src/HOL/Divides.thy
author wenzelm
Wed Aug 22 22:55:41 2012 +0200 (2012-08-22)
changeset 48891 c0eafbd55de3
parent 48561 12aa0cb2b447
child 49962 a8cc904a6820
permissions -rw-r--r--
prefer ML_file over old uses;
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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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ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
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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 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 left_distrib)
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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]: "(a + b * c) mod b = a mod b"
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  by (simp add: mult_commute [of b])
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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"
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proof -
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  from `c dvd b` obtain k where "b = c * k"
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    by (rule dvdE)
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  have "a mod b mod c = a mod (c * k) mod c"
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    by (simp only: `b = c * k`)
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  also have "\<dots> = (a mod (c * k) + a div (c * k) * k * c) mod c"
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    by (simp only: mod_mult_self1)
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  also have "\<dots> = (a div (c * k) * (c * k) + a mod (c * k)) mod c"
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    by (simp only: add_ac mult_ac)
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  also have "\<dots> = a mod c"
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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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haftmann@30930
   313
lemma div_mult_div_if_dvd:
haftmann@30930
   314
  "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
haftmann@30930
   315
  apply (cases "y = 0", simp)
haftmann@30930
   316
  apply (cases "z = 0", simp)
haftmann@30930
   317
  apply (auto elim!: dvdE simp add: algebra_simps)
nipkow@30476
   318
  apply (subst mult_assoc [symmetric])
nipkow@30476
   319
  apply (simp add: no_zero_divisors)
haftmann@30930
   320
  done
haftmann@30930
   321
haftmann@35367
   322
lemma div_mult_swap:
haftmann@35367
   323
  assumes "c dvd b"
haftmann@35367
   324
  shows "a * (b div c) = (a * b) div c"
haftmann@35367
   325
proof -
haftmann@35367
   326
  from assms have "b div c * (a div 1) = b * a div (c * 1)"
haftmann@35367
   327
    by (simp only: div_mult_div_if_dvd one_dvd)
haftmann@35367
   328
  then show ?thesis by (simp add: mult_commute)
haftmann@35367
   329
qed
haftmann@35367
   330
   
haftmann@30930
   331
lemma div_mult_mult2 [simp]:
haftmann@30930
   332
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@30930
   333
  by (drule div_mult_mult1) (simp add: mult_commute)
haftmann@30930
   334
haftmann@30930
   335
lemma div_mult_mult1_if [simp]:
haftmann@30930
   336
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   337
  by simp_all
nipkow@30476
   338
haftmann@30930
   339
lemma mod_mult_mult1:
haftmann@30930
   340
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   341
proof (cases "c = 0")
haftmann@30930
   342
  case True then show ?thesis by simp
haftmann@30930
   343
next
haftmann@30930
   344
  case False
haftmann@30930
   345
  from mod_div_equality
haftmann@30930
   346
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   347
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   348
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@30930
   349
  with mod_div_equality show ?thesis by simp 
haftmann@30930
   350
qed
haftmann@30930
   351
  
haftmann@30930
   352
lemma mod_mult_mult2:
haftmann@30930
   353
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@30930
   354
  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
haftmann@30930
   355
huffman@47159
   356
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159
   357
  by (fact mod_mult_mult2 [symmetric])
huffman@47159
   358
huffman@47159
   359
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159
   360
  by (fact mod_mult_mult1 [symmetric])
huffman@47159
   361
huffman@31662
   362
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   363
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   364
huffman@31662
   365
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   366
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   367
haftmann@31009
   368
lemma div_power:
huffman@31661
   369
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
nipkow@30476
   370
apply (induct n)
nipkow@30476
   371
 apply simp
nipkow@30476
   372
apply(simp add: div_mult_div_if_dvd dvd_power_same)
nipkow@30476
   373
done
nipkow@30476
   374
haftmann@35367
   375
lemma dvd_div_eq_mult:
haftmann@35367
   376
  assumes "a \<noteq> 0" and "a dvd b"  
haftmann@35367
   377
  shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@35367
   378
proof
haftmann@35367
   379
  assume "b = c * a"
haftmann@35367
   380
  then show "b div a = c" by (simp add: assms)
haftmann@35367
   381
next
haftmann@35367
   382
  assume "b div a = c"
haftmann@35367
   383
  then have "b div a * a = c * a" by simp
haftmann@35367
   384
  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
haftmann@35367
   385
  ultimately show "b = c * a" by simp
haftmann@35367
   386
qed
haftmann@35367
   387
   
haftmann@35367
   388
lemma dvd_div_div_eq_mult:
haftmann@35367
   389
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@35367
   390
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
haftmann@35367
   391
  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
   392
huffman@31661
   393
end
huffman@31661
   394
haftmann@35673
   395
class ring_div = semiring_div + comm_ring_1
huffman@29405
   396
begin
huffman@29405
   397
haftmann@36634
   398
subclass ring_1_no_zero_divisors ..
haftmann@36634
   399
huffman@29405
   400
text {* Negation respects modular equivalence. *}
huffman@29405
   401
huffman@29405
   402
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   403
proof -
huffman@29405
   404
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405
   405
    by (simp only: mod_div_equality)
huffman@29405
   406
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
huffman@29405
   407
    by (simp only: minus_add_distrib minus_mult_left add_ac)
huffman@29405
   408
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   409
    by (rule mod_mult_self1)
huffman@29405
   410
  finally show ?thesis .
huffman@29405
   411
qed
huffman@29405
   412
huffman@29405
   413
lemma mod_minus_cong:
huffman@29405
   414
  assumes "a mod b = a' mod b"
huffman@29405
   415
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   416
proof -
huffman@29405
   417
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   418
    unfolding assms ..
huffman@29405
   419
  thus ?thesis
huffman@29405
   420
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   421
qed
huffman@29405
   422
huffman@29405
   423
text {* Subtraction respects modular equivalence. *}
huffman@29405
   424
huffman@29405
   425
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
huffman@29405
   426
  unfolding diff_minus
huffman@29405
   427
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   428
huffman@29405
   429
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
huffman@29405
   430
  unfolding diff_minus
huffman@29405
   431
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   432
huffman@29405
   433
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
huffman@29405
   434
  unfolding diff_minus
huffman@29405
   435
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   436
huffman@29405
   437
lemma mod_diff_cong:
huffman@29405
   438
  assumes "a mod c = a' mod c"
huffman@29405
   439
  assumes "b mod c = b' mod c"
huffman@29405
   440
  shows "(a - b) mod c = (a' - b') mod c"
huffman@29405
   441
  unfolding diff_minus using assms
huffman@29405
   442
  by (intro mod_add_cong mod_minus_cong)
huffman@29405
   443
nipkow@30180
   444
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   445
apply (case_tac "y = 0") apply simp
nipkow@30180
   446
apply (auto simp add: dvd_def)
nipkow@30180
   447
apply (subgoal_tac "-(y * k) = y * - k")
nipkow@30180
   448
 apply (erule ssubst)
nipkow@30180
   449
 apply (erule div_mult_self1_is_id)
nipkow@30180
   450
apply simp
nipkow@30180
   451
done
nipkow@30180
   452
nipkow@30180
   453
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   454
apply (case_tac "y = 0") apply simp
nipkow@30180
   455
apply (auto simp add: dvd_def)
nipkow@30180
   456
apply (subgoal_tac "y * k = -y * -k")
nipkow@30180
   457
 apply (erule ssubst)
nipkow@30180
   458
 apply (rule div_mult_self1_is_id)
nipkow@30180
   459
 apply simp
nipkow@30180
   460
apply simp
nipkow@30180
   461
done
nipkow@30180
   462
huffman@47159
   463
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159
   464
  using div_mult_mult1 [of "- 1" a b]
huffman@47159
   465
  unfolding neg_equal_0_iff_equal by simp
huffman@47159
   466
huffman@47159
   467
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159
   468
  using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159
   469
huffman@47159
   470
lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159
   471
  using div_minus_minus [of "-a" b] by simp
huffman@47159
   472
huffman@47159
   473
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159
   474
  using mod_minus_minus [of "-a" b] by simp
huffman@47159
   475
huffman@47160
   476
lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160
   477
  using div_minus_right [of a 1] by simp
huffman@47160
   478
huffman@47160
   479
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160
   480
  using mod_minus_right [of a 1] by simp
huffman@47160
   481
huffman@29405
   482
end
huffman@29405
   483
haftmann@25942
   484
haftmann@26100
   485
subsection {* Division on @{typ nat} *}
haftmann@26100
   486
haftmann@26100
   487
text {*
haftmann@26100
   488
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   489
  of a characteristic relation with two input arguments
haftmann@26100
   490
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   491
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   492
*}
haftmann@26100
   493
haftmann@33340
   494
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   495
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   496
    m = fst qr * n + snd qr \<and>
haftmann@30923
   497
      (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
   498
haftmann@33340
   499
text {* @{const divmod_nat_rel} is total: *}
haftmann@26100
   500
haftmann@33340
   501
lemma divmod_nat_rel_ex:
haftmann@33340
   502
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   503
proof (cases "n = 0")
haftmann@30923
   504
  case True  with that show thesis
haftmann@33340
   505
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   506
next
haftmann@26100
   507
  case False
haftmann@26100
   508
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   509
  proof (induct m)
haftmann@26100
   510
    case 0 with `n \<noteq> 0`
haftmann@26100
   511
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   512
    then show ?case by blast
haftmann@26100
   513
  next
haftmann@26100
   514
    case (Suc m) then obtain q' r'
haftmann@26100
   515
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   516
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   517
      case True
haftmann@26100
   518
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   519
      with True show ?thesis by blast
haftmann@26100
   520
    next
haftmann@26100
   521
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   522
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   523
      ultimately have "n = Suc r'" by auto
haftmann@26100
   524
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   525
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   526
    qed
haftmann@26100
   527
  qed
haftmann@26100
   528
  with that show thesis
haftmann@33340
   529
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   530
qed
haftmann@26100
   531
haftmann@33340
   532
text {* @{const divmod_nat_rel} is injective: *}
haftmann@26100
   533
haftmann@33340
   534
lemma divmod_nat_rel_unique:
haftmann@33340
   535
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   536
    and "divmod_nat_rel m n qr'"
haftmann@30923
   537
  shows "qr = qr'"
haftmann@26100
   538
proof (cases "n = 0")
haftmann@26100
   539
  case True with assms show ?thesis
haftmann@30923
   540
    by (cases qr, cases qr')
haftmann@33340
   541
      (simp add: divmod_nat_rel_def)
haftmann@26100
   542
next
haftmann@26100
   543
  case False
haftmann@26100
   544
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   545
  apply (rule leI)
haftmann@26100
   546
  apply (subst less_iff_Suc_add)
haftmann@26100
   547
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   548
  done
haftmann@30923
   549
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
haftmann@33340
   550
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
haftmann@30923
   551
  moreover from this assms have "snd qr = snd qr'"
haftmann@33340
   552
    by (simp add: divmod_nat_rel_def)
haftmann@30923
   553
  ultimately show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   554
qed
haftmann@26100
   555
haftmann@26100
   556
text {*
haftmann@26100
   557
  We instantiate divisibility on the natural numbers by
haftmann@33340
   558
  means of @{const divmod_nat_rel}:
haftmann@26100
   559
*}
haftmann@25942
   560
haftmann@33340
   561
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767
   562
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   563
haftmann@33340
   564
lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   565
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   566
proof -
haftmann@33340
   567
  from divmod_nat_rel_ex
haftmann@33340
   568
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   569
  then show ?thesis
haftmann@33340
   570
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   571
qed
haftmann@30923
   572
huffman@47135
   573
lemma divmod_nat_unique:
haftmann@33340
   574
  assumes "divmod_nat_rel m n qr" 
haftmann@33340
   575
  shows "divmod_nat m n = qr"
haftmann@33340
   576
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   577
huffman@46551
   578
instantiation nat :: semiring_div
huffman@46551
   579
begin
huffman@46551
   580
haftmann@26100
   581
definition div_nat where
haftmann@33340
   582
  "m div n = fst (divmod_nat m n)"
haftmann@26100
   583
huffman@46551
   584
lemma fst_divmod_nat [simp]:
huffman@46551
   585
  "fst (divmod_nat m n) = m div n"
huffman@46551
   586
  by (simp add: div_nat_def)
huffman@46551
   587
haftmann@26100
   588
definition mod_nat where
haftmann@33340
   589
  "m mod n = snd (divmod_nat m n)"
haftmann@25571
   590
huffman@46551
   591
lemma snd_divmod_nat [simp]:
huffman@46551
   592
  "snd (divmod_nat m n) = m mod n"
huffman@46551
   593
  by (simp add: mod_nat_def)
huffman@46551
   594
haftmann@33340
   595
lemma divmod_nat_div_mod:
haftmann@33340
   596
  "divmod_nat m n = (m div n, m mod n)"
huffman@46551
   597
  by (simp add: prod_eq_iff)
haftmann@26100
   598
huffman@47135
   599
lemma div_nat_unique:
haftmann@33340
   600
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   601
  shows "m div n = q"
huffman@47135
   602
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   603
huffman@47135
   604
lemma mod_nat_unique:
haftmann@33340
   605
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   606
  shows "m mod n = r"
huffman@47135
   607
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   608
haftmann@33340
   609
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
huffman@46551
   610
  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
paulson@14267
   611
huffman@47136
   612
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
huffman@47136
   613
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
huffman@47136
   614
huffman@47136
   615
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
huffman@47136
   616
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   617
huffman@47137
   618
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
huffman@47137
   619
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   620
haftmann@33340
   621
lemma divmod_nat_step:
haftmann@26100
   622
  assumes "0 < n" and "n \<le> m"
haftmann@33340
   623
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
huffman@47135
   624
proof (rule divmod_nat_unique)
huffman@47134
   625
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
huffman@47134
   626
    by (rule divmod_nat_rel)
huffman@47134
   627
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
huffman@47134
   628
    unfolding divmod_nat_rel_def using assms by auto
haftmann@26100
   629
qed
haftmann@25942
   630
wenzelm@26300
   631
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   632
haftmann@26100
   633
lemma div_less [simp]:
haftmann@26100
   634
  fixes m n :: nat
haftmann@26100
   635
  assumes "m < n"
haftmann@26100
   636
  shows "m div n = 0"
huffman@46551
   637
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   638
haftmann@26100
   639
lemma le_div_geq:
haftmann@26100
   640
  fixes m n :: nat
haftmann@26100
   641
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   642
  shows "m div n = Suc ((m - n) div n)"
huffman@46551
   643
  using assms divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   644
haftmann@26100
   645
lemma mod_less [simp]:
haftmann@26100
   646
  fixes m n :: nat
haftmann@26100
   647
  assumes "m < n"
haftmann@26100
   648
  shows "m mod n = m"
huffman@46551
   649
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   650
haftmann@26100
   651
lemma le_mod_geq:
haftmann@26100
   652
  fixes m n :: nat
haftmann@26100
   653
  assumes "n \<le> m"
haftmann@26100
   654
  shows "m mod n = (m - n) mod n"
huffman@46551
   655
  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   656
huffman@47136
   657
instance proof
huffman@47136
   658
  fix m n :: nat
huffman@47136
   659
  show "m div n * n + m mod n = m"
huffman@47136
   660
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136
   661
next
huffman@47136
   662
  fix m n q :: nat
huffman@47136
   663
  assume "n \<noteq> 0"
huffman@47136
   664
  then show "(q + m * n) div n = m + q div n"
huffman@47136
   665
    by (induct m) (simp_all add: le_div_geq)
huffman@47136
   666
next
huffman@47136
   667
  fix m n q :: nat
huffman@47136
   668
  assume "m \<noteq> 0"
huffman@47136
   669
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136
   670
    unfolding divmod_nat_rel_def
huffman@47136
   671
    by (auto split: split_if_asm, simp_all add: algebra_simps)
huffman@47136
   672
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136
   673
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136
   674
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136
   675
next
huffman@47136
   676
  fix n :: nat show "n div 0 = 0"
haftmann@33340
   677
    by (simp add: div_nat_def divmod_nat_zero)
huffman@47136
   678
next
huffman@47136
   679
  fix n :: nat show "0 div n = 0"
huffman@47136
   680
    by (simp add: div_nat_def divmod_nat_zero_left)
haftmann@25942
   681
qed
haftmann@26100
   682
haftmann@25942
   683
end
paulson@14267
   684
haftmann@33361
   685
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@33361
   686
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
huffman@46551
   687
  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
haftmann@33361
   688
haftmann@26100
   689
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   690
haftmann@30934
   691
ML {*
wenzelm@43594
   692
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
   693
(
haftmann@30934
   694
  val div_name = @{const_name div};
haftmann@30934
   695
  val mod_name = @{const_name mod};
haftmann@30934
   696
  val mk_binop = HOLogic.mk_binop;
huffman@48561
   697
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
   698
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
   699
  fun mk_sum [] = HOLogic.zero
huffman@48561
   700
    | mk_sum [t] = t
huffman@48561
   701
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
   702
  fun dest_sum tm =
huffman@48561
   703
    if HOLogic.is_zero tm then []
huffman@48561
   704
    else
huffman@48561
   705
      (case try HOLogic.dest_Suc tm of
huffman@48561
   706
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
   707
      | NONE =>
huffman@48561
   708
          (case try dest_plus tm of
huffman@48561
   709
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
   710
          | NONE => [tm]));
haftmann@25942
   711
haftmann@30934
   712
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
   713
haftmann@30934
   714
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@35050
   715
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
wenzelm@41550
   716
)
haftmann@25942
   717
*}
haftmann@25942
   718
wenzelm@43594
   719
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
wenzelm@43594
   720
haftmann@26100
   721
haftmann@26100
   722
subsubsection {* Quotient *}
haftmann@26100
   723
haftmann@26100
   724
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   725
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   726
haftmann@26100
   727
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   728
by (simp add: div_geq)
haftmann@26100
   729
haftmann@26100
   730
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   731
by simp
haftmann@26100
   732
haftmann@26100
   733
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   734
by simp
haftmann@26100
   735
haftmann@25942
   736
haftmann@25942
   737
subsubsection {* Remainder *}
haftmann@25942
   738
haftmann@26100
   739
lemma mod_less_divisor [simp]:
haftmann@26100
   740
  fixes m n :: nat
haftmann@26100
   741
  assumes "n > 0"
haftmann@26100
   742
  shows "m mod n < (n::nat)"
haftmann@33340
   743
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
   744
haftmann@26100
   745
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   746
  fixes m n :: nat
haftmann@26100
   747
  shows "m mod n \<le> m"
haftmann@26100
   748
proof (rule add_leD2)
haftmann@26100
   749
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   750
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   751
qed
haftmann@26100
   752
haftmann@26100
   753
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
   754
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   755
haftmann@26100
   756
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
   757
by (simp add: le_mod_geq)
haftmann@26100
   758
paulson@14267
   759
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
   760
by (induct m) (simp_all add: mod_geq)
paulson@14267
   761
paulson@14267
   762
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   763
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
huffman@47138
   764
  using mod_div_equality2 [of n m] by arith
paulson@14267
   765
nipkow@15439
   766
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   767
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   768
  apply simp
wenzelm@22718
   769
  done
paulson@14267
   770
haftmann@26100
   771
subsubsection {* Quotient and Remainder *}
paulson@14267
   772
haftmann@33340
   773
lemma divmod_nat_rel_mult1_eq:
bulwahn@46552
   774
  "divmod_nat_rel b c (q, r)
haftmann@33340
   775
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
   776
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
   777
haftmann@30923
   778
lemma div_mult1_eq:
haftmann@30923
   779
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135
   780
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
   781
haftmann@33340
   782
lemma divmod_nat_rel_add1_eq:
bulwahn@46552
   783
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340
   784
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
   785
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
   786
paulson@14267
   787
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   788
lemma div_add1_eq:
nipkow@25134
   789
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135
   790
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
   791
paulson@14267
   792
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   793
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   794
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   795
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   796
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   797
  done
paulson@10559
   798
haftmann@33340
   799
lemma divmod_nat_rel_mult2_eq:
bulwahn@46552
   800
  "divmod_nat_rel a b (q, r)
haftmann@33340
   801
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@33340
   802
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   803
paulson@14267
   804
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
huffman@47135
   805
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267
   806
paulson@14267
   807
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
huffman@47135
   808
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267
   809
paulson@14267
   810
huffman@46551
   811
subsubsection {* Further Facts about Quotient and Remainder *}
paulson@14267
   812
paulson@14267
   813
lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667
   814
by (induct m) (simp_all add: div_geq)
paulson@14267
   815
paulson@14267
   816
(* Monotonicity of div in first argument *)
haftmann@30923
   817
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   818
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   819
apply (case_tac "k=0", simp)
paulson@15251
   820
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   821
apply (case_tac "n<k")
paulson@14267
   822
(* 1  case n<k *)
paulson@14267
   823
apply simp
paulson@14267
   824
(* 2  case n >= k *)
paulson@14267
   825
apply (case_tac "m<k")
paulson@14267
   826
(* 2.1  case m<k *)
paulson@14267
   827
apply simp
paulson@14267
   828
(* 2.2  case m>=k *)
nipkow@15439
   829
apply (simp add: div_geq diff_le_mono)
paulson@14267
   830
done
paulson@14267
   831
paulson@14267
   832
(* Antimonotonicity of div in second argument *)
paulson@14267
   833
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   834
apply (subgoal_tac "0<n")
wenzelm@22718
   835
 prefer 2 apply simp
paulson@15251
   836
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   837
apply (rename_tac "k")
paulson@14267
   838
apply (case_tac "k<n", simp)
paulson@14267
   839
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   840
 prefer 2 apply simp
paulson@14267
   841
apply (simp add: div_geq)
paulson@15251
   842
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   843
 prefer 2
paulson@14267
   844
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   845
apply (rule le_trans, simp)
nipkow@15439
   846
apply (simp)
paulson@14267
   847
done
paulson@14267
   848
paulson@14267
   849
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   850
apply (case_tac "n=0", simp)
paulson@14267
   851
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   852
apply (rule div_le_mono2)
paulson@14267
   853
apply (simp_all (no_asm_simp))
paulson@14267
   854
done
paulson@14267
   855
wenzelm@22718
   856
(* Similar for "less than" *)
huffman@47138
   857
lemma div_less_dividend [simp]:
huffman@47138
   858
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
   859
apply (induct m rule: nat_less_induct)
paulson@14267
   860
apply (rename_tac "m")
paulson@14267
   861
apply (case_tac "m<n", simp)
paulson@14267
   862
apply (subgoal_tac "0<n")
wenzelm@22718
   863
 prefer 2 apply simp
paulson@14267
   864
apply (simp add: div_geq)
paulson@14267
   865
apply (case_tac "n<m")
paulson@15251
   866
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   867
  apply (rule impI less_trans_Suc)+
paulson@14267
   868
apply assumption
nipkow@15439
   869
  apply (simp_all)
paulson@14267
   870
done
paulson@14267
   871
paulson@14267
   872
text{*A fact for the mutilated chess board*}
paulson@14267
   873
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   874
apply (case_tac "n=0", simp)
paulson@15251
   875
apply (induct "m" rule: nat_less_induct)
paulson@14267
   876
apply (case_tac "Suc (na) <n")
paulson@14267
   877
(* case Suc(na) < n *)
paulson@14267
   878
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   879
(* case n \<le> Suc(na) *)
paulson@16796
   880
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   881
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   882
done
paulson@14267
   883
paulson@14267
   884
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
   885
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   886
wenzelm@22718
   887
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   888
paulson@14267
   889
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   890
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
haftmann@27651
   891
  apply (cut_tac a = m in mod_div_equality)
wenzelm@22718
   892
  apply (simp only: add_ac)
wenzelm@22718
   893
  apply (blast intro: sym)
wenzelm@22718
   894
  done
paulson@14267
   895
nipkow@13152
   896
lemma split_div:
nipkow@13189
   897
 "P(n div k :: nat) =
nipkow@13189
   898
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   899
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   900
proof
nipkow@13189
   901
  assume P: ?P
nipkow@13189
   902
  show ?Q
nipkow@13189
   903
  proof (cases)
nipkow@13189
   904
    assume "k = 0"
haftmann@27651
   905
    with P show ?Q by simp
nipkow@13189
   906
  next
nipkow@13189
   907
    assume not0: "k \<noteq> 0"
nipkow@13189
   908
    thus ?Q
nipkow@13189
   909
    proof (simp, intro allI impI)
nipkow@13189
   910
      fix i j
nipkow@13189
   911
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   912
      show "P i"
nipkow@13189
   913
      proof (cases)
wenzelm@22718
   914
        assume "i = 0"
wenzelm@22718
   915
        with n j P show "P i" by simp
nipkow@13189
   916
      next
wenzelm@22718
   917
        assume "i \<noteq> 0"
wenzelm@22718
   918
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   919
      qed
nipkow@13189
   920
    qed
nipkow@13189
   921
  qed
nipkow@13189
   922
next
nipkow@13189
   923
  assume Q: ?Q
nipkow@13189
   924
  show ?P
nipkow@13189
   925
  proof (cases)
nipkow@13189
   926
    assume "k = 0"
haftmann@27651
   927
    with Q show ?P by simp
nipkow@13189
   928
  next
nipkow@13189
   929
    assume not0: "k \<noteq> 0"
nipkow@13189
   930
    with Q have R: ?R by simp
nipkow@13189
   931
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   932
    show ?P by simp
nipkow@13189
   933
  qed
nipkow@13189
   934
qed
nipkow@13189
   935
berghofe@13882
   936
lemma split_div_lemma:
haftmann@26100
   937
  assumes "0 < n"
haftmann@26100
   938
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
   939
proof
haftmann@26100
   940
  assume ?rhs
haftmann@26100
   941
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
   942
  then have A: "n * q \<le> m" by simp
haftmann@26100
   943
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
   944
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
   945
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
   946
  with nq have "m < n + n * q" by simp
haftmann@26100
   947
  then have B: "m < n * Suc q" by simp
haftmann@26100
   948
  from A B show ?lhs ..
haftmann@26100
   949
next
haftmann@26100
   950
  assume P: ?lhs
haftmann@33340
   951
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@33340
   952
    unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
haftmann@33340
   953
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
haftmann@30923
   954
  have "(q, m - n * q) = (m div n, m mod n)" by auto
haftmann@30923
   955
  then show ?rhs by simp
haftmann@26100
   956
qed
berghofe@13882
   957
berghofe@13882
   958
theorem split_div':
berghofe@13882
   959
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   960
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   961
  apply (case_tac "0 < n")
berghofe@13882
   962
  apply (simp only: add: split_div_lemma)
haftmann@27651
   963
  apply simp_all
berghofe@13882
   964
  done
berghofe@13882
   965
nipkow@13189
   966
lemma split_mod:
nipkow@13189
   967
 "P(n mod k :: nat) =
nipkow@13189
   968
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   969
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   970
proof
nipkow@13189
   971
  assume P: ?P
nipkow@13189
   972
  show ?Q
nipkow@13189
   973
  proof (cases)
nipkow@13189
   974
    assume "k = 0"
haftmann@27651
   975
    with P show ?Q by simp
nipkow@13189
   976
  next
nipkow@13189
   977
    assume not0: "k \<noteq> 0"
nipkow@13189
   978
    thus ?Q
nipkow@13189
   979
    proof (simp, intro allI impI)
nipkow@13189
   980
      fix i j
nipkow@13189
   981
      assume "n = k*i + j" "j < k"
nipkow@13189
   982
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   983
    qed
nipkow@13189
   984
  qed
nipkow@13189
   985
next
nipkow@13189
   986
  assume Q: ?Q
nipkow@13189
   987
  show ?P
nipkow@13189
   988
  proof (cases)
nipkow@13189
   989
    assume "k = 0"
haftmann@27651
   990
    with Q show ?P by simp
nipkow@13189
   991
  next
nipkow@13189
   992
    assume not0: "k \<noteq> 0"
nipkow@13189
   993
    with Q have R: ?R by simp
nipkow@13189
   994
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   995
    show ?P by simp
nipkow@13189
   996
  qed
nipkow@13189
   997
qed
nipkow@13189
   998
berghofe@13882
   999
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
huffman@47138
  1000
  using mod_div_equality [of m n] by arith
huffman@47138
  1001
huffman@47138
  1002
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
huffman@47138
  1003
  using mod_div_equality [of m n] by arith
huffman@47138
  1004
(* FIXME: very similar to mult_div_cancel *)
haftmann@22800
  1005
haftmann@22800
  1006
huffman@46551
  1007
subsubsection {* An ``induction'' law for modulus arithmetic. *}
paulson@14640
  1008
paulson@14640
  1009
lemma mod_induct_0:
paulson@14640
  1010
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1011
  and base: "P i" and i: "i<p"
paulson@14640
  1012
  shows "P 0"
paulson@14640
  1013
proof (rule ccontr)
paulson@14640
  1014
  assume contra: "\<not>(P 0)"
paulson@14640
  1015
  from i have p: "0<p" by simp
paulson@14640
  1016
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1017
  proof
paulson@14640
  1018
    fix k
paulson@14640
  1019
    show "?A k"
paulson@14640
  1020
    proof (induct k)
paulson@14640
  1021
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1022
    next
paulson@14640
  1023
      fix n
paulson@14640
  1024
      assume ih: "?A n"
paulson@14640
  1025
      show "?A (Suc n)"
paulson@14640
  1026
      proof (clarsimp)
wenzelm@22718
  1027
        assume y: "P (p - Suc n)"
wenzelm@22718
  1028
        have n: "Suc n < p"
wenzelm@22718
  1029
        proof (rule ccontr)
wenzelm@22718
  1030
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1031
          hence "p - Suc n = 0"
wenzelm@22718
  1032
            by simp
wenzelm@22718
  1033
          with y contra show "False"
wenzelm@22718
  1034
            by simp
wenzelm@22718
  1035
        qed
wenzelm@22718
  1036
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1037
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1038
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1039
          by blast
wenzelm@22718
  1040
        show "False"
wenzelm@22718
  1041
        proof (cases "n=0")
wenzelm@22718
  1042
          case True
wenzelm@22718
  1043
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1044
        next
wenzelm@22718
  1045
          case False
wenzelm@22718
  1046
          with p have "p-n < p" by arith
wenzelm@22718
  1047
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1048
        qed
paulson@14640
  1049
      qed
paulson@14640
  1050
    qed
paulson@14640
  1051
  qed
paulson@14640
  1052
  moreover
paulson@14640
  1053
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1054
    by (blast dest: less_imp_add_positive)
paulson@14640
  1055
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1056
  moreover
paulson@14640
  1057
  note base
paulson@14640
  1058
  ultimately
paulson@14640
  1059
  show "False" by blast
paulson@14640
  1060
qed
paulson@14640
  1061
paulson@14640
  1062
lemma mod_induct:
paulson@14640
  1063
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1064
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1065
  shows "P j"
paulson@14640
  1066
proof -
paulson@14640
  1067
  have "\<forall>j<p. P j"
paulson@14640
  1068
  proof
paulson@14640
  1069
    fix j
paulson@14640
  1070
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1071
    proof (induct j)
paulson@14640
  1072
      from step base i show "?A 0"
wenzelm@22718
  1073
        by (auto elim: mod_induct_0)
paulson@14640
  1074
    next
paulson@14640
  1075
      fix k
paulson@14640
  1076
      assume ih: "?A k"
paulson@14640
  1077
      show "?A (Suc k)"
paulson@14640
  1078
      proof
wenzelm@22718
  1079
        assume suc: "Suc k < p"
wenzelm@22718
  1080
        hence k: "k<p" by simp
wenzelm@22718
  1081
        with ih have "P k" ..
wenzelm@22718
  1082
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1083
          by blast
wenzelm@22718
  1084
        moreover
wenzelm@22718
  1085
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1086
          by simp
wenzelm@22718
  1087
        ultimately
wenzelm@22718
  1088
        show "P (Suc k)" by simp
paulson@14640
  1089
      qed
paulson@14640
  1090
    qed
paulson@14640
  1091
  qed
paulson@14640
  1092
  with j show ?thesis by blast
paulson@14640
  1093
qed
paulson@14640
  1094
haftmann@33296
  1095
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1096
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1097
huffman@47138
  1098
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1099
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1100
haftmann@33296
  1101
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1102
by (simp add: mult_2 [symmetric])
haftmann@33296
  1103
haftmann@33296
  1104
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1105
proof -
boehmes@35815
  1106
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1107
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1108
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1109
  then show ?thesis by auto
haftmann@33296
  1110
qed
haftmann@33296
  1111
haftmann@33296
  1112
text{*These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1113
    at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@33296
  1114
    We already have some rules to simplify operands smaller than 3.*}
haftmann@33296
  1115
haftmann@33296
  1116
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1117
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1118
haftmann@33296
  1119
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1120
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1121
haftmann@33296
  1122
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1123
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1124
haftmann@33296
  1125
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1126
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1127
huffman@47108
  1128
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1129
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1130
haftmann@33361
  1131
haftmann@33361
  1132
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
haftmann@33361
  1133
apply (induct "m")
haftmann@33361
  1134
apply (simp_all add: mod_Suc)
haftmann@33361
  1135
done
haftmann@33361
  1136
huffman@47108
  1137
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1138
huffman@47138
  1139
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1140
by (simp add: div_le_mono)
haftmann@33361
  1141
haftmann@33361
  1142
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1143
by (cases n) simp_all
haftmann@33361
  1144
boehmes@35815
  1145
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1146
proof -
boehmes@35815
  1147
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
boehmes@35815
  1148
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
boehmes@35815
  1149
qed
haftmann@33361
  1150
haftmann@33361
  1151
  (* Potential use of algebra : Equality modulo n*)
haftmann@33361
  1152
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
haftmann@33361
  1153
by (simp add: mult_ac add_ac)
haftmann@33361
  1154
haftmann@33361
  1155
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1156
proof -
haftmann@33361
  1157
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
haftmann@33361
  1158
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
haftmann@33361
  1159
  finally show ?thesis .
haftmann@33361
  1160
qed
haftmann@33361
  1161
haftmann@33361
  1162
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
haftmann@33361
  1163
apply (subst mod_Suc [of m]) 
haftmann@33361
  1164
apply (subst mod_Suc [of "m mod n"], simp) 
haftmann@33361
  1165
done
haftmann@33361
  1166
huffman@47108
  1167
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1168
  fixes n :: nat
huffman@47108
  1169
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
huffman@47108
  1170
  by simp
huffman@47108
  1171
haftmann@33361
  1172
haftmann@33361
  1173
subsection {* Division on @{typ int} *}
haftmann@33361
  1174
haftmann@33361
  1175
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
haftmann@33361
  1176
    --{*definition of quotient and remainder*}
huffman@47139
  1177
  "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139
  1178
    (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
  1179
haftmann@33361
  1180
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1181
    --{*for the division algorithm*}
huffman@47108
  1182
    "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@33361
  1183
                         else (2 * q, r))"
haftmann@33361
  1184
haftmann@33361
  1185
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@33361
  1186
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1187
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@33361
  1188
     else adjust b (posDivAlg a (2 * b)))"
haftmann@33361
  1189
by auto
haftmann@33361
  1190
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
haftmann@33361
  1191
  (auto simp add: mult_2)
haftmann@33361
  1192
haftmann@33361
  1193
text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@33361
  1194
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1195
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@33361
  1196
     else adjust b (negDivAlg a (2 * b)))"
haftmann@33361
  1197
by auto
haftmann@33361
  1198
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
haftmann@33361
  1199
  (auto simp add: mult_2)
haftmann@33361
  1200
haftmann@33361
  1201
text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@33361
  1202
haftmann@33361
  1203
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1204
    --{*The full division algorithm considers all possible signs for a, b
haftmann@33361
  1205
       including the special case @{text "a=0, b<0"} because 
haftmann@33361
  1206
       @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@33361
  1207
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@33361
  1208
                  else if a = 0 then (0, 0)
huffman@46560
  1209
                       else apsnd uminus (negDivAlg (-a) (-b))
haftmann@33361
  1210
               else 
haftmann@33361
  1211
                  if 0 < b then negDivAlg a b
huffman@46560
  1212
                  else apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1213
haftmann@33361
  1214
instantiation int :: Divides.div
haftmann@33361
  1215
begin
haftmann@33361
  1216
huffman@46551
  1217
definition div_int where
haftmann@33361
  1218
  "a div b = fst (divmod_int a b)"
haftmann@33361
  1219
huffman@46551
  1220
lemma fst_divmod_int [simp]:
huffman@46551
  1221
  "fst (divmod_int a b) = a div b"
huffman@46551
  1222
  by (simp add: div_int_def)
huffman@46551
  1223
huffman@46551
  1224
definition mod_int where
huffman@46560
  1225
  "a mod b = snd (divmod_int a b)"
haftmann@33361
  1226
huffman@46551
  1227
lemma snd_divmod_int [simp]:
huffman@46551
  1228
  "snd (divmod_int a b) = a mod b"
huffman@46551
  1229
  by (simp add: mod_int_def)
huffman@46551
  1230
haftmann@33361
  1231
instance ..
haftmann@33361
  1232
paulson@3366
  1233
end
haftmann@33361
  1234
haftmann@33361
  1235
lemma divmod_int_mod_div:
haftmann@33361
  1236
  "divmod_int p q = (p div q, p mod q)"
huffman@46551
  1237
  by (simp add: prod_eq_iff)
haftmann@33361
  1238
haftmann@33361
  1239
text{*
haftmann@33361
  1240
Here is the division algorithm in ML:
haftmann@33361
  1241
haftmann@33361
  1242
\begin{verbatim}
haftmann@33361
  1243
    fun posDivAlg (a,b) =
haftmann@33361
  1244
      if a<b then (0,a)
haftmann@33361
  1245
      else let val (q,r) = posDivAlg(a, 2*b)
haftmann@33361
  1246
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1247
           end
haftmann@33361
  1248
haftmann@33361
  1249
    fun negDivAlg (a,b) =
haftmann@33361
  1250
      if 0\<le>a+b then (~1,a+b)
haftmann@33361
  1251
      else let val (q,r) = negDivAlg(a, 2*b)
haftmann@33361
  1252
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1253
           end;
haftmann@33361
  1254
haftmann@33361
  1255
    fun negateSnd (q,r:int) = (q,~r);
haftmann@33361
  1256
haftmann@33361
  1257
    fun divmod (a,b) = if 0\<le>a then 
haftmann@33361
  1258
                          if b>0 then posDivAlg (a,b) 
haftmann@33361
  1259
                           else if a=0 then (0,0)
haftmann@33361
  1260
                                else negateSnd (negDivAlg (~a,~b))
haftmann@33361
  1261
                       else 
haftmann@33361
  1262
                          if 0<b then negDivAlg (a,b)
haftmann@33361
  1263
                          else        negateSnd (posDivAlg (~a,~b));
haftmann@33361
  1264
\end{verbatim}
haftmann@33361
  1265
*}
haftmann@33361
  1266
haftmann@33361
  1267
huffman@46551
  1268
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
haftmann@33361
  1269
haftmann@33361
  1270
lemma unique_quotient_lemma:
haftmann@33361
  1271
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
haftmann@33361
  1272
      ==> q' \<le> (q::int)"
haftmann@33361
  1273
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1274
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1275
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1276
apply (erule_tac [2] order_le_less_trans)
haftmann@33361
  1277
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
haftmann@33361
  1278
apply (subgoal_tac "b * q' < b * (1 + q) ")
haftmann@33361
  1279
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
haftmann@33361
  1280
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1281
done
haftmann@33361
  1282
haftmann@33361
  1283
lemma unique_quotient_lemma_neg:
haftmann@33361
  1284
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
haftmann@33361
  1285
      ==> q \<le> (q'::int)"
haftmann@33361
  1286
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
haftmann@33361
  1287
    auto)
haftmann@33361
  1288
haftmann@33361
  1289
lemma unique_quotient:
bulwahn@46552
  1290
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1291
      ==> q = q'"
haftmann@33361
  1292
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361
  1293
apply (blast intro: order_antisym
haftmann@33361
  1294
             dest: order_eq_refl [THEN unique_quotient_lemma] 
haftmann@33361
  1295
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1296
done
haftmann@33361
  1297
haftmann@33361
  1298
haftmann@33361
  1299
lemma unique_remainder:
bulwahn@46552
  1300
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1301
      ==> r = r'"
haftmann@33361
  1302
apply (subgoal_tac "q = q'")
haftmann@33361
  1303
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1304
apply (blast intro: unique_quotient)
haftmann@33361
  1305
done
haftmann@33361
  1306
haftmann@33361
  1307
huffman@46551
  1308
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
haftmann@33361
  1309
haftmann@33361
  1310
text{*And positive divisors*}
haftmann@33361
  1311
haftmann@33361
  1312
lemma adjust_eq [simp]:
huffman@47108
  1313
     "adjust b (q, r) = 
huffman@47108
  1314
      (let diff = r - b in  
huffman@47108
  1315
        if 0 \<le> diff then (2 * q + 1, diff)   
haftmann@33361
  1316
                     else (2*q, r))"
huffman@47108
  1317
  by (simp add: Let_def adjust_def)
haftmann@33361
  1318
haftmann@33361
  1319
declare posDivAlg.simps [simp del]
haftmann@33361
  1320
haftmann@33361
  1321
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1322
lemma posDivAlg_eqn:
haftmann@33361
  1323
     "0 < b ==>  
haftmann@33361
  1324
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
haftmann@33361
  1325
by (rule posDivAlg.simps [THEN trans], simp)
haftmann@33361
  1326
haftmann@33361
  1327
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
haftmann@33361
  1328
theorem posDivAlg_correct:
haftmann@33361
  1329
  assumes "0 \<le> a" and "0 < b"
haftmann@33361
  1330
  shows "divmod_int_rel a b (posDivAlg a b)"
wenzelm@41550
  1331
  using assms
wenzelm@41550
  1332
  apply (induct a b rule: posDivAlg.induct)
wenzelm@41550
  1333
  apply auto
wenzelm@41550
  1334
  apply (simp add: divmod_int_rel_def)
wenzelm@41550
  1335
  apply (subst posDivAlg_eqn, simp add: right_distrib)
wenzelm@41550
  1336
  apply (case_tac "a < b")
wenzelm@41550
  1337
  apply simp_all
wenzelm@41550
  1338
  apply (erule splitE)
wenzelm@41550
  1339
  apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
wenzelm@41550
  1340
  done
haftmann@33361
  1341
haftmann@33361
  1342
huffman@46551
  1343
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
haftmann@33361
  1344
haftmann@33361
  1345
text{*And positive divisors*}
haftmann@33361
  1346
haftmann@33361
  1347
declare negDivAlg.simps [simp del]
haftmann@33361
  1348
haftmann@33361
  1349
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1350
lemma negDivAlg_eqn:
haftmann@33361
  1351
     "0 < b ==>  
haftmann@33361
  1352
      negDivAlg a b =       
haftmann@33361
  1353
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
haftmann@33361
  1354
by (rule negDivAlg.simps [THEN trans], simp)
haftmann@33361
  1355
haftmann@33361
  1356
(*Correctness of negDivAlg: it computes quotients correctly
haftmann@33361
  1357
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
haftmann@33361
  1358
lemma negDivAlg_correct:
haftmann@33361
  1359
  assumes "a < 0" and "b > 0"
haftmann@33361
  1360
  shows "divmod_int_rel a b (negDivAlg a b)"
wenzelm@41550
  1361
  using assms
wenzelm@41550
  1362
  apply (induct a b rule: negDivAlg.induct)
wenzelm@41550
  1363
  apply (auto simp add: linorder_not_le)
wenzelm@41550
  1364
  apply (simp add: divmod_int_rel_def)
wenzelm@41550
  1365
  apply (subst negDivAlg_eqn, assumption)
wenzelm@41550
  1366
  apply (case_tac "a + b < (0\<Colon>int)")
wenzelm@41550
  1367
  apply simp_all
wenzelm@41550
  1368
  apply (erule splitE)
wenzelm@41550
  1369
  apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
wenzelm@41550
  1370
  done
haftmann@33361
  1371
haftmann@33361
  1372
huffman@46551
  1373
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
haftmann@33361
  1374
haftmann@33361
  1375
(*the case a=0*)
huffman@47139
  1376
lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
haftmann@33361
  1377
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1378
haftmann@33361
  1379
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
haftmann@33361
  1380
by (subst posDivAlg.simps, auto)
haftmann@33361
  1381
huffman@47139
  1382
lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
huffman@47139
  1383
by (subst posDivAlg.simps, auto)
huffman@47139
  1384
haftmann@33361
  1385
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
haftmann@33361
  1386
by (subst negDivAlg.simps, auto)
haftmann@33361
  1387
huffman@46560
  1388
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
huffman@47139
  1389
by (auto simp add: divmod_int_rel_def)
huffman@47139
  1390
huffman@47139
  1391
lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
huffman@47139
  1392
apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
haftmann@33361
  1393
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
haftmann@33361
  1394
                    posDivAlg_correct negDivAlg_correct)
haftmann@33361
  1395
huffman@47141
  1396
lemma divmod_int_unique:
huffman@47141
  1397
  assumes "divmod_int_rel a b qr" 
huffman@47141
  1398
  shows "divmod_int a b = qr"
huffman@47141
  1399
  using assms divmod_int_correct [of a b]
huffman@47141
  1400
  using unique_quotient [of a b] unique_remainder [of a b]
huffman@47141
  1401
  by (metis pair_collapse)
huffman@47141
  1402
huffman@47141
  1403
lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
huffman@47141
  1404
  using divmod_int_correct by (simp add: divmod_int_mod_div)
huffman@47141
  1405
huffman@47141
  1406
lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
huffman@47141
  1407
  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
huffman@47141
  1408
huffman@47141
  1409
lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
huffman@47141
  1410
  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
huffman@47141
  1411
huffman@47141
  1412
instance int :: ring_div
huffman@47141
  1413
proof
huffman@47141
  1414
  fix a b :: int
huffman@47141
  1415
  show "a div b * b + a mod b = a"
huffman@47141
  1416
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1417
    unfolding divmod_int_rel_def by (simp add: mult_commute)
huffman@47141
  1418
next
huffman@47141
  1419
  fix a b c :: int
huffman@47141
  1420
  assume "b \<noteq> 0"
huffman@47141
  1421
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
huffman@47141
  1422
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1423
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141
  1424
  thus "(a + c * b) div b = c + a div b"
huffman@47141
  1425
    by (rule div_int_unique)
huffman@47141
  1426
next
huffman@47141
  1427
  fix a b c :: int
huffman@47141
  1428
  assume "c \<noteq> 0"
huffman@47141
  1429
  hence "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141
  1430
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141
  1431
    unfolding divmod_int_rel_def
huffman@47141
  1432
    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
huffman@47141
  1433
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
huffman@47141
  1434
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
huffman@47141
  1435
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
huffman@47141
  1436
    using divmod_int_rel_div_mod [of a b] .
huffman@47141
  1437
  thus "(c * a) div (c * b) = a div b"
huffman@47141
  1438
    by (rule div_int_unique)
huffman@47141
  1439
next
huffman@47141
  1440
  fix a :: int show "a div 0 = 0"
huffman@47141
  1441
    by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141
  1442
next
huffman@47141
  1443
  fix a :: int show "0 div a = 0"
huffman@47141
  1444
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141
  1445
qed
huffman@47141
  1446
haftmann@33361
  1447
text{*Basic laws about division and remainder*}
haftmann@33361
  1448
haftmann@33361
  1449
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
huffman@47141
  1450
  by (fact mod_div_equality2 [symmetric])
haftmann@33361
  1451
haftmann@33361
  1452
text {* Tool setup *}
haftmann@33361
  1453
huffman@47108
  1454
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
huffman@47108
  1455
lemmas add_0s = add_0_left add_0_right
huffman@47108
  1456
haftmann@33361
  1457
ML {*
wenzelm@43594
  1458
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550
  1459
(
haftmann@33361
  1460
  val div_name = @{const_name div};
haftmann@33361
  1461
  val mod_name = @{const_name mod};
haftmann@33361
  1462
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1463
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1464
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1465
huffman@47165
  1466
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361
  1467
haftmann@33361
  1468
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
haftmann@33361
  1469
    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
wenzelm@41550
  1470
)
haftmann@33361
  1471
*}
haftmann@33361
  1472
wenzelm@43594
  1473
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
wenzelm@43594
  1474
huffman@47141
  1475
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
huffman@47141
  1476
  using divmod_int_correct [of a b]
huffman@47141
  1477
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1478
wenzelm@45607
  1479
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1480
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1481
huffman@47141
  1482
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
huffman@47141
  1483
  using divmod_int_correct [of a b]
huffman@47141
  1484
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1485
wenzelm@45607
  1486
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1487
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1488
haftmann@33361
  1489
huffman@46551
  1490
subsubsection {* General Properties of div and mod *}
haftmann@33361
  1491
haftmann@33361
  1492
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1493
apply (rule div_int_unique)
haftmann@33361
  1494
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1495
done
haftmann@33361
  1496
haftmann@33361
  1497
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1498
apply (rule div_int_unique)
haftmann@33361
  1499
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1500
done
haftmann@33361
  1501
haftmann@33361
  1502
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1503
apply (rule div_int_unique)
haftmann@33361
  1504
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1505
done
haftmann@33361
  1506
haftmann@33361
  1507
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1508
haftmann@33361
  1509
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1510
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1511
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1512
done
haftmann@33361
  1513
haftmann@33361
  1514
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1515
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1516
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1517
done
haftmann@33361
  1518
haftmann@33361
  1519
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1520
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361
  1521
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1522
done
haftmann@33361
  1523
haftmann@33361
  1524
text{*There is no @{text mod_neg_pos_trivial}.*}
haftmann@33361
  1525
haftmann@33361
  1526
huffman@46551
  1527
subsubsection {* Laws for div and mod with Unary Minus *}
haftmann@33361
  1528
haftmann@33361
  1529
lemma zminus1_lemma:
huffman@47139
  1530
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
haftmann@33361
  1531
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@33361
  1532
                          if r=0 then 0 else b-r)"
haftmann@33361
  1533
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1534
haftmann@33361
  1535
haftmann@33361
  1536
lemma zdiv_zminus1_eq_if:
haftmann@33361
  1537
     "b \<noteq> (0::int)  
haftmann@33361
  1538
      ==> (-a) div b =  
haftmann@33361
  1539
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47140
  1540
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1541
haftmann@33361
  1542
lemma zmod_zminus1_eq_if:
haftmann@33361
  1543
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1544
apply (case_tac "b = 0", simp)
huffman@47140
  1545
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1546
done
haftmann@33361
  1547
haftmann@33361
  1548
lemma zmod_zminus1_not_zero:
haftmann@33361
  1549
  fixes k l :: int
haftmann@33361
  1550
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1551
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1552
haftmann@33361
  1553
lemma zdiv_zminus2_eq_if:
haftmann@33361
  1554
     "b \<noteq> (0::int)  
haftmann@33361
  1555
      ==> a div (-b) =  
haftmann@33361
  1556
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1557
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1558
haftmann@33361
  1559
lemma zmod_zminus2_eq_if:
haftmann@33361
  1560
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1561
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1562
haftmann@33361
  1563
lemma zmod_zminus2_not_zero:
haftmann@33361
  1564
  fixes k l :: int
haftmann@33361
  1565
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1566
  unfolding zmod_zminus2_eq_if by auto 
haftmann@33361
  1567
haftmann@33361
  1568
huffman@46551
  1569
subsubsection {* Computation of Division and Remainder *}
haftmann@33361
  1570
haftmann@33361
  1571
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@33361
  1572
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1573
haftmann@33361
  1574
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@33361
  1575
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1576
haftmann@33361
  1577
text{*a positive, b positive *}
haftmann@33361
  1578
haftmann@33361
  1579
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@33361
  1580
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1581
haftmann@33361
  1582
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@33361
  1583
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1584
haftmann@33361
  1585
text{*a negative, b positive *}
haftmann@33361
  1586
haftmann@33361
  1587
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@33361
  1588
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1589
haftmann@33361
  1590
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@33361
  1591
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1592
haftmann@33361
  1593
text{*a positive, b negative *}
haftmann@33361
  1594
haftmann@33361
  1595
lemma div_pos_neg:
huffman@46560
  1596
     "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1597
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1598
haftmann@33361
  1599
lemma mod_pos_neg:
huffman@46560
  1600
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1601
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1602
haftmann@33361
  1603
text{*a negative, b negative *}
haftmann@33361
  1604
haftmann@33361
  1605
lemma div_neg_neg:
huffman@46560
  1606
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1607
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1608
haftmann@33361
  1609
lemma mod_neg_neg:
huffman@46560
  1610
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1611
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1612
haftmann@33361
  1613
text {*Simplify expresions in which div and mod combine numerical constants*}
haftmann@33361
  1614
huffman@45530
  1615
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1616
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
huffman@45530
  1617
huffman@45530
  1618
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1619
  by (rule div_int_unique [of a b q r],
bulwahn@46552
  1620
    simp add: divmod_int_rel_def)
huffman@45530
  1621
huffman@45530
  1622
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1623
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1624
    simp add: divmod_int_rel_def)
huffman@45530
  1625
huffman@45530
  1626
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1627
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1628
    simp add: divmod_int_rel_def)
huffman@45530
  1629
haftmann@33361
  1630
(* simprocs adapted from HOL/ex/Binary.thy *)
haftmann@33361
  1631
ML {*
haftmann@33361
  1632
local
huffman@45530
  1633
  val mk_number = HOLogic.mk_number HOLogic.intT
huffman@45530
  1634
  val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1635
  val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1636
  val zero = @{term "0 :: int"}
huffman@45530
  1637
  val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1638
  val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1639
  val simps = @{thms arith_simps} @ @{thms rel_simps} @
huffman@47108
  1640
    map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
huffman@45530
  1641
  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
huffman@45530
  1642
    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps simps))));
haftmann@33361
  1643
  fun binary_proc proc ss ct =
haftmann@33361
  1644
    (case Thm.term_of ct of
haftmann@33361
  1645
      _ $ t $ u =>
haftmann@33361
  1646
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
haftmann@33361
  1647
        SOME args => proc (Simplifier.the_context ss) args
haftmann@33361
  1648
      | NONE => NONE)
haftmann@33361
  1649
    | _ => NONE);
haftmann@33361
  1650
in
huffman@45530
  1651
  fun divmod_proc posrule negrule =
huffman@45530
  1652
    binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
huffman@45530
  1653
      if b = 0 then NONE else let
huffman@45530
  1654
        val (q, r) = pairself mk_number (Integer.div_mod a b)
huffman@45530
  1655
        val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
huffman@45530
  1656
        val (goal2, goal3, rule) = if b > 0
huffman@45530
  1657
          then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
huffman@45530
  1658
          else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
huffman@45530
  1659
      in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
haftmann@33361
  1660
end
haftmann@33361
  1661
*}
haftmann@33361
  1662
huffman@47108
  1663
simproc_setup binary_int_div
huffman@47108
  1664
  ("numeral m div numeral n :: int" |
huffman@47108
  1665
   "numeral m div neg_numeral n :: int" |
huffman@47108
  1666
   "neg_numeral m div numeral n :: int" |
huffman@47108
  1667
   "neg_numeral m div neg_numeral n :: int") =
huffman@45530
  1668
  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
haftmann@33361
  1669
huffman@47108
  1670
simproc_setup binary_int_mod
huffman@47108
  1671
  ("numeral m mod numeral n :: int" |
huffman@47108
  1672
   "numeral m mod neg_numeral n :: int" |
huffman@47108
  1673
   "neg_numeral m mod numeral n :: int" |
huffman@47108
  1674
   "neg_numeral m mod neg_numeral n :: int") =
huffman@45530
  1675
  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
haftmann@33361
  1676
huffman@47108
  1677
lemmas posDivAlg_eqn_numeral [simp] =
huffman@47108
  1678
    posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
huffman@47108
  1679
huffman@47108
  1680
lemmas negDivAlg_eqn_numeral [simp] =
huffman@47108
  1681
    negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
haftmann@33361
  1682
haftmann@33361
  1683
haftmann@33361
  1684
text{*Special-case simplification *}
haftmann@33361
  1685
haftmann@33361
  1686
(** The last remaining special cases for constant arithmetic:
haftmann@33361
  1687
    1 div z and 1 mod z **)
haftmann@33361
  1688
huffman@47108
  1689
lemmas div_pos_pos_1_numeral [simp] =
huffman@47108
  1690
  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
huffman@47108
  1691
huffman@47108
  1692
lemmas div_pos_neg_1_numeral [simp] =
huffman@47108
  1693
  div_pos_neg [OF zero_less_one, of "neg_numeral w",
huffman@47108
  1694
  OF neg_numeral_less_zero] for w
huffman@47108
  1695
huffman@47108
  1696
lemmas mod_pos_pos_1_numeral [simp] =
huffman@47108
  1697
  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
huffman@47108
  1698
huffman@47108
  1699
lemmas mod_pos_neg_1_numeral [simp] =
huffman@47108
  1700
  mod_pos_neg [OF zero_less_one, of "neg_numeral w",
huffman@47108
  1701
  OF neg_numeral_less_zero] for w
huffman@47108
  1702
huffman@47108
  1703
lemmas posDivAlg_eqn_1_numeral [simp] =
huffman@47108
  1704
    posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
huffman@47108
  1705
huffman@47108
  1706
lemmas negDivAlg_eqn_1_numeral [simp] =
huffman@47108
  1707
    negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
haftmann@33361
  1708
haftmann@33361
  1709
huffman@46551
  1710
subsubsection {* Monotonicity in the First Argument (Dividend) *}
haftmann@33361
  1711
haftmann@33361
  1712
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361
  1713
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1714
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1715
apply (rule unique_quotient_lemma)
haftmann@33361
  1716
apply (erule subst)
haftmann@33361
  1717
apply (erule subst, simp_all)
haftmann@33361
  1718
done
haftmann@33361
  1719
haftmann@33361
  1720
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361
  1721
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1722
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1723
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1724
apply (erule subst)
haftmann@33361
  1725
apply (erule subst, simp_all)
haftmann@33361
  1726
done
haftmann@33361
  1727
haftmann@33361
  1728
huffman@46551
  1729
subsubsection {* Monotonicity in the Second Argument (Divisor) *}
haftmann@33361
  1730
haftmann@33361
  1731
lemma q_pos_lemma:
haftmann@33361
  1732
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1733
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1734
 apply (simp add: zero_less_mult_iff)
haftmann@33361
  1735
apply (simp add: right_distrib)
haftmann@33361
  1736
done
haftmann@33361
  1737
haftmann@33361
  1738
lemma zdiv_mono2_lemma:
haftmann@33361
  1739
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
haftmann@33361
  1740
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
haftmann@33361
  1741
      ==> q \<le> (q'::int)"
haftmann@33361
  1742
apply (frule q_pos_lemma, assumption+) 
haftmann@33361
  1743
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  1744
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1745
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  1746
 prefer 2 apply simp
haftmann@33361
  1747
apply (simp (no_asm_simp) add: right_distrib)
huffman@44766
  1748
apply (subst add_commute, rule add_less_le_mono, arith)
haftmann@33361
  1749
apply (rule mult_right_mono, auto)
haftmann@33361
  1750
done
haftmann@33361
  1751
haftmann@33361
  1752
lemma zdiv_mono2:
haftmann@33361
  1753
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  1754
apply (subgoal_tac "b \<noteq> 0")
haftmann@33361
  1755
 prefer 2 apply arith
haftmann@33361
  1756
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1757
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1758
apply (rule zdiv_mono2_lemma)
haftmann@33361
  1759
apply (erule subst)
haftmann@33361
  1760
apply (erule subst, simp_all)
haftmann@33361
  1761
done
haftmann@33361
  1762
haftmann@33361
  1763
lemma q_neg_lemma:
haftmann@33361
  1764
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  1765
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  1766
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  1767
done
haftmann@33361
  1768
haftmann@33361
  1769
lemma zdiv_mono2_neg_lemma:
haftmann@33361
  1770
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
haftmann@33361
  1771
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
haftmann@33361
  1772
      ==> q' \<le> (q::int)"
haftmann@33361
  1773
apply (frule q_neg_lemma, assumption+) 
haftmann@33361
  1774
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  1775
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1776
apply (simp add: right_distrib)
haftmann@33361
  1777
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  1778
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  1779
done
haftmann@33361
  1780
haftmann@33361
  1781
lemma zdiv_mono2_neg:
haftmann@33361
  1782
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361
  1783
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1784
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1785
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  1786
apply (erule subst)
haftmann@33361
  1787
apply (erule subst, simp_all)
haftmann@33361
  1788
done
haftmann@33361
  1789
haftmann@33361
  1790
huffman@46551
  1791
subsubsection {* More Algebraic Laws for div and mod *}
haftmann@33361
  1792
haftmann@33361
  1793
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
haftmann@33361
  1794
haftmann@33361
  1795
lemma zmult1_lemma:
bulwahn@46552
  1796
     "[| divmod_int_rel b c (q, r) |]  
haftmann@33361
  1797
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@33361
  1798
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
haftmann@33361
  1799
haftmann@33361
  1800
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  1801
apply (case_tac "c = 0", simp)
huffman@47140
  1802
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  1803
done
haftmann@33361
  1804
haftmann@33361
  1805
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@33361
  1806
haftmann@33361
  1807
lemma zadd1_lemma:
bulwahn@46552
  1808
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
haftmann@33361
  1809
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
haftmann@33361
  1810
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
haftmann@33361
  1811
haftmann@33361
  1812
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  1813
lemma zdiv_zadd1_eq:
haftmann@33361
  1814
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  1815
apply (case_tac "c = 0", simp)
huffman@47140
  1816
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
haftmann@33361
  1817
done
haftmann@33361
  1818
haftmann@33361
  1819
lemma posDivAlg_div_mod:
haftmann@33361
  1820
  assumes "k \<ge> 0"
haftmann@33361
  1821
  and "l \<ge> 0"
haftmann@33361
  1822
  shows "posDivAlg k l = (k div l, k mod l)"
haftmann@33361
  1823
proof (cases "l = 0")
haftmann@33361
  1824
  case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@33361
  1825
next
haftmann@33361
  1826
  case False with assms posDivAlg_correct
haftmann@33361
  1827
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@33361
  1828
    by simp
huffman@47140
  1829
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  1830
  show ?thesis by simp
haftmann@33361
  1831
qed
haftmann@33361
  1832
haftmann@33361
  1833
lemma negDivAlg_div_mod:
haftmann@33361
  1834
  assumes "k < 0"
haftmann@33361
  1835
  and "l > 0"
haftmann@33361
  1836
  shows "negDivAlg k l = (k div l, k mod l)"
haftmann@33361
  1837
proof -
haftmann@33361
  1838
  from assms have "l \<noteq> 0" by simp
haftmann@33361
  1839
  from assms negDivAlg_correct
haftmann@33361
  1840
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@33361
  1841
    by simp
huffman@47140
  1842
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  1843
  show ?thesis by simp
haftmann@33361
  1844
qed
haftmann@33361
  1845
haftmann@33361
  1846
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  1847
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  1848
haftmann@33361
  1849
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  1850
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  1851
huffman@47108
  1852
lemma zmod_zdiv_equality':
huffman@47108
  1853
  "(m\<Colon>int) mod n = m - (m div n) * n"
huffman@47141
  1854
  using mod_div_equality [of m n] by arith
huffman@47108
  1855
haftmann@33361
  1856
huffman@46551
  1857
subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
haftmann@33361
  1858
haftmann@33361
  1859
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  1860
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  1861
  to cause particular problems.*)
haftmann@33361
  1862
haftmann@33361
  1863
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
haftmann@33361
  1864
haftmann@33361
  1865
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
haftmann@33361
  1866
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  1867
 apply (simp add: algebra_simps)
haftmann@33361
  1868
apply (rule order_le_less_trans)
haftmann@33361
  1869
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  1870
 apply (rule mult_left_mono_neg)
huffman@35216
  1871
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  1872
 apply (simp)
haftmann@33361
  1873
apply (simp)
haftmann@33361
  1874
done
haftmann@33361
  1875
haftmann@33361
  1876
lemma zmult2_lemma_aux2:
haftmann@33361
  1877
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  1878
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  1879
 apply arith
haftmann@33361
  1880
apply (simp add: mult_le_0_iff)
haftmann@33361
  1881
done
haftmann@33361
  1882
haftmann@33361
  1883
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  1884
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  1885
apply arith
haftmann@33361
  1886
apply (simp add: zero_le_mult_iff)
haftmann@33361
  1887
done
haftmann@33361
  1888
haftmann@33361
  1889
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  1890
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  1891
 apply (simp add: right_diff_distrib)
haftmann@33361
  1892
apply (rule order_less_le_trans)
haftmann@33361
  1893
 apply (erule mult_strict_right_mono)
haftmann@33361
  1894
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  1895
  apply simp
huffman@35216
  1896
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  1897
apply simp
haftmann@33361
  1898
done
haftmann@33361
  1899
bulwahn@46552
  1900
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
haftmann@33361
  1901
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@33361
  1902
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
haftmann@33361
  1903
                   zero_less_mult_iff right_distrib [symmetric] 
huffman@47139
  1904
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
haftmann@33361
  1905
haftmann@33361
  1906
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
haftmann@33361
  1907
apply (case_tac "b = 0", simp)
huffman@47140
  1908
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  1909
done
haftmann@33361
  1910
haftmann@33361
  1911
lemma zmod_zmult2_eq:
haftmann@33361
  1912
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
haftmann@33361
  1913
apply (case_tac "b = 0", simp)
huffman@47140
  1914
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  1915
done
haftmann@33361
  1916
huffman@47108
  1917
lemma div_pos_geq:
huffman@47108
  1918
  fixes k l :: int
huffman@47108
  1919
  assumes "0 < l" and "l \<le> k"
huffman@47108
  1920
  shows "k div l = (k - l) div l + 1"
huffman@47108
  1921
proof -
huffman@47108
  1922
  have "k = (k - l) + l" by simp
huffman@47108
  1923
  then obtain j where k: "k = j + l" ..
huffman@47108
  1924
  with assms show ?thesis by simp
huffman@47108
  1925
qed
huffman@47108
  1926
huffman@47108
  1927
lemma mod_pos_geq:
huffman@47108
  1928
  fixes k l :: int
huffman@47108
  1929
  assumes "0 < l" and "l \<le> k"
huffman@47108
  1930
  shows "k mod l = (k - l) mod l"
huffman@47108
  1931
proof -
huffman@47108
  1932
  have "k = (k - l) + l" by simp
huffman@47108
  1933
  then obtain j where k: "k = j + l" ..
huffman@47108
  1934
  with assms show ?thesis by simp
huffman@47108
  1935
qed
huffman@47108
  1936
haftmann@33361
  1937
huffman@46551
  1938
subsubsection {* Splitting Rules for div and mod *}
haftmann@33361
  1939
haftmann@33361
  1940
text{*The proofs of the two lemmas below are essentially identical*}
haftmann@33361
  1941
haftmann@33361
  1942
lemma split_pos_lemma:
haftmann@33361
  1943
 "0<k ==> 
haftmann@33361
  1944
    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
  1945
apply (rule iffI, clarify)
haftmann@33361
  1946
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  1947
 apply (subst mod_add_eq) 
haftmann@33361
  1948
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  1949
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
haftmann@33361
  1950
txt{*converse direction*}
haftmann@33361
  1951
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  1952
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  1953
done
haftmann@33361
  1954
haftmann@33361
  1955
lemma split_neg_lemma:
haftmann@33361
  1956
 "k<0 ==>
haftmann@33361
  1957
    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
  1958
apply (rule iffI, clarify)
haftmann@33361
  1959
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  1960
 apply (subst mod_add_eq) 
haftmann@33361
  1961
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  1962
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
haftmann@33361
  1963
txt{*converse direction*}
haftmann@33361
  1964
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  1965
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  1966
done
haftmann@33361
  1967
haftmann@33361
  1968
lemma split_zdiv:
haftmann@33361
  1969
 "P(n div k :: int) =
haftmann@33361
  1970
  ((k = 0 --> P 0) & 
haftmann@33361
  1971
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
haftmann@33361
  1972
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  1973
apply (case_tac "k=0", simp)
haftmann@33361
  1974
apply (simp only: linorder_neq_iff)
haftmann@33361
  1975
apply (erule disjE) 
haftmann@33361
  1976
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
haftmann@33361
  1977
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  1978
done
haftmann@33361
  1979
haftmann@33361
  1980
lemma split_zmod:
haftmann@33361
  1981
 "P(n mod k :: int) =
haftmann@33361
  1982
  ((k = 0 --> P n) & 
haftmann@33361
  1983
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
haftmann@33361
  1984
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  1985
apply (case_tac "k=0", simp)
haftmann@33361
  1986
apply (simp only: linorder_neq_iff)
haftmann@33361
  1987
apply (erule disjE) 
haftmann@33361
  1988
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
haftmann@33361
  1989
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  1990
done
haftmann@33361
  1991
webertj@33730
  1992
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
webertj@33730
  1993
  when these are applied to some constant that is of the form
huffman@47108
  1994
  @{term "numeral k"}: *}
huffman@47108
  1995
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  1996
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  1997
haftmann@33361
  1998
huffman@47166
  1999
subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
huffman@47166
  2000
huffman@47166
  2001
lemma pos_divmod_int_rel_mult_2:
huffman@47166
  2002
  assumes "0 \<le> b"
huffman@47166
  2003
  assumes "divmod_int_rel a b (q, r)"
huffman@47166
  2004
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166
  2005
  using assms unfolding divmod_int_rel_def by auto
huffman@47166
  2006
huffman@47166
  2007
lemma neg_divmod_int_rel_mult_2:
huffman@47166
  2008
  assumes "b \<le> 0"
huffman@47166
  2009
  assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166
  2010
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166
  2011
  using assms unfolding divmod_int_rel_def by auto
haftmann@33361
  2012
haftmann@33361
  2013
text{*computing div by shifting *}
haftmann@33361
  2014
haftmann@33361
  2015
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
huffman@47166
  2016
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
huffman@47166
  2017
  by (rule div_int_unique)
haftmann@33361
  2018
boehmes@35815
  2019
lemma neg_zdiv_mult_2: 
boehmes@35815
  2020
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
huffman@47166
  2021
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
huffman@47166
  2022
  by (rule div_int_unique)
haftmann@33361
  2023
huffman@47108
  2024
(* FIXME: add rules for negative numerals *)
huffman@47108
  2025
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  2026
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  2027
    numeral v div (numeral w :: int)"
huffman@47108
  2028
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  2029
  by (rule div_mult_mult1, simp)
huffman@47108
  2030
huffman@47108
  2031
lemma zdiv_numeral_Bit1 [simp]:
huffman@47108
  2032
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
huffman@47108
  2033
    (numeral v div (numeral w :: int))"
huffman@47108
  2034
  unfolding numeral.simps
huffman@47108
  2035
  unfolding mult_2 [symmetric] add_commute [of _ 1]
huffman@47108
  2036
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  2037
haftmann@33361
  2038
lemma pos_zmod_mult_2:
haftmann@33361
  2039
  fixes a b :: int
haftmann@33361
  2040
  assumes "0 \<le> a"
haftmann@33361
  2041
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
huffman@47166
  2042
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2043
  by (rule mod_int_unique)
haftmann@33361
  2044
haftmann@33361
  2045
lemma neg_zmod_mult_2:
haftmann@33361
  2046
  fixes a b :: int
haftmann@33361
  2047
  assumes "a \<le> 0"
haftmann@33361
  2048
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
huffman@47166
  2049
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2050
  by (rule mod_int_unique)
haftmann@33361
  2051
huffman@47108
  2052
(* FIXME: add rules for negative numerals *)
huffman@47108
  2053
lemma zmod_numeral_Bit0 [simp]:
huffman@47108
  2054
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
huffman@47108
  2055
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  2056
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  2057
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  2058
huffman@47108
  2059
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  2060
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2061
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  2062
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
huffman@47108
  2063
  unfolding mult_2 [symmetric] add_commute [of _ 1]
huffman@47108
  2064
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  2065
nipkow@39489
  2066
lemma zdiv_eq_0_iff:
nipkow@39489
  2067
 "(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
  2068
proof
nipkow@39489
  2069
  assume ?L
nipkow@39489
  2070
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
nipkow@39489
  2071
  with `?L` show ?R by blast
nipkow@39489
  2072
next
nipkow@39489
  2073
  assume ?R thus ?L
nipkow@39489
  2074
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  2075
qed
nipkow@39489
  2076
nipkow@39489
  2077
huffman@46551
  2078
subsubsection {* Quotients of Signs *}
haftmann@33361
  2079
haftmann@33361
  2080
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2081
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2082
apply (rule order_trans)
haftmann@33361
  2083
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2084
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2085
done
haftmann@33361
  2086
haftmann@33361
  2087
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2088
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2089
haftmann@33361
  2090
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2091
by (drule zdiv_mono1, auto)
haftmann@33361
  2092
nipkow@33804
  2093
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
nipkow@33804
  2094
conditional upon the sign of @{text a} or @{text b}. There are many more.
nipkow@33804
  2095
They should all be simp rules unless that causes too much search. *}
nipkow@33804
  2096
haftmann@33361
  2097
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2098
apply auto
haftmann@33361
  2099
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2100
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2101
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2102
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2103
done
haftmann@33361
  2104
haftmann@33361
  2105
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2106
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2107
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2108
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2109
done
haftmann@33361
  2110
haftmann@33361
  2111
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2112
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2113
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2114
nipkow@39489
  2115
lemma pos_imp_zdiv_pos_iff:
nipkow@39489
  2116
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
nipkow@39489
  2117
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
nipkow@39489
  2118
by arith
nipkow@39489
  2119
haftmann@33361
  2120
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2121
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2122
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2123
nipkow@33804
  2124
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2125
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2126
apply rule
nipkow@33804
  2127
 apply rule
nipkow@33804
  2128
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2129
 apply(cases "b=0")apply simp
nipkow@33804
  2130
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2131
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2132
done
nipkow@33804
  2133
nipkow@39489
  2134
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2135
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2136
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2137
done
nipkow@39489
  2138
nipkow@39489
  2139
haftmann@33361
  2140
subsubsection {* The Divides Relation *}
haftmann@33361
  2141
huffman@47268
  2142
lemma dvd_neg_numeral_left [simp]:
huffman@47268
  2143
  fixes y :: "'a::comm_ring_1"
huffman@47268
  2144
  shows "(neg_numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
huffman@47268
  2145
  unfolding neg_numeral_def minus_dvd_iff ..
huffman@47268
  2146
huffman@47268
  2147
lemma dvd_neg_numeral_right [simp]:
huffman@47268
  2148
  fixes x :: "'a::comm_ring_1"
huffman@47268
  2149
  shows "x dvd (neg_numeral k) \<longleftrightarrow> x dvd (numeral k)"
huffman@47268
  2150
  unfolding neg_numeral_def dvd_minus_iff ..
haftmann@33361
  2151
huffman@47108
  2152
lemmas dvd_eq_mod_eq_0_numeral [simp] =
huffman@47108
  2153
  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
huffman@47108
  2154
huffman@47108
  2155
huffman@47108
  2156
subsubsection {* Further properties *}
huffman@47108
  2157
haftmann@33361
  2158
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
haftmann@33361
  2159
  using zmod_zdiv_equality[where a="m" and b="n"]
huffman@47142
  2160
  by (simp add: algebra_simps) (* FIXME: generalize *)
haftmann@33361
  2161
haftmann@33361
  2162
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
haftmann@33361
  2163
apply (subst split_div, auto)
haftmann@33361
  2164
apply (subst split_zdiv, auto)
haftmann@33361
  2165
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
  2166
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2167
done
haftmann@33361
  2168
haftmann@33361
  2169
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
haftmann@33361
  2170
apply (subst split_mod, auto)
haftmann@33361
  2171
apply (subst split_zmod, auto)
haftmann@33361
  2172
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
haftmann@33361
  2173
       in unique_remainder)
haftmann@33361
  2174
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2175
done
haftmann@33361
  2176
haftmann@33361
  2177
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
haftmann@33361
  2178
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2179
haftmann@33361
  2180
text{*Suggested by Matthias Daum*}
haftmann@33361
  2181
lemma int_power_div_base:
haftmann@33361
  2182
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2183
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2184
 apply (erule ssubst)
haftmann@33361
  2185
 apply (simp only: power_add)
haftmann@33361
  2186
 apply simp_all
haftmann@33361
  2187
done
haftmann@33361
  2188
haftmann@33361
  2189
text {* by Brian Huffman *}
haftmann@33361
  2190
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2191
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2192
haftmann@33361
  2193
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2194
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2195
haftmann@33361
  2196
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2197
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2198
haftmann@33361
  2199
lemmas zmod_simps =
haftmann@33361
  2200
  mod_add_left_eq  [symmetric]
haftmann@33361
  2201
  mod_add_right_eq [symmetric]
huffman@47142
  2202
  mod_mult_right_eq[symmetric]
haftmann@33361
  2203
  mod_mult_left_eq [symmetric]
huffman@47164
  2204
  power_mod
haftmann@33361
  2205
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2206
haftmann@33361
  2207
text {* Distributive laws for function @{text nat}. *}
haftmann@33361
  2208
haftmann@33361
  2209
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2210
apply (rule linorder_cases [of y 0])
haftmann@33361
  2211
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2212
apply simp
haftmann@33361
  2213
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2214
done
haftmann@33361
  2215
haftmann@33361
  2216
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2217
lemma nat_mod_distrib:
haftmann@33361
  2218
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2219
apply (case_tac "y = 0", simp)
haftmann@33361
  2220
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2221
done
haftmann@33361
  2222
haftmann@33361
  2223
text  {* transfer setup *}
haftmann@33361
  2224
haftmann@33361
  2225
lemma transfer_nat_int_functions:
haftmann@33361
  2226
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2227
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2228
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2229
haftmann@33361
  2230
lemma transfer_nat_int_function_closures:
haftmann@33361
  2231
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2232
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2233
  apply (cases "y = 0")
haftmann@33361
  2234
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2235
  apply (cases "y = 0")
haftmann@33361
  2236
  apply auto
haftmann@33361
  2237
done
haftmann@33361
  2238
haftmann@35644
  2239
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2240
  transfer_nat_int_functions
haftmann@33361
  2241
  transfer_nat_int_function_closures
haftmann@33361
  2242
]
haftmann@33361
  2243
haftmann@33361
  2244
lemma transfer_int_nat_functions:
haftmann@33361
  2245
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2246
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2247
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2248
haftmann@33361
  2249
lemma transfer_int_nat_function_closures:
haftmann@33361
  2250
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  2251
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  2252
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  2253
haftmann@35644
  2254
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  2255
  transfer_int_nat_functions
haftmann@33361
  2256
  transfer_int_nat_function_closures
haftmann@33361
  2257
]
haftmann@33361
  2258
haftmann@33361
  2259
text{*Suggested by Matthias Daum*}
haftmann@33361
  2260
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  2261
apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225
  2262
 apply (simp add: nat_div_distrib [symmetric])
haftmann@33361
  2263
apply (rule Divides.div_less_dividend, simp_all)
haftmann@33361
  2264
done
haftmann@33361
  2265
haftmann@33361
  2266
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
haftmann@33361
  2267
proof
haftmann@33361
  2268
  assume H: "x mod n = y mod n"
haftmann@33361
  2269
  hence "x mod n - y mod n = 0" by simp
haftmann@33361
  2270
  hence "(x mod n - y mod n) mod n = 0" by simp 
haftmann@33361
  2271
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
haftmann@33361
  2272
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
haftmann@33361
  2273
next
haftmann@33361
  2274
  assume H: "n dvd x - y"
haftmann@33361
  2275
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
haftmann@33361
  2276
  hence "x = n*k + y" by simp
haftmann@33361
  2277
  hence "x mod n = (n*k + y) mod n" by simp
haftmann@33361
  2278
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
haftmann@33361
  2279
qed
haftmann@33361
  2280
haftmann@33361
  2281
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
haftmann@33361
  2282
  shows "\<exists>q. x = y + n * q"
haftmann@33361
  2283
proof-
haftmann@33361
  2284
  from xy have th: "int x - int y = int (x - y)" by simp 
haftmann@33361
  2285
  from xyn have "int x mod int n = int y mod int n" 
huffman@46551
  2286
    by (simp add: zmod_int [symmetric])
haftmann@33361
  2287
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
haftmann@33361
  2288
  hence "n dvd x - y" by (simp add: th zdvd_int)
haftmann@33361
  2289
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
haftmann@33361
  2290
qed
haftmann@33361
  2291
haftmann@33361
  2292
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
haftmann@33361
  2293
  (is "?lhs = ?rhs")
haftmann@33361
  2294
proof
haftmann@33361
  2295
  assume H: "x mod n = y mod n"
haftmann@33361
  2296
  {assume xy: "x \<le> y"
haftmann@33361
  2297
    from H have th: "y mod n = x mod n" by simp
haftmann@33361
  2298
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
haftmann@33361
  2299
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361
  2300
  moreover
haftmann@33361
  2301
  {assume xy: "y \<le> x"
haftmann@33361
  2302
    from nat_mod_eq_lemma[OF H xy] have ?rhs 
haftmann@33361
  2303
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
haftmann@33361
  2304
  ultimately  show ?rhs using linear[of x y] by blast  
haftmann@33361
  2305
next
haftmann@33361
  2306
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
haftmann@33361
  2307
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
haftmann@33361
  2308
  thus  ?lhs by simp
haftmann@33361
  2309
qed
haftmann@33361
  2310
huffman@47108
  2311
lemma div_nat_numeral [simp]:
huffman@47108
  2312
  "(numeral v :: nat) div numeral v' = nat (numeral v div numeral v')"
haftmann@33361
  2313
  by (simp add: nat_div_distrib)
haftmann@33361
  2314
huffman@47108
  2315
lemma one_div_nat_numeral [simp]:
huffman@47108
  2316
  "Suc 0 div numeral v' = nat (1 div numeral v')"
huffman@47108
  2317
  by (subst nat_div_distrib, simp_all)
huffman@47108
  2318
huffman@47108
  2319
lemma mod_nat_numeral [simp]:
huffman@47108
  2320
  "(numeral v :: nat) mod numeral v' = nat (numeral v mod numeral v')"
haftmann@33361
  2321
  by (simp add: nat_mod_distrib)
haftmann@33361
  2322
huffman@47108
  2323
lemma one_mod_nat_numeral [simp]:
huffman@47108
  2324
  "Suc 0 mod numeral v' = nat (1 mod numeral v')"
huffman@47108
  2325
  by (subst nat_mod_distrib) simp_all
huffman@47108
  2326
huffman@47108
  2327
lemma mod_2_not_eq_zero_eq_one_int:
huffman@47108
  2328
  fixes k :: int
huffman@47108
  2329
  shows "k mod 2 \<noteq> 0 \<longleftrightarrow> k mod 2 = 1"
huffman@47108
  2330
  by auto
huffman@47108
  2331
huffman@47108
  2332
huffman@47108
  2333
subsubsection {* Tools setup *}
huffman@47108
  2334
huffman@47108
  2335
text {* Nitpick *}
blanchet@34126
  2336
blanchet@41792
  2337
lemmas [nitpick_unfold] = dvd_eq_mod_eq_0 mod_div_equality' zmod_zdiv_equality'
blanchet@34126
  2338
haftmann@35673
  2339
haftmann@33361
  2340
subsubsection {* Code generation *}
haftmann@33361
  2341
haftmann@33361
  2342
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  2343
  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@33361
  2344
haftmann@33361
  2345
lemma pdivmod_posDivAlg [code]:
haftmann@33361
  2346
  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
haftmann@33361
  2347
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
haftmann@33361
  2348
haftmann@33361
  2349
lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@33361
  2350
  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
haftmann@33361
  2351
    then pdivmod k l
haftmann@33361
  2352
    else (let (r, s) = pdivmod k l in
huffman@47108
  2353
       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@33361
  2354
proof -
haftmann@33361
  2355
  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
haftmann@33361
  2356
  show ?thesis
haftmann@33361
  2357
    by (simp add: divmod_int_mod_div pdivmod_def)
haftmann@33361
  2358
      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
haftmann@33361
  2359
      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
haftmann@33361
  2360
qed
haftmann@33361
  2361
haftmann@33361
  2362
lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@33361
  2363
  apsnd ((op *) (sgn l)) (if sgn k = sgn l
haftmann@33361
  2364
    then pdivmod k l
haftmann@33361
  2365
    else (let (r, s) = pdivmod k l in
haftmann@33361
  2366
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@33361
  2367
proof -
haftmann@33361
  2368
  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
haftmann@33361
  2369
    by (auto simp add: not_less sgn_if)
haftmann@33361
  2370
  then show ?thesis by (simp add: divmod_int_pdivmod)
haftmann@33361
  2371
qed
haftmann@33361
  2372
haftmann@33364
  2373
code_modulename SML
haftmann@33364
  2374
  Divides Arith
haftmann@33364
  2375
haftmann@33364
  2376
code_modulename OCaml
haftmann@33364
  2377
  Divides Arith
haftmann@33364
  2378
haftmann@33364
  2379
code_modulename Haskell
haftmann@33364
  2380
  Divides Arith
haftmann@33364
  2381
haftmann@33361
  2382
end