067
   582
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067
   583
    and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067
   584
proof -
haftmann@53067
   585
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   586
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   587
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   588
  moreover have "b \<le> a mod b + b"
haftmann@53067
   589
  proof -
haftmann@53067
   590
    from `0 < b` pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067
   591
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067
   592
    then show ?thesis by simp
haftmann@53067
   593
  qed
haftmann@53067
   594
  moreover note assms w_exhaust
haftmann@53067
   595
  ultimately have "w = 0" by auto
haftmann@53067
   596
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067
   597
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   598
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067
   599
  with `w = 0` have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067
   600
  then show ?P and ?Q
haftmann@53067
   601
    by (simp_all add: div mod)
haftmann@53067
   602
qed
haftmann@53067
   603
haftmann@53067
   604
definition divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@53067
   605
where
haftmann@53067
   606
  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@53067
   607
haftmann@53067
   608
lemma fst_divmod [simp]:
haftmann@53067
   609
  "fst (divmod m n) = numeral m div numeral n"
haftmann@53067
   610
  by (simp add: divmod_def)
haftmann@53067
   611
haftmann@53067
   612
lemma snd_divmod [simp]:
haftmann@53067
   613
  "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067
   614
  by (simp add: divmod_def)
haftmann@53067
   615
haftmann@53067
   616
definition divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@53067
   617
where
haftmann@53067
   618
  "divmod_step l qr = (let (q, r) = qr
haftmann@53067
   619
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@53067
   620
    else (2 * q, r))"
haftmann@53067
   621
haftmann@53067
   622
text {*
haftmann@53067
   623
  This is a formulation of one step (referring to one digit position)
haftmann@53067
   624
  in school-method division: compare the dividend at the current
haftmann@53070
   625
  digit position with the remainder from previous division steps
haftmann@53067
   626
  and evaluate accordingly.
haftmann@53067
   627
*}
haftmann@53067
   628
haftmann@53067
   629
lemma divmod_step_eq [code]:
haftmann@53067
   630
  "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067
   631
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067
   632
  by (simp add: divmod_step_def)
haftmann@53067
   633
haftmann@53067
   634
lemma divmod_step_simps [simp]:
haftmann@53067
   635
  "r < numeral l \<Longrightarrow> divmod_step l (q, r) = (2 * q, r)"
haftmann@53067
   636
  "numeral l \<le> r \<Longrightarrow> divmod_step l (q, r) = (2 * q + 1, r - numeral l)"
haftmann@53067
   637
  by (auto simp add: divmod_step_eq not_le)
haftmann@53067
   638
haftmann@53067
   639
text {*
haftmann@53067
   640
  This is a formulation of school-method division.
haftmann@53067
   641
  If the divisor is smaller than the dividend, terminate.
haftmann@53067
   642
  If not, shift the dividend to the right until termination
haftmann@53067
   643
  occurs and then reiterate single division steps in the
haftmann@53067
   644
  opposite direction.
haftmann@53067
   645
*}
haftmann@53067
   646
haftmann@53067
   647
lemma divmod_divmod_step [code]:
haftmann@53067
   648
  "divmod m n = (if m < n then (0, numeral m)
haftmann@53067
   649
    else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067
   650
proof (cases "m < n")
haftmann@53067
   651
  case True then have "numeral m < numeral n" by simp
haftmann@53067
   652
  then show ?thesis
haftmann@53067
   653
    by (simp add: prod_eq_iff div_less mod_less)
haftmann@53067
   654
next
haftmann@53067
   655
  case False
haftmann@53067
   656
  have "divmod m n =
haftmann@53067
   657
    divmod_step n (numeral m div (2 * numeral n),
haftmann@53067
   658
      numeral m mod (2 * numeral n))"
haftmann@53067
   659
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067
   660
    case True
haftmann@53067
   661
    with divmod_step_simps
haftmann@53067
   662
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   663
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@53067
   664
        by blast
haftmann@53067
   665
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067
   666
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067
   667
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067
   668
      by simp_all
haftmann@53067
   669
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   670
  next
haftmann@53067
   671
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067
   672
      by (simp add: not_le)
haftmann@53067
   673
    with divmod_step_simps
haftmann@53067
   674
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   675
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@53067
   676
        by blast
haftmann@53067
   677
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067
   678
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067
   679
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067
   680
      by (simp_all only: zero_less_numeral)
haftmann@53067
   681
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   682
  qed
haftmann@53067
   683
  then have "divmod m n =
haftmann@53067
   684
    divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067
   685
      numeral m mod numeral (Num.Bit0 n))"
haftmann@53067
   686
    by (simp only: numeral.simps distrib mult_1) 
haftmann@53067
   687
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067
   688
    by (simp add: divmod_def)
haftmann@53067
   689
  with False show ?thesis by simp
haftmann@53067
   690
qed
haftmann@53067
   691
haftmann@53069
   692
lemma divmod_cancel [code]:
haftmann@53069
   693
  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
haftmann@53069
   694
  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
haftmann@53069
   695
proof -
haftmann@53069
   696
  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069
   697
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069
   698
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069
   699
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069
   700
  then show ?P and ?Q
haftmann@53069
   701
    by (simp_all add: prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@53069
   702
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2] add.commute del: numeral_times_numeral)
haftmann@53069
   703
 qed
haftmann@53069
   704
haftmann@53067
   705
end
haftmann@53067
   706
haftmann@53067
   707
hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero
haftmann@53067
   708
  -- {* restore simple accesses for more general variants of theorems *}
haftmann@53067
   709
haftmann@53067
   710
  
haftmann@26100
   711
subsection {* Division on @{typ nat} *}
haftmann@26100
   712
haftmann@26100
   713
text {*
haftmann@26100
   714
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   715
  of a characteristic relation with two input arguments
haftmann@26100
   716
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   717
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   718
*}
haftmann@26100
   719
haftmann@33340
   720
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   721
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   722
    m = fst qr * n + snd qr \<and>
haftmann@30923
   723
      (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
   724
haftmann@33340
   725
text {* @{const divmod_nat_rel} is total: *}
haftmann@26100
   726
haftmann@33340
   727
lemma divmod_nat_rel_ex:
haftmann@33340
   728
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   729
proof (cases "n = 0")
haftmann@30923
   730
  case True  with that show thesis
haftmann@33340
   731
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   732
next
haftmann@26100
   733
  case False
haftmann@26100
   734
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   735
  proof (induct m)
haftmann@26100
   736
    case 0 with `n \<noteq> 0`
haftmann@26100
   737
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   738
    then show ?case by blast
haftmann@26100
   739
  next
haftmann@26100
   740
    case (Suc m) then obtain q' r'
haftmann@26100
   741
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   742
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   743
      case True
haftmann@26100
   744
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   745
      with True show ?thesis by blast
haftmann@26100
   746
    next
haftmann@26100
   747
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   748
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   749
      ultimately have "n = Suc r'" by auto
haftmann@26100
   750
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   751
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   752
    qed
haftmann@26100
   753
  qed
haftmann@26100
   754
  with that show thesis
haftmann@33340
   755
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   756
qed
haftmann@26100
   757
haftmann@33340
   758
text {* @{const divmod_nat_rel} is injective: *}
haftmann@26100
   759
haftmann@33340
   760
lemma divmod_nat_rel_unique:
haftmann@33340
   761
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   762
    and "divmod_nat_rel m n qr'"
haftmann@30923
   763
  shows "qr = qr'"
haftmann@26100
   764
proof (cases "n = 0")
haftmann@26100
   765
  case True with assms show ?thesis
haftmann@30923
   766
    by (cases qr, cases qr')
haftmann@33340
   767
      (simp add: divmod_nat_rel_def)
haftmann@26100
   768
next
haftmann@26100
   769
  case False
haftmann@26100
   770
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   771
  apply (rule leI)
haftmann@26100
   772
  apply (subst less_iff_Suc_add)
haftmann@26100
   773
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   774
  done
wenzelm@53374
   775
  from `n \<noteq> 0` assms have *: "fst qr = fst qr'"
haftmann@33340
   776
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
wenzelm@53374
   777
  with assms have "snd qr = snd qr'"
haftmann@33340
   778
    by (simp add: divmod_nat_rel_def)
wenzelm@53374
   779
  with * show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   780
qed
haftmann@26100
   781
haftmann@26100
   782
text {*
haftmann@26100
   783
  We instantiate divisibility on the natural numbers by
haftmann@33340
   784
  means of @{const divmod_nat_rel}:
haftmann@26100
   785
*}
haftmann@25942
   786
haftmann@33340
   787
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767
   788
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   789
haftmann@33340
   790
lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   791
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   792
proof -
haftmann@33340
   793
  from divmod_nat_rel_ex
haftmann@33340
   794
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   795
  then show ?thesis
haftmann@33340
   796
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   797
qed
haftmann@30923
   798
huffman@47135
   799
lemma divmod_nat_unique:
haftmann@33340
   800
  assumes "divmod_nat_rel m n qr" 
haftmann@33340
   801
  shows "divmod_nat m n = qr"
haftmann@33340
   802
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   803
huffman@46551
   804
instantiation nat :: semiring_div
huffman@46551
   805
begin
huffman@46551
   806
haftmann@26100
   807
definition div_nat where
haftmann@33340
   808
  "m div n = fst (divmod_nat m n)"
haftmann@26100
   809
huffman@46551
   810
lemma fst_divmod_nat [simp]:
huffman@46551
   811
  "fst (divmod_nat m n) = m div n"
huffman@46551
   812
  by (simp add: div_nat_def)
huffman@46551
   813
haftmann@26100
   814
definition mod_nat where
haftmann@33340
   815
  "m mod n = snd (divmod_nat m n)"
haftmann@25571
   816
huffman@46551
   817
lemma snd_divmod_nat [simp]:
huffman@46551
   818
  "snd (divmod_nat m n) = m mod n"
huffman@46551
   819
  by (simp add: mod_nat_def)
huffman@46551
   820
haftmann@33340
   821
lemma divmod_nat_div_mod:
haftmann@33340
   822
  "divmod_nat m n = (m div n, m mod n)"
huffman@46551
   823
  by (simp add: prod_eq_iff)
haftmann@26100
   824
huffman@47135
   825
lemma div_nat_unique:
haftmann@33340
   826
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   827
  shows "m div n = q"
huffman@47135
   828
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   829
huffman@47135
   830
lemma mod_nat_unique:
haftmann@33340
   831
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   832
  shows "m mod n = r"
huffman@47135
   833
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   834
haftmann@33340
   835
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
huffman@46551
   836
  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
paulson@14267
   837
huffman@47136
   838
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
huffman@47136
   839
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
huffman@47136
   840
huffman@47136
   841
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
huffman@47136
   842
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   843
huffman@47137
   844
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
huffman@47137
   845
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   846
haftmann@33340
   847
lemma divmod_nat_step:
haftmann@26100
   848
  assumes "0 < n" and "n \<le> m"
haftmann@33340
   849
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
huffman@47135
   850
proof (rule divmod_nat_unique)
huffman@47134
   851
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
huffman@47134
   852
    by (rule divmod_nat_rel)
huffman@47134
   853
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
huffman@47134
   854
    unfolding divmod_nat_rel_def using assms by auto
haftmann@26100
   855
qed
haftmann@25942
   856
wenzelm@26300
   857
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   858
haftmann@26100
   859
lemma div_less [simp]:
haftmann@26100
   860
  fixes m n :: nat
haftmann@26100
   861
  assumes "m < n"
haftmann@26100
   862
  shows "m div n = 0"
huffman@46551
   863
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   864
haftmann@26100
   865
lemma le_div_geq:
haftmann@26100
   866
  fixes m n :: nat
haftmann@26100
   867
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   868
  shows "m div n = Suc ((m - n) div n)"
huffman@46551
   869
  using assms divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   870
haftmann@26100
   871
lemma mod_less [simp]:
haftmann@26100
   872
  fixes m n :: nat
haftmann@26100
   873
  assumes "m < n"
haftmann@26100
   874
  shows "m mod n = m"
huffman@46551
   875
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   876
haftmann@26100
   877
lemma le_mod_geq:
haftmann@26100
   878
  fixes m n :: nat
haftmann@26100
   879
  assumes "n \<le> m"
haftmann@26100
   880
  shows "m mod n = (m - n) mod n"
huffman@46551
   881
  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   882
huffman@47136
   883
instance proof
huffman@47136
   884
  fix m n :: nat
huffman@47136
   885
  show "m div n * n + m mod n = m"
huffman@47136
   886
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136
   887
next
huffman@47136
   888
  fix m n q :: nat
huffman@47136
   889
  assume "n \<noteq> 0"
huffman@47136
   890
  then show "(q + m * n) div n = m + q div n"
huffman@47136
   891
    by (induct m) (simp_all add: le_div_geq)
huffman@47136
   892
next
huffman@47136
   893
  fix m n q :: nat
huffman@47136
   894
  assume "m \<noteq> 0"
huffman@47136
   895
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136
   896
    unfolding divmod_nat_rel_def
huffman@47136
   897
    by (auto split: split_if_asm, simp_all add: algebra_simps)
huffman@47136
   898
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136
   899
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136
   900
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136
   901
next
huffman@47136
   902
  fix n :: nat show "n div 0 = 0"
haftmann@33340
   903
    by (simp add: div_nat_def divmod_nat_zero)
huffman@47136
   904
next
huffman@47136
   905
  fix n :: nat show "0 div n = 0"
huffman@47136
   906
    by (simp add: div_nat_def divmod_nat_zero_left)
haftmann@25942
   907
qed
haftmann@26100
   908
haftmann@25942
   909
end
paulson@14267
   910
haftmann@33361
   911
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@33361
   912
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
huffman@46551
   913
  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
haftmann@33361
   914
haftmann@26100
   915
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   916
wenzelm@51299
   917
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299
   918
haftmann@30934
   919
ML {*
wenzelm@43594
   920
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
   921
(
haftmann@30934
   922
  val div_name = @{const_name div};
haftmann@30934
   923
  val mod_name = @{const_name mod};
haftmann@30934
   924
  val mk_binop = HOLogic.mk_binop;
huffman@48561
   925
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
   926
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
   927
  fun mk_sum [] = HOLogic.zero
huffman@48561
   928
    | mk_sum [t] = t
huffman@48561
   929
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
   930
  fun dest_sum tm =
huffman@48561
   931
    if HOLogic.is_zero tm then []
huffman@48561
   932
    else
huffman@48561
   933
      (case try HOLogic.dest_Suc tm of
huffman@48561
   934
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
   935
      | NONE =>
huffman@48561
   936
          (case try dest_plus tm of
huffman@48561
   937
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
   938
          | NONE => [tm]));
haftmann@25942
   939
haftmann@30934
   940
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
   941
haftmann@30934
   942
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@35050
   943
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
wenzelm@41550
   944
)
haftmann@25942
   945
*}
haftmann@25942
   946
wenzelm@43594
   947
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
wenzelm@43594
   948
haftmann@26100
   949
haftmann@26100
   950
subsubsection {* Quotient *}
haftmann@26100
   951
haftmann@26100
   952
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   953
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   954
haftmann@26100
   955
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   956
by (simp add: div_geq)
haftmann@26100
   957
haftmann@26100
   958
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   959
by simp
haftmann@26100
   960
haftmann@26100
   961
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   962
by simp
haftmann@26100
   963
haftmann@53066
   964
lemma div_positive:
haftmann@53066
   965
  fixes m n :: nat
haftmann@53066
   966
  assumes "n > 0"
haftmann@53066
   967
  assumes "m \<ge> n"
haftmann@53066
   968
  shows "m div n > 0"
haftmann@53066
   969
proof -
haftmann@53066
   970
  from `m \<ge> n` obtain q where "m = n + q"
haftmann@53066
   971
    by (auto simp add: le_iff_add)
haftmann@53066
   972
  with `n > 0` show ?thesis by simp
haftmann@53066
   973
qed
haftmann@53066
   974
haftmann@25942
   975
haftmann@25942
   976
subsubsection {* Remainder *}
haftmann@25942
   977
haftmann@26100
   978
lemma mod_less_divisor [simp]:
haftmann@26100
   979
  fixes m n :: nat
haftmann@26100
   980
  assumes "n > 0"
haftmann@26100
   981
  shows "m mod n < (n::nat)"
haftmann@33340
   982
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
   983
haftmann@51173
   984
lemma mod_Suc_le_divisor [simp]:
haftmann@51173
   985
  "m mod Suc n \<le> n"
haftmann@51173
   986
  using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173
   987
haftmann@26100
   988
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   989
  fixes m n :: nat
haftmann@26100
   990
  shows "m mod n \<le> m"
haftmann@26100
   991
proof (rule add_leD2)
haftmann@26100
   992
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   993
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   994
qed
haftmann@26100
   995
haftmann@26100
   996
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
   997
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   998
haftmann@26100
   999
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
  1000
by (simp add: le_mod_geq)
haftmann@26100
  1001
paulson@14267
  1002
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
  1003
by (induct m) (simp_all add: mod_geq)
paulson@14267
  1004
paulson@14267
  1005
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
  1006
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
huffman@47138
  1007
  using mod_div_equality2 [of n m] by arith
paulson@14267
  1008
nipkow@15439
  1009
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
  1010
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
  1011
  apply simp
wenzelm@22718
  1012
  done
paulson@14267
  1013
haftmann@26100
  1014
subsubsection {* Quotient and Remainder *}
paulson@14267
  1015
haftmann@33340
  1016
lemma divmod_nat_rel_mult1_eq:
bulwahn@46552
  1017
  "divmod_nat_rel b c (q, r)
haftmann@33340
  1018
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
  1019
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1020
haftmann@30923
  1021
lemma div_mult1_eq:
haftmann@30923
  1022
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135
  1023
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1024
haftmann@33340
  1025
lemma divmod_nat_rel_add1_eq:
bulwahn@46552
  1026
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340
  1027
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
  1028
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
  1029
paulson@14267
  1030
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
  1031
lemma div_add1_eq:
nipkow@25134
  1032
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135
  1033
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
  1034
paulson@14267
  1035
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
  1036
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
  1037
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
  1038
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
  1039
  apply (simp add: add_mult_distrib2)
wenzelm@22718
  1040
  done
paulson@10559
  1041
haftmann@33340
  1042
lemma divmod_nat_rel_mult2_eq:
bulwahn@46552
  1043
  "divmod_nat_rel a b (q, r)
haftmann@33340
  1044
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@33340
  1045
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
  1046
paulson@14267
  1047
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
huffman@47135
  1048
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267
  1049
paulson@14267
  1050
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
huffman@47135
  1051
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267
  1052
paulson@14267
  1053
huffman@46551
  1054
subsubsection {* Further Facts about Quotient and Remainder *}
paulson@14267
  1055
paulson@14267
  1056
lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667
  1057
by (induct m) (simp_all add: div_geq)
paulson@14267
  1058
paulson@14267
  1059
(* Monotonicity of div in first argument *)
haftmann@30923
  1060
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
  1061
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
  1062
apply (case_tac "k=0", simp)
paulson@15251
  1063
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
  1064
apply (case_tac "n<k")
paulson@14267
  1065
(* 1  case n<k *)
paulson@14267
  1066
apply simp
paulson@14267
  1067
(* 2  case n >= k *)
paulson@14267
  1068
apply (case_tac "m<k")
paulson@14267
  1069
(* 2.1  case m<k *)
paulson@14267
  1070
apply simp
paulson@14267
  1071
(* 2.2  case m>=k *)
nipkow@15439
  1072
apply (simp add: div_geq diff_le_mono)
paulson@14267
  1073
done
paulson@14267
  1074
paulson@14267
  1075
(* Antimonotonicity of div in second argument *)
paulson@14267
  1076
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
  1077
apply (subgoal_tac "0<n")
wenzelm@22718
  1078
 prefer 2 apply simp
paulson@15251
  1079
apply (induct_tac k rule: nat_less_induct)
paulson@14267
  1080
apply (rename_tac "k")
paulson@14267
  1081
apply (case_tac "k<n", simp)
paulson@14267
  1082
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
  1083
 prefer 2 apply simp
paulson@14267
  1084
apply (simp add: div_geq)
paulson@15251
  1085
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
  1086
 prefer 2
paulson@14267
  1087
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
  1088
apply (rule le_trans, simp)
nipkow@15439
  1089
apply (simp)
paulson@14267
  1090
done
paulson@14267
  1091
paulson@14267
  1092
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
  1093
apply (case_tac "n=0", simp)
paulson@14267
  1094
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
  1095
apply (rule div_le_mono2)
paulson@14267
  1096
apply (simp_all (no_asm_simp))
paulson@14267
  1097
done
paulson@14267
  1098
wenzelm@22718
  1099
(* Similar for "less than" *)
huffman@47138
  1100
lemma div_less_dividend [simp]:
huffman@47138
  1101
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
  1102
apply (induct m rule: nat_less_induct)
paulson@14267
  1103
apply (rename_tac "m")
paulson@14267
  1104
apply (case_tac "m<n", simp)
paulson@14267
  1105
apply (subgoal_tac "0<n")
wenzelm@22718
  1106
 prefer 2 apply simp
paulson@14267
  1107
apply (simp add: div_geq)
paulson@14267
  1108
apply (case_tac "n<m")
paulson@15251
  1109
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
  1110
  apply (rule impI less_trans_Suc)+
paulson@14267
  1111
apply assumption
nipkow@15439
  1112
  apply (simp_all)
paulson@14267
  1113
done
paulson@14267
  1114
paulson@14267
  1115
text{*A fact for the mutilated chess board*}
paulson@14267
  1116
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
  1117
apply (case_tac "n=0", simp)
paulson@15251
  1118
apply (induct "m" rule: nat_less_induct)
paulson@14267
  1119
apply (case_tac "Suc (na) <n")
paulson@14267
  1120
(* case Suc(na) < n *)
paulson@14267
  1121
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
  1122
(* case n \<le> Suc(na) *)
paulson@16796
  1123
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
  1124
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
  1125
done
paulson@14267
  1126
paulson@14267
  1127
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
  1128
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
  1129
wenzelm@22718
  1130
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
  1131
paulson@14267
  1132
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
  1133
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
haftmann@27651
  1134
  apply (cut_tac a = m in mod_div_equality)
wenzelm@22718
  1135
  apply (simp only: add_ac)
wenzelm@22718
  1136
  apply (blast intro: sym)
wenzelm@22718
  1137
  done
paulson@14267
  1138
nipkow@13152
  1139
lemma split_div:
nipkow@13189
  1140
 "P(n div k :: nat) =
nipkow@13189
  1141
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
  1142
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1143
proof
nipkow@13189
  1144
  assume P: ?P
nipkow@13189
  1145
  show ?Q
nipkow@13189
  1146
  proof (cases)
nipkow@13189
  1147
    assume "k = 0"
haftmann@27651
  1148
    with P show ?Q by simp
nipkow@13189
  1149
  next
nipkow@13189
  1150
    assume not0: "k \<noteq> 0"
nipkow@13189
  1151
    thus ?Q
nipkow@13189
  1152
    proof (simp, intro allI impI)
nipkow@13189
  1153
      fix i j
nipkow@13189
  1154
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
  1155
      show "P i"
nipkow@13189
  1156
      proof (cases)
wenzelm@22718
  1157
        assume "i = 0"
wenzelm@22718
  1158
        with n j P show "P i" by simp
nipkow@13189
  1159
      next
wenzelm@22718
  1160
        assume "i \<noteq> 0"
wenzelm@22718
  1161
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
  1162
      qed
nipkow@13189
  1163
    qed
nipkow@13189
  1164
  qed
nipkow@13189
  1165
next
nipkow@13189
  1166
  assume Q: ?Q
nipkow@13189
  1167
  show ?P
nipkow@13189
  1168
  proof (cases)
nipkow@13189
  1169
    assume "k = 0"
haftmann@27651
  1170
    with Q show ?P by simp
nipkow@13189
  1171
  next
nipkow@13189
  1172
    assume not0: "k \<noteq> 0"
nipkow@13189
  1173
    with Q have R: ?R by simp
nipkow@13189
  1174
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1175
    show ?P by simp
nipkow@13189
  1176
  qed
nipkow@13189
  1177
qed
nipkow@13189
  1178
berghofe@13882
  1179
lemma split_div_lemma:
haftmann@26100
  1180
  assumes "0 < n"
haftmann@26100
  1181
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
  1182
proof
haftmann@26100
  1183
  assume ?rhs
haftmann@26100
  1184
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
  1185
  then have A: "n * q \<le> m" by simp
haftmann@26100
  1186
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
  1187
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
  1188
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
  1189
  with nq have "m < n + n * q" by simp
haftmann@26100
  1190
  then have B: "m < n * Suc q" by simp
haftmann@26100
  1191
  from A B show ?lhs ..
haftmann@26100
  1192
next
haftmann@26100
  1193
  assume P: ?lhs
haftmann@33340
  1194
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@33340
  1195
    unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
haftmann@33340
  1196
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
haftmann@30923
  1197
  have "(q, m - n * q) = (m div n, m mod n)" by auto
haftmann@30923
  1198
  then show ?rhs by simp
haftmann@26100
  1199
qed
berghofe@13882
  1200
berghofe@13882
  1201
theorem split_div':
berghofe@13882
  1202
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
  1203
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
  1204
  apply (case_tac "0 < n")
berghofe@13882
  1205
  apply (simp only: add: split_div_lemma)
haftmann@27651
  1206
  apply simp_all
berghofe@13882
  1207
  done
berghofe@13882
  1208
nipkow@13189
  1209
lemma split_mod:
nipkow@13189
  1210
 "P(n mod k :: nat) =
nipkow@13189
  1211
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
  1212
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1213
proof
nipkow@13189
  1214
  assume P: ?P
nipkow@13189
  1215
  show ?Q
nipkow@13189
  1216
  proof (cases)
nipkow@13189
  1217
    assume "k = 0"
haftmann@27651
  1218
    with P show ?Q by simp
nipkow@13189
  1219
  next
nipkow@13189
  1220
    assume not0: "k \<noteq> 0"
nipkow@13189
  1221
    thus ?Q
nipkow@13189
  1222
    proof (simp, intro allI impI)
nipkow@13189
  1223
      fix i j
nipkow@13189
  1224
      assume "n = k*i + j" "j < k"
nipkow@13189
  1225
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
  1226
    qed
nipkow@13189
  1227
  qed
nipkow@13189
  1228
next
nipkow@13189
  1229
  assume Q: ?Q
nipkow@13189
  1230
  show ?P
nipkow@13189
  1231
  proof (cases)
nipkow@13189
  1232
    assume "k = 0"
haftmann@27651
  1233
    with Q show ?P by simp
nipkow@13189
  1234
  next
nipkow@13189
  1235
    assume not0: "k \<noteq> 0"
nipkow@13189
  1236
    with Q have R: ?R by simp
nipkow@13189
  1237
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1238
    show ?P by simp
nipkow@13189
  1239
  qed
nipkow@13189
  1240
qed
nipkow@13189
  1241
berghofe@13882
  1242
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
huffman@47138
  1243
  using mod_div_equality [of m n] by arith
huffman@47138
  1244
huffman@47138
  1245
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
huffman@47138
  1246
  using mod_div_equality [of m n] by arith
huffman@47138
  1247
(* FIXME: very similar to mult_div_cancel *)
haftmann@22800
  1248
noschinl@52398
  1249
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398
  1250
  apply rule
noschinl@52398
  1251
  apply (cases "b = 0")
noschinl@52398
  1252
  apply simp_all
noschinl@52398
  1253
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398
  1254
  done
noschinl@52398
  1255
haftmann@22800
  1256
huffman@46551
  1257
subsubsection {* An ``induction'' law for modulus arithmetic. *}
paulson@14640
  1258
paulson@14640
  1259
lemma mod_induct_0:
paulson@14640
  1260
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1261
  and base: "P i" and i: "i<p"
paulson@14640
  1262
  shows "P 0"
paulson@14640
  1263
proof (rule ccontr)
paulson@14640
  1264
  assume contra: "\<not>(P 0)"
paulson@14640
  1265
  from i have p: "0<p" by simp
paulson@14640
  1266
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1267
  proof
paulson@14640
  1268
    fix k
paulson@14640
  1269
    show "?A k"
paulson@14640
  1270
    proof (induct k)
paulson@14640
  1271
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1272
    next
paulson@14640
  1273
      fix n
paulson@14640
  1274
      assume ih: "?A n"
paulson@14640
  1275
      show "?A (Suc n)"
paulson@14640
  1276
      proof (clarsimp)
wenzelm@22718
  1277
        assume y: "P (p - Suc n)"
wenzelm@22718
  1278
        have n: "Suc n < p"
wenzelm@22718
  1279
        proof (rule ccontr)
wenzelm@22718
  1280
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1281
          hence "p - Suc n = 0"
wenzelm@22718
  1282
            by simp
wenzelm@22718
  1283
          with y contra show "False"
wenzelm@22718
  1284
            by simp
wenzelm@22718
  1285
        qed
wenzelm@22718
  1286
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1287
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1288
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1289
          by blast
wenzelm@22718
  1290
        show "False"
wenzelm@22718
  1291
        proof (cases "n=0")
wenzelm@22718
  1292
          case True
wenzelm@22718
  1293
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1294
        next
wenzelm@22718
  1295
          case False
wenzelm@22718
  1296
          with p have "p-n < p" by arith
wenzelm@22718
  1297
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1298
        qed
paulson@14640
  1299
      qed
paulson@14640
  1300
    qed
paulson@14640
  1301
  qed
paulson@14640
  1302
  moreover
paulson@14640
  1303
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1304
    by (blast dest: less_imp_add_positive)
paulson@14640
  1305
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1306
  moreover
paulson@14640
  1307
  note base
paulson@14640
  1308
  ultimately
paulson@14640
  1309
  show "False" by blast
paulson@14640
  1310
qed
paulson@14640
  1311
paulson@14640
  1312
lemma mod_induct:
paulson@14640
  1313
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1314
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1315
  shows "P j"
paulson@14640
  1316
proof -
paulson@14640
  1317
  have "\<forall>j<p. P j"
paulson@14640
  1318
  proof
paulson@14640
  1319
    fix j
paulson@14640
  1320
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1321
    proof (induct j)
paulson@14640
  1322
      from step base i show "?A 0"
wenzelm@22718
  1323
        by (auto elim: mod_induct_0)
paulson@14640
  1324
    next
paulson@14640
  1325
      fix k
paulson@14640
  1326
      assume ih: "?A k"
paulson@14640
  1327
      show "?A (Suc k)"
paulson@14640
  1328
      proof
wenzelm@22718
  1329
        assume suc: "Suc k < p"
wenzelm@22718
  1330
        hence k: "k<p" by simp
wenzelm@22718
  1331
        with ih have "P k" ..
wenzelm@22718
  1332
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1333
          by blast
wenzelm@22718
  1334
        moreover
wenzelm@22718
  1335
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1336
          by simp
wenzelm@22718
  1337
        ultimately
wenzelm@22718
  1338
        show "P (Suc k)" by simp
paulson@14640
  1339
      qed
paulson@14640
  1340
    qed
paulson@14640
  1341
  qed
paulson@14640
  1342
  with j show ?thesis by blast
paulson@14640
  1343
qed
paulson@14640
  1344
haftmann@33296
  1345
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1346
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1347
huffman@47138
  1348
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1349
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1350
haftmann@33296
  1351
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1352
by (simp add: mult_2 [symmetric])
haftmann@33296
  1353
haftmann@33296
  1354
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1355
proof -
boehmes@35815
  1356
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1357
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1358
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1359
  then show ?thesis by auto
haftmann@33296
  1360
qed
haftmann@33296
  1361
haftmann@33296
  1362
text{*These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1363
    at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@33296
  1364
    We already have some rules to simplify operands smaller than 3.*}
haftmann@33296
  1365
haftmann@33296
  1366
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1367
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1368
haftmann@33296
  1369
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1370
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1371
haftmann@33296
  1372
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1373
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1374
haftmann@33296
  1375
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1376
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1377
huffman@47108
  1378
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1379
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1380
haftmann@33361
  1381
haftmann@33361
  1382
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
haftmann@33361
  1383
apply (induct "m")
haftmann@33361
  1384
apply (simp_all add: mod_Suc)
haftmann@33361
  1385
done
haftmann@33361
  1386
huffman@47108
  1387
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1388
huffman@47138
  1389
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1390
by (simp add: div_le_mono)
haftmann@33361
  1391
haftmann@33361
  1392
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1393
by (cases n) simp_all
haftmann@33361
  1394
boehmes@35815
  1395
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1396
proof -
boehmes@35815
  1397
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
boehmes@35815
  1398
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
boehmes@35815
  1399
qed
haftmann@33361
  1400
haftmann@33361
  1401
  (* Potential use of algebra : Equality modulo n*)
haftmann@33361
  1402
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
haftmann@33361
  1403
by (simp add: mult_ac add_ac)
haftmann@33361
  1404
haftmann@33361
  1405
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1406
proof -
haftmann@33361
  1407
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
haftmann@33361
  1408
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
haftmann@33361
  1409
  finally show ?thesis .
haftmann@33361
  1410
qed
haftmann@33361
  1411
haftmann@33361
  1412
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
haftmann@33361
  1413
apply (subst mod_Suc [of m]) 
haftmann@33361
  1414
apply (subst mod_Suc [of "m mod n"], simp) 
haftmann@33361
  1415
done
haftmann@33361
  1416
huffman@47108
  1417
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1418
  fixes n :: nat
huffman@47108
  1419
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
huffman@47108
  1420
  by simp
huffman@47108
  1421
haftmann@53067
  1422
instance nat :: semiring_numeral_div
haftmann@53067
  1423
  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)
haftmann@53067
  1424
haftmann@33361
  1425
haftmann@33361
  1426
subsection {* Division on @{typ int} *}
haftmann@33361
  1427
haftmann@33361
  1428
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
haftmann@33361
  1429
    --{*definition of quotient and remainder*}
huffman@47139
  1430
  "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139
  1431
    (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
  1432
haftmann@53067
  1433
text {*
haftmann@53067
  1434
  The following algorithmic devlopment actually echos what has already
haftmann@53067
  1435
  been developed in class @{class semiring_numeral_div}.  In the long
haftmann@53067
  1436
  run it seems better to derive division on @{typ int} just from
haftmann@53067
  1437
  division on @{typ nat} and instantiate @{class semiring_numeral_div}
haftmann@53067
  1438
  accordingly.
haftmann@53067
  1439
*}
haftmann@53067
  1440
haftmann@33361
  1441
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1442
    --{*for the division algorithm*}
huffman@47108
  1443
    "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@33361
  1444
                         else (2 * q, r))"
haftmann@33361
  1445
haftmann@33361
  1446
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@33361
  1447
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1448
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@33361
  1449
     else adjust b (posDivAlg a (2 * b)))"
haftmann@33361
  1450
by auto
haftmann@33361
  1451
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
haftmann@33361
  1452
  (auto simp add: mult_2)
haftmann@33361
  1453
haftmann@33361
  1454
text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@33361
  1455
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1456
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@33361
  1457
     else adjust b (negDivAlg a (2 * b)))"
haftmann@33361
  1458
by auto
haftmann@33361
  1459
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
haftmann@33361
  1460
  (auto simp add: mult_2)
haftmann@33361
  1461
haftmann@33361
  1462
text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@33361
  1463
haftmann@33361
  1464
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1465
    --{*The full division algorithm considers all possible signs for a, b
haftmann@33361
  1466
       including the special case @{text "a=0, b<0"} because 
haftmann@33361
  1467
       @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@33361
  1468
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@33361
  1469
                  else if a = 0 then (0, 0)
huffman@46560
  1470
                       else apsnd uminus (negDivAlg (-a) (-b))
haftmann@33361
  1471
               else 
haftmann@33361
  1472
                  if 0 < b then negDivAlg a b
huffman@46560
  1473
                  else apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1474
haftmann@33361
  1475
instantiation int :: Divides.div
haftmann@33361
  1476
begin
haftmann@33361
  1477
huffman@46551
  1478
definition div_int where
haftmann@33361
  1479
  "a div b = fst (divmod_int a b)"
haftmann@33361
  1480
huffman@46551
  1481
lemma fst_divmod_int [simp]:
huffman@46551
  1482
  "fst (divmod_int a b) = a div b"
huffman@46551
  1483
  by (simp add: div_int_def)
huffman@46551
  1484
huffman@46551
  1485
definition mod_int where
huffman@46560
  1486
  "a mod b = snd (divmod_int a b)"
haftmann@33361
  1487
huffman@46551
  1488
lemma snd_divmod_int [simp]:
huffman@46551
  1489
  "snd (divmod_int a b) = a mod b"
huffman@46551
  1490
  by (simp add: mod_int_def)
huffman@46551
  1491
haftmann@33361
  1492
instance ..
haftmann@33361
  1493
paulson@3366
  1494
end
haftmann@33361
  1495
haftmann@33361
  1496
lemma divmod_int_mod_div:
haftmann@33361
  1497
  "divmod_int p q = (p div q, p mod q)"
huffman@46551
  1498
  by (simp add: prod_eq_iff)
haftmann@33361
  1499
haftmann@33361
  1500
text{*
haftmann@33361
  1501
Here is the division algorithm in ML:
haftmann@33361
  1502
haftmann@33361
  1503
\begin{verbatim}
haftmann@33361
  1504
    fun posDivAlg (a,b) =
haftmann@33361
  1505
      if a<b then (0,a)
haftmann@33361
  1506
      else let val (q,r) = posDivAlg(a, 2*b)
haftmann@33361
  1507
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1508
           end
haftmann@33361
  1509
haftmann@33361
  1510
    fun negDivAlg (a,b) =
haftmann@33361
  1511
      if 0\<le>a+b then (~1,a+b)
haftmann@33361
  1512
      else let val (q,r) = negDivAlg(a, 2*b)
haftmann@33361
  1513
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1514
           end;
haftmann@33361
  1515
haftmann@33361
  1516
    fun negateSnd (q,r:int) = (q,~r);
haftmann@33361
  1517
haftmann@33361
  1518
    fun divmod (a,b) = if 0\<le>a then 
haftmann@33361
  1519
                          if b>0 then posDivAlg (a,b) 
haftmann@33361
  1520
                           else if a=0 then (0,0)
haftmann@33361
  1521
                                else negateSnd (negDivAlg (~a,~b))
haftmann@33361
  1522
                       else 
haftmann@33361
  1523
                          if 0<b then negDivAlg (a,b)
haftmann@33361
  1524
                          else        negateSnd (posDivAlg (~a,~b));
haftmann@33361
  1525
\end{verbatim}
haftmann@33361
  1526
*}
haftmann@33361
  1527
haftmann@33361
  1528
huffman@46551
  1529
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
haftmann@33361
  1530
haftmann@33361
  1531
lemma unique_quotient_lemma:
haftmann@33361
  1532
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
haftmann@33361
  1533
      ==> q' \<le> (q::int)"
haftmann@33361
  1534
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1535
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1536
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1537
apply (erule_tac [2] order_le_less_trans)
webertj@49962
  1538
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1539
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
  1540
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1541
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1542
done
haftmann@33361
  1543
haftmann@33361
  1544
lemma unique_quotient_lemma_neg:
haftmann@33361
  1545
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
haftmann@33361
  1546
      ==> q \<le> (q'::int)"
haftmann@33361
  1547
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
haftmann@33361
  1548
    auto)
haftmann@33361
  1549
haftmann@33361
  1550
lemma unique_quotient:
bulwahn@46552
  1551
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1552
      ==> q = q'"
haftmann@33361
  1553
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361
  1554
apply (blast intro: order_antisym
haftmann@33361
  1555
             dest: order_eq_refl [THEN unique_quotient_lemma] 
haftmann@33361
  1556
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1557
done
haftmann@33361
  1558
haftmann@33361
  1559
haftmann@33361
  1560
lemma unique_remainder:
bulwahn@46552
  1561
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1562
      ==> r = r'"
haftmann@33361
  1563
apply (subgoal_tac "q = q'")
haftmann@33361
  1564
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1565
apply (blast intro: unique_quotient)
haftmann@33361
  1566
done
haftmann@33361
  1567
haftmann@33361
  1568
huffman@46551
  1569
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
haftmann@33361
  1570
haftmann@33361
  1571
text{*And positive divisors*}
haftmann@33361
  1572
haftmann@33361
  1573
lemma adjust_eq [simp]:
huffman@47108
  1574
     "adjust b (q, r) = 
huffman@47108
  1575
      (let diff = r - b in  
huffman@47108
  1576
        if 0 \<le> diff then (2 * q + 1, diff)   
haftmann@33361
  1577
                     else (2*q, r))"
huffman@47108
  1578
  by (simp add: Let_def adjust_def)
haftmann@33361
  1579
haftmann@33361
  1580
declare posDivAlg.simps [simp del]
haftmann@33361
  1581
haftmann@33361
  1582
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1583
lemma posDivAlg_eqn:
haftmann@33361
  1584
     "0 < b ==>  
haftmann@33361
  1585
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
haftmann@33361
  1586
by (rule posDivAlg.simps [THEN trans], simp)
haftmann@33361
  1587
haftmann@33361
  1588
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
haftmann@33361
  1589
theorem posDivAlg_correct:
haftmann@33361
  1590
  assumes "0 \<le> a" and "0 < b"
haftmann@33361
  1591
  shows "divmod_int_rel a b (posDivAlg a b)"
wenzelm@41550
  1592
  using assms
wenzelm@41550
  1593
  apply (induct a b rule: posDivAlg.induct)
wenzelm@41550
  1594
  apply auto
wenzelm@41550
  1595
  apply (simp add: divmod_int_rel_def)
webertj@49962
  1596
  apply (subst posDivAlg_eqn, simp add: distrib_left)
wenzelm@41550
  1597
  apply (case_tac "a < b")
wenzelm@41550
  1598
  apply simp_all
wenzelm@41550
  1599
  apply (erule splitE)
webertj@49962
  1600
  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)
wenzelm@41550
  1601
  done
haftmann@33361
  1602
haftmann@33361
  1603
huffman@46551
  1604
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
haftmann@33361
  1605
haftmann@33361
  1606
text{*And positive divisors*}
haftmann@33361
  1607
haftmann@33361
  1608
declare negDivAlg.simps [simp del]
haftmann@33361
  1609
haftmann@33361
  1610
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1611
lemma negDivAlg_eqn:
haftmann@33361
  1612
     "0 < b ==>  
haftmann@33361
  1613
      negDivAlg a b =       
haftmann@33361
  1614
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
haftmann@33361
  1615
by (rule negDivAlg.simps [THEN trans], simp)
haftmann@33361
  1616
haftmann@33361
  1617
(*Correctness of negDivAlg: it computes quotients correctly
haftmann@33361
  1618
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
haftmann@33361
  1619
lemma negDivAlg_correct:
haftmann@33361
  1620
  assumes "a < 0" and "b > 0"
haftmann@33361
  1621
  shows "divmod_int_rel a b (negDivAlg a b)"
wenzelm@41550
  1622
  using assms
wenzelm@41550
  1623
  apply (induct a b rule: negDivAlg.induct)
wenzelm@41550
  1624
  apply (auto simp add: linorder_not_le)
wenzelm@41550
  1625
  apply (simp add: divmod_int_rel_def)
wenzelm@41550
  1626
  apply (subst negDivAlg_eqn, assumption)
wenzelm@41550
  1627
  apply (case_tac "a + b < (0\<Colon>int)")
wenzelm@41550
  1628
  apply simp_all
wenzelm@41550
  1629
  apply (erule splitE)
webertj@49962
  1630
  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)
wenzelm@41550
  1631
  done
haftmann@33361
  1632
haftmann@33361
  1633
huffman@46551
  1634
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
haftmann@33361
  1635
haftmann@33361
  1636
(*the case a=0*)
huffman@47139
  1637
lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
haftmann@33361
  1638
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1639
haftmann@33361
  1640
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
haftmann@33361
  1641
by (subst posDivAlg.simps, auto)
haftmann@33361
  1642
huffman@47139
  1643
lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
huffman@47139
  1644
by (subst posDivAlg.simps, auto)
huffman@47139
  1645
haftmann@33361
  1646
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
haftmann@33361
  1647
by (subst negDivAlg.simps, auto)
haftmann@33361
  1648
huffman@46560
  1649
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
huffman@47139
  1650
by (auto simp add: divmod_int_rel_def)
huffman@47139
  1651
huffman@47139
  1652
lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
huffman@47139
  1653
apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
haftmann@33361
  1654
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
haftmann@33361
  1655
                    posDivAlg_correct negDivAlg_correct)
haftmann@33361
  1656
huffman@47141
  1657
lemma divmod_int_unique:
huffman@47141
  1658
  assumes "divmod_int_rel a b qr" 
huffman@47141
  1659
  shows "divmod_int a b = qr"
huffman@47141
  1660
  using assms divmod_int_correct [of a b]
huffman@47141
  1661
  using unique_quotient [of a b] unique_remainder [of a b]
huffman@47141
  1662
  by (metis pair_collapse)
huffman@47141
  1663
huffman@47141
  1664
lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
huffman@47141
  1665
  using divmod_int_correct by (simp add: divmod_int_mod_div)
huffman@47141
  1666
huffman@47141
  1667
lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
huffman@47141
  1668
  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
huffman@47141
  1669
huffman@47141
  1670
lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
huffman@47141
  1671
  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
huffman@47141
  1672
huffman@47141
  1673
instance int :: ring_div
huffman@47141
  1674
proof
huffman@47141
  1675
  fix a b :: int
huffman@47141
  1676
  show "a div b * b + a mod b = a"
huffman@47141
  1677
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1678
    unfolding divmod_int_rel_def by (simp add: mult_commute)
huffman@47141
  1679
next
huffman@47141
  1680
  fix a b c :: int
huffman@47141
  1681
  assume "b \<noteq> 0"
huffman@47141
  1682
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
huffman@47141
  1683
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1684
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141
  1685
  thus "(a + c * b) div b = c + a div b"
huffman@47141
  1686
    by (rule div_int_unique)
huffman@47141
  1687
next
huffman@47141
  1688
  fix a b c :: int
huffman@47141
  1689
  assume "c \<noteq> 0"
huffman@47141
  1690
  hence "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141
  1691
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141
  1692
    unfolding divmod_int_rel_def
huffman@47141
  1693
    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
huffman@47141
  1694
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
huffman@47141
  1695
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
huffman@47141
  1696
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
huffman@47141
  1697
    using divmod_int_rel_div_mod [of a b] .
huffman@47141
  1698
  thus "(c * a) div (c * b) = a div b"
huffman@47141
  1699
    by (rule div_int_unique)
huffman@47141
  1700
next
huffman@47141
  1701
  fix a :: int show "a div 0 = 0"
huffman@47141
  1702
    by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141
  1703
next
huffman@47141
  1704
  fix a :: int show "0 div a = 0"
huffman@47141
  1705
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141
  1706
qed
huffman@47141
  1707
haftmann@33361
  1708
text{*Basic laws about division and remainder*}
haftmann@33361
  1709
haftmann@33361
  1710
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
huffman@47141
  1711
  by (fact mod_div_equality2 [symmetric])
haftmann@33361
  1712
haftmann@33361
  1713
text {* Tool setup *}
haftmann@33361
  1714
huffman@47108
  1715
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
huffman@47108
  1716
lemmas add_0s = add_0_left add_0_right
huffman@47108
  1717
haftmann@33361
  1718
ML {*
wenzelm@43594
  1719
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550
  1720
(
haftmann@33361
  1721
  val div_name = @{const_name div};
haftmann@33361
  1722
  val mod_name = @{const_name mod};
haftmann@33361
  1723
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1724
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1725
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1726
huffman@47165
  1727
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361
  1728
haftmann@33361
  1729
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
haftmann@33361
  1730
    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
wenzelm@41550
  1731
)
haftmann@33361
  1732
*}
haftmann@33361
  1733
wenzelm@43594
  1734
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
wenzelm@43594
  1735
huffman@47141
  1736
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
huffman@47141
  1737
  using divmod_int_correct [of a b]
huffman@47141
  1738
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1739
wenzelm@45607
  1740
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1741
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1742
huffman@47141
  1743
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
huffman@47141
  1744
  using divmod_int_correct [of a b]
huffman@47141
  1745
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1746
wenzelm@45607
  1747
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1748
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1749
haftmann@33361
  1750
huffman@46551
  1751
subsubsection {* General Properties of div and mod *}
haftmann@33361
  1752
haftmann@33361
  1753
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1754
apply (rule div_int_unique)
haftmann@33361
  1755
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1756
done
haftmann@33361
  1757
haftmann@33361
  1758
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1759
apply (rule div_int_unique)
haftmann@33361
  1760
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1761
done
haftmann@33361
  1762
haftmann@33361
  1763
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1764
apply (rule div_int_unique)
haftmann@33361
  1765
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1766
done
haftmann@33361
  1767
haftmann@33361
  1768
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1769
haftmann@33361
  1770
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1771
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1772
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1773
done
haftmann@33361
  1774
haftmann@33361
  1775
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1776
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1777
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1778
done
haftmann@33361
  1779
haftmann@33361
  1780
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1781
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361
  1782
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1783
done
haftmann@33361
  1784
haftmann@33361
  1785
text{*There is no @{text mod_neg_pos_trivial}.*}
haftmann@33361
  1786
haftmann@33361
  1787
huffman@46551
  1788
subsubsection {* Laws for div and mod with Unary Minus *}
haftmann@33361
  1789
haftmann@33361
  1790
lemma zminus1_lemma:
huffman@47139
  1791
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
haftmann@33361
  1792
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@33361
  1793
                          if r=0 then 0 else b-r)"
haftmann@33361
  1794
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1795
haftmann@33361
  1796
haftmann@33361
  1797
lemma zdiv_zminus1_eq_if:
haftmann@33361
  1798
     "b \<noteq> (0::int)  
haftmann@33361
  1799
      ==> (-a) div b =  
haftmann@33361
  1800
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47140
  1801
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1802
haftmann@33361
  1803
lemma zmod_zminus1_eq_if:
haftmann@33361
  1804
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1805
apply (case_tac "b = 0", simp)
huffman@47140
  1806
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1807
done
haftmann@33361
  1808
haftmann@33361
  1809
lemma zmod_zminus1_not_zero:
haftmann@33361
  1810
  fixes k l :: int
haftmann@33361
  1811
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1812
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1813
haftmann@33361
  1814
lemma zdiv_zminus2_eq_if:
haftmann@33361
  1815
     "b \<noteq> (0::int)  
haftmann@33361
  1816
      ==> a div (-b) =  
haftmann@33361
  1817
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1818
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1819
haftmann@33361
  1820
lemma zmod_zminus2_eq_if:
haftmann@33361
  1821
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1822
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1823
haftmann@33361
  1824
lemma zmod_zminus2_not_zero:
haftmann@33361
  1825
  fixes k l :: int
haftmann@33361
  1826
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1827
  unfolding zmod_zminus2_eq_if by auto 
haftmann@33361
  1828
haftmann@33361
  1829
huffman@46551
  1830
subsubsection {* Computation of Division and Remainder *}
haftmann@33361
  1831
haftmann@33361
  1832
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@33361
  1833
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1834
haftmann@33361
  1835
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@33361
  1836
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1837
haftmann@33361
  1838
text{*a positive, b positive *}
haftmann@33361
  1839
haftmann@33361
  1840
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@33361
  1841
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1842
haftmann@33361
  1843
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@33361
  1844
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1845
haftmann@33361
  1846
text{*a negative, b positive *}
haftmann@33361
  1847
haftmann@33361
  1848
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@33361
  1849
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1850
haftmann@33361
  1851
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@33361
  1852
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1853
haftmann@33361
  1854
text{*a positive, b negative *}
haftmann@33361
  1855
haftmann@33361
  1856
lemma div_pos_neg:
huffman@46560
  1857
     "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1858
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1859
haftmann@33361
  1860
lemma mod_pos_neg:
huffman@46560
  1861
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1862
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1863
haftmann@33361
  1864
text{*a negative, b negative *}
haftmann@33361
  1865
haftmann@33361
  1866
lemma div_neg_neg:
huffman@46560
  1867
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1868
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1869
haftmann@33361
  1870
lemma mod_neg_neg:
huffman@46560
  1871
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1872
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1873
haftmann@33361
  1874
text {*Simplify expresions in which div and mod combine numerical constants*}
haftmann@33361
  1875
huffman@45530
  1876
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1877
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
huffman@45530
  1878
huffman@45530
  1879
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1880
  by (rule div_int_unique [of a b q r],
bulwahn@46552
  1881
    simp add: divmod_int_rel_def)
huffman@45530
  1882
huffman@45530
  1883
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1884
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1885
    simp add: divmod_int_rel_def)
huffman@45530
  1886
huffman@45530
  1887
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1888
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1889
    simp add: divmod_int_rel_def)
huffman@45530
  1890
haftmann@53069
  1891
text {*
haftmann@53069
  1892
  numeral simprocs -- high chance that these can be replaced
haftmann@53069
  1893
  by divmod algorithm from @{class semiring_numeral_div}
haftmann@53069
  1894
*}
haftmann@53069
  1895
haftmann@33361
  1896
ML {*
haftmann@33361
  1897
local
huffman@45530
  1898
  val mk_number = HOLogic.mk_number HOLogic.intT
huffman@45530
  1899
  val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1900
  val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1901
  val zero = @{term "0 :: int"}
huffman@45530
  1902
  val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1903
  val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1904
  val simps = @{thms arith_simps} @ @{thms rel_simps} @
huffman@47108
  1905
    map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
huffman@45530
  1906
  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51717
  1907
    (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));
wenzelm@51717
  1908
  fun binary_proc proc ctxt ct =
haftmann@33361
  1909
    (case Thm.term_of ct of
haftmann@33361
  1910
      _ $ t $ u =>
haftmann@33361
  1911
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
wenzelm@51717
  1912
        SOME args => proc ctxt args
haftmann@33361
  1913
      | NONE => NONE)
haftmann@33361
  1914
    | _ => NONE);
haftmann@33361
  1915
in
huffman@45530
  1916
  fun divmod_proc posrule negrule =
huffman@45530
  1917
    binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
huffman@45530
  1918
      if b = 0 then NONE else let
huffman@45530
  1919
        val (q, r) = pairself mk_number (Integer.div_mod a b)
huffman@45530
  1920
        val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
huffman@45530
  1921
        val (goal2, goal3, rule) = if b > 0
huffman@45530
  1922
          then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
huffman@45530
  1923
          else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
huffman@45530
  1924
      in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
haftmann@33361
  1925
end
haftmann@33361
  1926
*}
haftmann@33361
  1927
huffman@47108
  1928
simproc_setup binary_int_div
huffman@47108
  1929
  ("numeral m div numeral n :: int" |
huffman@47108
  1930
   "numeral m div neg_numeral n :: int" |
huffman@47108
  1931
   "neg_numeral m div numeral n :: int" |
huffman@47108
  1932
   "neg_numeral m div neg_numeral n :: int") =
huffman@45530
  1933
  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
haftmann@33361
  1934
huffman@47108
  1935
simproc_setup binary_int_mod
huffman@47108
  1936
  ("numeral m mod numeral n :: int" |
huffman@47108
  1937
   "numeral m mod neg_numeral n :: int" |
huffman@47108
  1938
   "neg_numeral m mod numeral n :: int" |
huffman@47108
  1939
   "neg_numeral m mod neg_numeral n :: int") =
huffman@45530
  1940
  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
haftmann@33361
  1941
huffman@47108
  1942
lemmas posDivAlg_eqn_numeral [simp] =
huffman@47108
  1943
    posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
huffman@47108
  1944
huffman@47108
  1945
lemmas negDivAlg_eqn_numeral [simp] =
huffman@47108
  1946
    negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
haftmann@33361
  1947
haftmann@33361
  1948
haftmann@33361
  1949
text{*Special-case simplification *}
haftmann@33361
  1950
haftmann@33361
  1951
(** The last remaining special cases for constant arithmetic:
haftmann@33361
  1952
    1 div z and 1 mod z **)
haftmann@33361
  1953
huffman@47108
  1954
lemmas div_pos_pos_1_numeral [simp] =
huffman@47108
  1955
  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
huffman@47108
  1956
huffman@47108
  1957
lemmas div_pos_neg_1_numeral [simp] =
huffman@47108
  1958
  div_pos_neg [OF zero_less_one, of "neg_numeral w",
huffman@47108
  1959
  OF neg_numeral_less_zero] for w
huffman@47108
  1960
huffman@47108
  1961
lemmas mod_pos_pos_1_numeral [simp] =
huffman@47108
  1962
  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
huffman@47108
  1963
huffman@47108
  1964
lemmas mod_pos_neg_1_numeral [simp] =
huffman@47108
  1965
  mod_pos_neg [OF zero_less_one, of "neg_numeral w",
huffman@47108
  1966
  OF neg_numeral_less_zero] for w
huffman@47108
  1967
huffman@47108
  1968
lemmas posDivAlg_eqn_1_numeral [simp] =
huffman@47108
  1969
    posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
huffman@47108
  1970
huffman@47108
  1971
lemmas negDivAlg_eqn_1_numeral [simp] =
huffman@47108
  1972
    negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
haftmann@33361
  1973
haftmann@33361
  1974
huffman@46551
  1975
subsubsection {* Monotonicity in the First Argument (Dividend) *}
haftmann@33361
  1976
haftmann@33361
  1977
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361
  1978
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1979
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1980
apply (rule unique_quotient_lemma)
haftmann@33361
  1981
apply (erule subst)
haftmann@33361
  1982
apply (erule subst, simp_all)
haftmann@33361
  1983
done
haftmann@33361
  1984
haftmann@33361
  1985
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361
  1986
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1987
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1988
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1989
apply (erule subst)
haftmann@33361
  1990
apply (erule subst, simp_all)
haftmann@33361
  1991
done
haftmann@33361
  1992
haftmann@33361
  1993
huffman@46551
  1994
subsubsection {* Monotonicity in the Second Argument (Divisor) *}
haftmann@33361
  1995
haftmann@33361
  1996
lemma q_pos_lemma:
haftmann@33361
  1997
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1998
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1999
 apply (simp add: zero_less_mult_iff)
webertj@49962
  2000
apply (simp add: distrib_left)
haftmann@33361
  2001
done
haftmann@33361
  2002
haftmann@33361
  2003
lemma zdiv_mono2_lemma:
haftmann@33361
  2004
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
haftmann@33361
  2005
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
haftmann@33361
  2006
      ==> q \<le> (q'::int)"
haftmann@33361
  2007
apply (frule q_pos_lemma, assumption+) 
haftmann@33361
  2008
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  2009
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  2010
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  2011
 prefer 2 apply simp
webertj@49962
  2012
apply (simp (no_asm_simp) add: distrib_left)
huffman@44766
  2013
apply (subst add_commute, rule add_less_le_mono, arith)
haftmann@33361
  2014
apply (rule mult_right_mono, auto)
haftmann@33361
  2015
done
haftmann@33361
  2016
haftmann@33361
  2017
lemma zdiv_mono2:
haftmann@33361
  2018
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  2019
apply (subgoal_tac "b \<noteq> 0")
haftmann@33361
  2020
 prefer 2 apply arith
haftmann@33361
  2021
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2022
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  2023
apply (rule zdiv_mono2_lemma)
haftmann@33361
  2024
apply (erule subst)
haftmann@33361
  2025
apply (erule subst, simp_all)
haftmann@33361
  2026
done
haftmann@33361
  2027
haftmann@33361
  2028
lemma q_neg_lemma:
haftmann@33361
  2029
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  2030
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  2031
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  2032
done
haftmann@33361
  2033
haftmann@33361
  2034
lemma zdiv_mono2_neg_lemma:
haftmann@33361
  2035
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
haftmann@33361
  2036
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
haftmann@33361
  2037
      ==> q' \<le> (q::int)"
haftmann@33361
  2038
apply (frule q_neg_lemma, assumption+) 
haftmann@33361
  2039
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  2040
 apply (simp add: mult_less_cancel_left)
webertj@49962
  2041
apply (simp add: distrib_left)
haftmann@33361
  2042
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  2043
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  2044
done
haftmann@33361
  2045
haftmann@33361
  2046
lemma zdiv_mono2_neg:
haftmann@33361
  2047
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361
  2048
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2049
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  2050
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  2051
apply (erule subst)
haftmann@33361
  2052
apply (erule subst, simp_all)
haftmann@33361
  2053
done
haftmann@33361
  2054
haftmann@33361
  2055
huffman@46551
  2056
subsubsection {* More Algebraic Laws for div and mod *}
haftmann@33361
  2057
haftmann@33361
  2058
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
haftmann@33361
  2059
haftmann@33361
  2060
lemma zmult1_lemma:
bulwahn@46552
  2061
     "[| divmod_int_rel b c (q, r) |]  
haftmann@33361
  2062
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
webertj@49962
  2063
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac)
haftmann@33361
  2064
haftmann@33361
  2065
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  2066
apply (case_tac "c = 0", simp)
huffman@47140
  2067
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  2068
done
haftmann@33361
  2069
haftmann@33361
  2070
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@33361
  2071
haftmann@33361
  2072
lemma zadd1_lemma:
bulwahn@46552
  2073
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
haftmann@33361
  2074
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
webertj@49962
  2075
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
haftmann@33361
  2076
haftmann@33361
  2077
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  2078
lemma zdiv_zadd1_eq:
haftmann@33361
  2079
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  2080
apply (case_tac "c = 0", simp)
huffman@47140
  2081
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
haftmann@33361
  2082
done
haftmann@33361
  2083
haftmann@33361
  2084
lemma posDivAlg_div_mod:
haftmann@33361
  2085
  assumes "k \<ge> 0"
haftmann@33361
  2086
  and "l \<ge> 0"
haftmann@33361
  2087
  shows "posDivAlg k l = (k div l, k mod l)"
haftmann@33361
  2088
proof (cases "l = 0")
haftmann@33361
  2089
  case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@33361
  2090
next
haftmann@33361
  2091
  case False with assms posDivAlg_correct
haftmann@33361
  2092
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@33361
  2093
    by simp
huffman@47140
  2094
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  2095
  show ?thesis by simp
haftmann@33361
  2096
qed
haftmann@33361
  2097
haftmann@33361
  2098
lemma negDivAlg_div_mod:
haftmann@33361
  2099
  assumes "k < 0"
haftmann@33361
  2100
  and "l > 0"
haftmann@33361
  2101
  shows "negDivAlg k l = (k div l, k mod l)"
haftmann@33361
  2102
proof -
haftmann@33361
  2103
  from assms have "l \<noteq> 0" by simp
haftmann@33361
  2104
  from assms negDivAlg_correct
haftmann@33361
  2105
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@33361
  2106
    by simp
huffman@47140
  2107
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  2108
  show ?thesis by simp
haftmann@33361
  2109
qed
haftmann@33361
  2110
haftmann@33361
  2111
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  2112
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  2113
haftmann@33361
  2114
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  2115
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  2116
huffman@47108
  2117
lemma zmod_zdiv_equality':
huffman@47108
  2118
  "(m\<Colon>int) mod n = m - (m div n) * n"
huffman@47141
  2119
  using mod_div_equality [of m n] by arith
huffman@47108
  2120
haftmann@33361
  2121
huffman@46551
  2122
subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
haftmann@33361
  2123
haftmann@33361
  2124
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  2125
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  2126
  to cause particular problems.*)
haftmann@33361
  2127
haftmann@33361
  2128
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
haftmann@33361
  2129
haftmann@33361
  2130
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
haftmann@33361
  2131
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  2132
 apply (simp add: algebra_simps)
haftmann@33361
  2133
apply (rule order_le_less_trans)
haftmann@33361
  2134
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  2135
 apply (rule mult_left_mono_neg)
huffman@35216
  2136
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  2137
 apply (simp)
haftmann@33361
  2138
apply (simp)
haftmann@33361
  2139
done
haftmann@33361
  2140
haftmann@33361
  2141
lemma zmult2_lemma_aux2:
haftmann@33361
  2142
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  2143
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  2144
 apply arith
haftmann@33361
  2145
apply (simp add: mult_le_0_iff)
haftmann@33361
  2146
done
haftmann@33361
  2147
haftmann@33361
  2148
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  2149
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  2150
apply arith
haftmann@33361
  2151
apply (simp add: zero_le_mult_iff)
haftmann@33361
  2152
done
haftmann@33361
  2153
haftmann@33361
  2154
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  2155
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  2156
 apply (simp add: right_diff_distrib)
haftmann@33361
  2157
apply (rule order_less_le_trans)
haftmann@33361
  2158
 apply (erule mult_strict_right_mono)
haftmann@33361
  2159
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  2160
  apply simp
huffman@35216
  2161
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  2162
apply simp
haftmann@33361
  2163
done
haftmann@33361
  2164
bulwahn@46552
  2165
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
haftmann@33361
  2166
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@33361
  2167
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
webertj@49962
  2168
                   zero_less_mult_iff distrib_left [symmetric] 
huffman@47139
  2169
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
haftmann@33361
  2170
haftmann@53068
  2171
lemma zdiv_zmult2_eq:
haftmann@53068
  2172
  fixes a b c :: int
haftmann@53068
  2173
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
  2174
apply (case_tac "b = 0", simp)
haftmann@53068
  2175
apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  2176
done
haftmann@33361
  2177
haftmann@33361
  2178
lemma zmod_zmult2_eq:
haftmann@53068
  2179
  fixes a b c :: int
haftmann@53068
  2180
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
  2181
apply (case_tac "b = 0", simp)
haftmann@53068
  2182
apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  2183
done
haftmann@33361
  2184
huffman@47108
  2185
lemma div_pos_geq:
huffman@47108
  2186
  fixes k l :: int
huffman@47108
  2187
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2188
  shows "k div l = (k - l) div l + 1"
huffman@47108
  2189
proof -
huffman@47108
  2190
  have "k = (k - l) + l" by simp
huffman@47108
  2191
  then obtain j where k: "k = j + l" ..
huffman@47108
  2192
  with assms show ?thesis by simp
huffman@47108
  2193
qed
huffman@47108
  2194
huffman@47108
  2195
lemma mod_pos_geq:
huffman@47108
  2196
  fixes k l :: int
huffman@47108
  2197
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2198
  shows "k mod l = (k - l) mod l"
huffman@47108
  2199
proof -
huffman@47108
  2200
  have "k = (k - l) + l" by simp
huffman@47108
  2201
  then obtain j where k: "k = j + l" ..
huffman@47108
  2202
  with assms show ?thesis by simp
huffman@47108
  2203
qed
huffman@47108
  2204
haftmann@33361
  2205
huffman@46551
  2206
subsubsection {* Splitting Rules for div and mod *}
haftmann@33361
  2207
haftmann@33361
  2208
text{*The proofs of the two lemmas below are essentially identical*}
haftmann@33361
  2209
haftmann@33361
  2210
lemma split_pos_lemma:
haftmann@33361
  2211
 "0<k ==> 
haftmann@33361
  2212
    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
  2213
apply (rule iffI, clarify)
haftmann@33361
  2214
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2215
 apply (subst mod_add_eq) 
haftmann@33361
  2216
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2217
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
haftmann@33361
  2218
txt{*converse direction*}
haftmann@33361
  2219
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2220
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2221
done
haftmann@33361
  2222
haftmann@33361
  2223
lemma split_neg_lemma:
haftmann@33361
  2224
 "k<0 ==>
haftmann@33361
  2225
    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
  2226
apply (rule iffI, clarify)
haftmann@33361
  2227
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2228
 apply (subst mod_add_eq) 
haftmann@33361
  2229
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2230
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
haftmann@33361
  2231
txt{*converse direction*}
haftmann@33361
  2232
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2233
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2234
done
haftmann@33361
  2235
haftmann@33361
  2236
lemma split_zdiv:
haftmann@33361
  2237
 "P(n div k :: int) =
haftmann@33361
  2238
  ((k = 0 --> P 0) & 
haftmann@33361
  2239
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
haftmann@33361
  2240
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  2241
apply (case_tac "k=0", simp)
haftmann@33361
  2242
apply (simp only: linorder_neq_iff)
haftmann@33361
  2243
apply (erule disjE) 
haftmann@33361
  2244
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
haftmann@33361
  2245
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  2246
done
haftmann@33361
  2247
haftmann@33361
  2248
lemma split_zmod:
haftmann@33361
  2249
 "P(n mod k :: int) =
haftmann@33361
  2250
  ((k = 0 --> P n) & 
haftmann@33361
  2251
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
haftmann@33361
  2252
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  2253
apply (case_tac "k=0", simp)
haftmann@33361
  2254
apply (simp only: linorder_neq_iff)
haftmann@33361
  2255
apply (erule disjE) 
haftmann@33361
  2256
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
haftmann@33361
  2257
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  2258
done
haftmann@33361
  2259
webertj@33730
  2260
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
webertj@33730
  2261
  when these are applied to some constant that is of the form
huffman@47108
  2262
  @{term "numeral k"}: *}
huffman@47108
  2263
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  2264
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  2265
haftmann@33361
  2266
huffman@47166
  2267
subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
huffman@47166
  2268
huffman@47166
  2269
lemma pos_divmod_int_rel_mult_2:
huffman@47166
  2270
  assumes "0 \<le> b"
huffman@47166
  2271
  assumes "divmod_int_rel a b (q, r)"
huffman@47166
  2272
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166
  2273
  using assms unfolding divmod_int_rel_def by auto
huffman@47166
  2274
huffman@47166
  2275
lemma neg_divmod_int_rel_mult_2:
huffman@47166
  2276
  assumes "b \<le> 0"
huffman@47166
  2277
  assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166
  2278
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166
  2279
  using assms unfolding divmod_int_rel_def by auto
haftmann@33361
  2280
haftmann@33361
  2281
text{*computing div by shifting *}
haftmann@33361
  2282
haftmann@33361
  2283
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
huffman@47166
  2284
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
huffman@47166
  2285
  by (rule div_int_unique)
haftmann@33361
  2286
boehmes@35815
  2287
lemma neg_zdiv_mult_2: 
boehmes@35815
  2288
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
huffman@47166
  2289
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
huffman@47166
  2290
  by (rule div_int_unique)
haftmann@33361
  2291
huffman@47108
  2292
(* FIXME: add rules for negative numerals *)
huffman@47108
  2293
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  2294
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  2295
    numeral v div (numeral w :: int)"
huffman@47108
  2296
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  2297
  by (rule div_mult_mult1, simp)
huffman@47108
  2298
huffman@47108
  2299
lemma zdiv_numeral_Bit1 [simp]:
huffman@47108
  2300
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
huffman@47108
  2301
    (numeral v div (numeral w :: int))"
huffman@47108
  2302
  unfolding numeral.simps
huffman@47108
  2303
  unfolding mult_2 [symmetric] add_commute [of _ 1]
huffman@47108
  2304
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  2305
haftmann@33361
  2306
lemma pos_zmod_mult_2:
haftmann@33361
  2307
  fixes a b :: int
haftmann@33361
  2308
  assumes "0 \<le> a"
haftmann@33361
  2309
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
huffman@47166
  2310
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2311
  by (rule mod_int_unique)
haftmann@33361
  2312
haftmann@33361
  2313
lemma neg_zmod_mult_2:
haftmann@33361
  2314
  fixes a b :: int
haftmann@33361
  2315
  assumes "a \<le> 0"
haftmann@33361
  2316
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
huffman@47166
  2317
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2318
  by (rule mod_int_unique)
haftmann@33361
  2319
huffman@47108
  2320
(* FIXME: add rules for negative numerals *)
huffman@47108
  2321
lemma zmod_numeral_Bit0 [simp]:
huffman@47108
  2322
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
huffman@47108
  2323
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  2324
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  2325
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  2326
huffman@47108
  2327
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  2328
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2329
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  2330
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
huffman@47108
  2331
  unfolding mult_2 [symmetric] add_commute [of _ 1]
huffman@47108
  2332
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  2333
nipkow@39489
  2334
lemma zdiv_eq_0_iff:
nipkow@39489
  2335
 "(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
  2336
proof
nipkow@39489
  2337
  assume ?L
nipkow@39489
  2338
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
nipkow@39489
  2339
  with `?L` show ?R by blast
nipkow@39489
  2340
next
nipkow@39489
  2341
  assume ?R thus ?L
nipkow@39489
  2342
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  2343
qed
nipkow@39489
  2344
nipkow@39489
  2345
huffman@46551
  2346
subsubsection {* Quotients of Signs *}
haftmann@33361
  2347
haftmann@33361
  2348
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2349
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2350
apply (rule order_trans)
haftmann@33361
  2351
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2352
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2353
done
haftmann@33361
  2354
haftmann@33361
  2355
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2356
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2357
haftmann@33361
  2358
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2359
by (drule zdiv_mono1, auto)
haftmann@33361
  2360
nipkow@33804
  2361
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
nipkow@33804
  2362
conditional upon the sign of @{text a} or @{text b}. There are many more.
nipkow@33804
  2363
They should all be simp rules unless that causes too much search. *}
nipkow@33804
  2364
haftmann@33361
  2365
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2366
apply auto
haftmann@33361
  2367
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2368
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2369
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2370
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2371
done
haftmann@33361
  2372
haftmann@33361
  2373
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2374
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2375
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2376
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2377
done
haftmann@33361
  2378
haftmann@33361
  2379
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2380
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2381
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2382
nipkow@39489
  2383
lemma pos_imp_zdiv_pos_iff:
nipkow@39489
  2384
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
nipkow@39489
  2385
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
nipkow@39489
  2386
by arith
nipkow@39489
  2387
haftmann@33361
  2388
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2389
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2390
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2391
nipkow@33804
  2392
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2393
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2394
apply rule
nipkow@33804
  2395
 apply rule
nipkow@33804
  2396
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2397
 apply(cases "b=0")apply simp
nipkow@33804
  2398
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2399
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2400
done
nipkow@33804
  2401
nipkow@39489
  2402
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2403
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2404
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2405
done
nipkow@39489
  2406
nipkow@39489
  2407
haftmann@33361
  2408
subsubsection {* The Divides Relation *}
haftmann@33361
  2409
huffman@47268
  2410
lemma dvd_neg_numeral_left [simp]:
huffman@47268
  2411
  fixes y :: "'a::comm_ring_1"
huffman@47268
  2412
  shows "(neg_numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
huffman@47268
  2413
  unfolding neg_numeral_def minus_dvd_iff ..
huffman@47268
  2414
huffman@47268
  2415
lemma dvd_neg_numeral_right [simp]:
huffman@47268
  2416
  fixes x :: "'a::comm_ring_1"
huffman@47268
  2417
  shows "x dvd (neg_numeral k) \<longleftrightarrow> x dvd (numeral k)"
huffman@47268
  2418
  unfolding neg_numeral_def dvd_minus_iff ..
haftmann@33361
  2419
huffman@47108
  2420
lemmas dvd_eq_mod_eq_0_numeral [simp] =
huffman@47108
  2421
  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
huffman@47108
  2422
huffman@47108
  2423
huffman@47108
  2424
subsubsection {* Further properties *}
huffman@47108
  2425
haftmann@33361
  2426
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
haftmann@33361
  2427
  using zmod_zdiv_equality[where a="m" and b="n"]
huffman@47142
  2428
  by (simp add: algebra_simps) (* FIXME: generalize *)
haftmann@33361
  2429
haftmann@33361
  2430
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
haftmann@33361
  2431
apply (subst split_div, auto)
haftmann@33361
  2432
apply (subst split_zdiv, auto)
haftmann@33361
  2433
apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in unique_quotient)
haftmann@33361
  2434
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2435
done
haftmann@33361
  2436
haftmann@33361
  2437
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
haftmann@33361
  2438
apply (subst split_mod, auto)
haftmann@33361
  2439
apply (subst split_zmod, auto)
haftmann@33361
  2440
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
haftmann@33361
  2441
       in unique_remainder)
haftmann@33361
  2442
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2443
done
haftmann@33361
  2444
haftmann@33361
  2445
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
haftmann@33361
  2446
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2447
haftmann@33361
  2448
text{*Suggested by Matthias Daum*}
haftmann@33361
  2449
lemma int_power_div_base:
haftmann@33361
  2450
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2451
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2452
 apply (erule ssubst)
haftmann@33361
  2453
 apply (simp only: power_add)
haftmann@33361
  2454
 apply simp_all
haftmann@33361
  2455
done
haftmann@33361
  2456
haftmann@33361
  2457
text {* by Brian Huffman *}
haftmann@33361
  2458
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2459
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2460
haftmann@33361
  2461
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2462
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2463
haftmann@33361
  2464
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2465
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2466
haftmann@33361
  2467
lemmas zmod_simps =
haftmann@33361
  2468
  mod_add_left_eq  [symmetric]
haftmann@33361
  2469
  mod_add_right_eq [symmetric]
huffman@47142
  2470
  mod_mult_right_eq[symmetric]
haftmann@33361
  2471
  mod_mult_left_eq [symmetric]
huffman@47164
  2472
  power_mod
haftmann@33361
  2473
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2474
haftmann@33361
  2475
text {* Distributive laws for function @{text nat}. *}
haftmann@33361
  2476
haftmann@33361
  2477
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2478
apply (rule linorder_cases [of y 0])
haftmann@33361
  2479
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2480
apply simp
haftmann@33361
  2481
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2482
done
haftmann@33361
  2483
haftmann@33361
  2484
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2485
lemma nat_mod_distrib:
haftmann@33361
  2486
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2487
apply (case_tac "y = 0", simp)
haftmann@33361
  2488
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2489
done
haftmann@33361
  2490
haftmann@33361
  2491
text  {* transfer setup *}
haftmann@33361
  2492
haftmann@33361
  2493
lemma transfer_nat_int_functions:
haftmann@33361
  2494
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2495
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2496
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2497
haftmann@33361
  2498
lemma transfer_nat_int_function_closures:
haftmann@33361
  2499
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2500
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2501
  apply (cases "y = 0")
haftmann@33361
  2502
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2503
  apply (cases "y = 0")
haftmann@33361
  2504
  apply auto
haftmann@33361
  2505
done
haftmann@33361
  2506
haftmann@35644
  2507
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2508
  transfer_nat_int_functions
haftmann@33361
  2509
  transfer_nat_int_function_closures
haftmann@33361
  2510
]
haftmann@33361
  2511
haftmann@33361
  2512
lemma transfer_int_nat_functions:
haftmann@33361
  2513
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2514
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2515
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2516
haftmann@33361
  2517
lemma transfer_int_nat_function_closures:
haftmann@33361
  2518
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  2519
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  2520
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  2521
haftmann@35644
  2522
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  2523
  transfer_int_nat_functions
haftmann@33361
  2524
  transfer_int_nat_function_closures
haftmann@33361
  2525
]
haftmann@33361
  2526
haftmann@33361
  2527
text{*Suggested by Matthias Daum*}