src/HOL/Divides.thy
author haftmann
Sun Aug 18 15:29:50 2013 +0200 (2013-08-18)
changeset 53068 41fc65da66f1
parent 53067 ee0b7c2315d2
child 53069 d165213e3924
permissions -rw-r--r--
relaxed preconditions
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(*  Title:      HOL/Divides.thy
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1999  University of Cambridge
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*)
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header {* The division operators div and mod *}
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theory Divides
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imports Nat_Transfer
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begin
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subsection {* Syntactic division operations *}
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class div = dvd +
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  fixes div :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "div" 70)
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    and mod :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "mod" 70)
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subsection {* Abstract division in commutative semirings. *}
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class semiring_div = comm_semiring_1_cancel + no_zero_divisors + div +
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  assumes mod_div_equality: "a div b * b + a mod b = a"
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    and div_by_0 [simp]: "a div 0 = 0"
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    and div_0 [simp]: "0 div a = 0"
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    and div_mult_self1 [simp]: "b \<noteq> 0 \<Longrightarrow> (a + c * b) div b = c + a div b"
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    and div_mult_mult1 [simp]: "c \<noteq> 0 \<Longrightarrow> (c * a) div (c * b) = a div b"
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begin
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text {* @{const div} and @{const mod} *}
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lemma mod_div_equality2: "b * (a div b) + a mod b = a"
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  unfolding mult_commute [of b]
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  by (rule mod_div_equality)
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lemma mod_div_equality': "a mod b + a div b * b = a"
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  using mod_div_equality [of a b]
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  by (simp only: add_ac)
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lemma div_mod_equality: "((a div b) * b + a mod b) + c = a + c"
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  by (simp add: mod_div_equality)
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lemma div_mod_equality2: "(b * (a div b) + a mod b) + c = a + c"
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  by (simp add: mod_div_equality2)
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lemma mod_by_0 [simp]: "a mod 0 = a"
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  using mod_div_equality [of a zero] by simp
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lemma mod_0 [simp]: "0 mod a = 0"
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  using mod_div_equality [of zero a] div_0 by simp
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lemma div_mult_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b * c) div b = c + a div b"
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  using assms div_mult_self1 [of b a c] by (simp add: mult_commute)
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lemma mod_mult_self1 [simp]: "(a + c * b) mod b = a mod b"
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proof (cases "b = 0")
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  case True then show ?thesis by simp
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next
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  case False
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  have "a + c * b = (a + c * b) div b * b + (a + c * b) mod b"
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    by (simp add: mod_div_equality)
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  also from False div_mult_self1 [of b a c] have
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    "\<dots> = (c + a div b) * b + (a + c * b) mod b"
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      by (simp add: algebra_simps)
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  finally have "a = a div b * b + (a + c * b) mod b"
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    by (simp add: add_commute [of a] add_assoc distrib_right)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: "(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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lemma div_mult_div_if_dvd:
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  "y dvd x \<Longrightarrow> z dvd w \<Longrightarrow> (x div y) * (w div z) = (x * w) div (y * z)"
haftmann@30930
   313
  apply (cases "y = 0", simp)
haftmann@30930
   314
  apply (cases "z = 0", simp)
haftmann@30930
   315
  apply (auto elim!: dvdE simp add: algebra_simps)
nipkow@30476
   316
  apply (subst mult_assoc [symmetric])
nipkow@30476
   317
  apply (simp add: no_zero_divisors)
haftmann@30930
   318
  done
haftmann@30930
   319
haftmann@35367
   320
lemma div_mult_swap:
haftmann@35367
   321
  assumes "c dvd b"
haftmann@35367
   322
  shows "a * (b div c) = (a * b) div c"
haftmann@35367
   323
proof -
haftmann@35367
   324
  from assms have "b div c * (a div 1) = b * a div (c * 1)"
haftmann@35367
   325
    by (simp only: div_mult_div_if_dvd one_dvd)
haftmann@35367
   326
  then show ?thesis by (simp add: mult_commute)
haftmann@35367
   327
qed
haftmann@35367
   328
   
haftmann@30930
   329
lemma div_mult_mult2 [simp]:
haftmann@30930
   330
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@30930
   331
  by (drule div_mult_mult1) (simp add: mult_commute)
haftmann@30930
   332
haftmann@30930
   333
lemma div_mult_mult1_if [simp]:
haftmann@30930
   334
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   335
  by simp_all
nipkow@30476
   336
haftmann@30930
   337
lemma mod_mult_mult1:
haftmann@30930
   338
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   339
proof (cases "c = 0")
haftmann@30930
   340
  case True then show ?thesis by simp
haftmann@30930
   341
next
haftmann@30930
   342
  case False
haftmann@30930
   343
  from mod_div_equality
haftmann@30930
   344
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   345
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   346
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@30930
   347
  with mod_div_equality show ?thesis by simp 
haftmann@30930
   348
qed
haftmann@30930
   349
  
haftmann@30930
   350
lemma mod_mult_mult2:
haftmann@30930
   351
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@30930
   352
  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
haftmann@30930
   353
huffman@47159
   354
lemma mult_mod_left: "(a mod b) * c = (a * c) mod (b * c)"
huffman@47159
   355
  by (fact mod_mult_mult2 [symmetric])
huffman@47159
   356
huffman@47159
   357
lemma mult_mod_right: "c * (a mod b) = (c * a) mod (c * b)"
huffman@47159
   358
  by (fact mod_mult_mult1 [symmetric])
huffman@47159
   359
huffman@31662
   360
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   361
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   362
huffman@31662
   363
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   364
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   365
haftmann@31009
   366
lemma div_power:
huffman@31661
   367
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
nipkow@30476
   368
apply (induct n)
nipkow@30476
   369
 apply simp
nipkow@30476
   370
apply(simp add: div_mult_div_if_dvd dvd_power_same)
nipkow@30476
   371
done
nipkow@30476
   372
haftmann@35367
   373
lemma dvd_div_eq_mult:
haftmann@35367
   374
  assumes "a \<noteq> 0" and "a dvd b"  
haftmann@35367
   375
  shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@35367
   376
proof
haftmann@35367
   377
  assume "b = c * a"
haftmann@35367
   378
  then show "b div a = c" by (simp add: assms)
haftmann@35367
   379
next
haftmann@35367
   380
  assume "b div a = c"
haftmann@35367
   381
  then have "b div a * a = c * a" by simp
haftmann@35367
   382
  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
haftmann@35367
   383
  ultimately show "b = c * a" by simp
haftmann@35367
   384
qed
haftmann@35367
   385
   
haftmann@35367
   386
lemma dvd_div_div_eq_mult:
haftmann@35367
   387
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@35367
   388
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
haftmann@35367
   389
  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
   390
huffman@31661
   391
end
huffman@31661
   392
haftmann@35673
   393
class ring_div = semiring_div + comm_ring_1
huffman@29405
   394
begin
huffman@29405
   395
haftmann@36634
   396
subclass ring_1_no_zero_divisors ..
haftmann@36634
   397
huffman@29405
   398
text {* Negation respects modular equivalence. *}
huffman@29405
   399
huffman@29405
   400
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   401
proof -
huffman@29405
   402
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405
   403
    by (simp only: mod_div_equality)
huffman@29405
   404
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
huffman@29405
   405
    by (simp only: minus_add_distrib minus_mult_left add_ac)
huffman@29405
   406
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   407
    by (rule mod_mult_self1)
huffman@29405
   408
  finally show ?thesis .
huffman@29405
   409
qed
huffman@29405
   410
huffman@29405
   411
lemma mod_minus_cong:
huffman@29405
   412
  assumes "a mod b = a' mod b"
huffman@29405
   413
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   414
proof -
huffman@29405
   415
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   416
    unfolding assms ..
huffman@29405
   417
  thus ?thesis
huffman@29405
   418
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   419
qed
huffman@29405
   420
huffman@29405
   421
text {* Subtraction respects modular equivalence. *}
huffman@29405
   422
huffman@29405
   423
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
huffman@29405
   424
  unfolding diff_minus
huffman@29405
   425
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   426
huffman@29405
   427
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
huffman@29405
   428
  unfolding diff_minus
huffman@29405
   429
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   430
huffman@29405
   431
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
huffman@29405
   432
  unfolding diff_minus
huffman@29405
   433
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   434
huffman@29405
   435
lemma mod_diff_cong:
huffman@29405
   436
  assumes "a mod c = a' mod c"
huffman@29405
   437
  assumes "b mod c = b' mod c"
huffman@29405
   438
  shows "(a - b) mod c = (a' - b') mod c"
huffman@29405
   439
  unfolding diff_minus using assms
huffman@29405
   440
  by (intro mod_add_cong mod_minus_cong)
huffman@29405
   441
nipkow@30180
   442
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   443
apply (case_tac "y = 0") apply simp
nipkow@30180
   444
apply (auto simp add: dvd_def)
nipkow@30180
   445
apply (subgoal_tac "-(y * k) = y * - k")
nipkow@30180
   446
 apply (erule ssubst)
nipkow@30180
   447
 apply (erule div_mult_self1_is_id)
nipkow@30180
   448
apply simp
nipkow@30180
   449
done
nipkow@30180
   450
nipkow@30180
   451
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   452
apply (case_tac "y = 0") apply simp
nipkow@30180
   453
apply (auto simp add: dvd_def)
nipkow@30180
   454
apply (subgoal_tac "y * k = -y * -k")
nipkow@30180
   455
 apply (erule ssubst)
nipkow@30180
   456
 apply (rule div_mult_self1_is_id)
nipkow@30180
   457
 apply simp
nipkow@30180
   458
apply simp
nipkow@30180
   459
done
nipkow@30180
   460
huffman@47159
   461
lemma div_minus_minus [simp]: "(-a) div (-b) = a div b"
huffman@47159
   462
  using div_mult_mult1 [of "- 1" a b]
huffman@47159
   463
  unfolding neg_equal_0_iff_equal by simp
huffman@47159
   464
huffman@47159
   465
lemma mod_minus_minus [simp]: "(-a) mod (-b) = - (a mod b)"
huffman@47159
   466
  using mod_mult_mult1 [of "- 1" a b] by simp
huffman@47159
   467
huffman@47159
   468
lemma div_minus_right: "a div (-b) = (-a) div b"
huffman@47159
   469
  using div_minus_minus [of "-a" b] by simp
huffman@47159
   470
huffman@47159
   471
lemma mod_minus_right: "a mod (-b) = - ((-a) mod b)"
huffman@47159
   472
  using mod_minus_minus [of "-a" b] by simp
huffman@47159
   473
huffman@47160
   474
lemma div_minus1_right [simp]: "a div (-1) = -a"
huffman@47160
   475
  using div_minus_right [of a 1] by simp
huffman@47160
   476
huffman@47160
   477
lemma mod_minus1_right [simp]: "a mod (-1) = 0"
huffman@47160
   478
  using mod_minus_right [of a 1] by simp
huffman@47160
   479
huffman@29405
   480
end
huffman@29405
   481
haftmann@25942
   482
haftmann@53067
   483
subsection {* Generic numeral division with a pragmatic type class *}
haftmann@53067
   484
haftmann@53067
   485
text {*
haftmann@53067
   486
  The following type class contains everything necessary to formulate
haftmann@53067
   487
  a division algorithm in ring structures with numerals, restricted
haftmann@53067
   488
  to its positive segments.  This is its primary motiviation, and it
haftmann@53067
   489
  could surely be formulated using a more fine-grained, more algebraic
haftmann@53067
   490
  and less technical class hierarchy.
haftmann@53067
   491
*}
haftmann@53067
   492
haftmann@53067
   493
haftmann@53067
   494
class semiring_numeral_div = linordered_semidom + minus + semiring_div +
haftmann@53067
   495
  assumes diff_invert_add1: "a + b = c \<Longrightarrow> a = c - b"
haftmann@53067
   496
    and le_add_diff_inverse2: "b \<le> a \<Longrightarrow> a - b + b = a"
haftmann@53067
   497
  assumes mult_div_cancel: "b * (a div b) = a - a mod b"
haftmann@53067
   498
    and div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
haftmann@53067
   499
    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
haftmann@53067
   500
    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
haftmann@53067
   501
    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
haftmann@53067
   502
    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
haftmann@53067
   503
    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
haftmann@53067
   504
    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@53067
   505
    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
haftmann@53067
   506
  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
haftmann@53067
   507
begin
haftmann@53067
   508
haftmann@53067
   509
lemma diff_zero [simp]:
haftmann@53067
   510
  "a - 0 = a"
haftmann@53067
   511
  by (rule diff_invert_add1 [symmetric]) simp
haftmann@53067
   512
haftmann@53067
   513
lemma parity:
haftmann@53067
   514
  "a mod 2 = 0 \<or> a mod 2 = 1"
haftmann@53067
   515
proof (rule ccontr)
haftmann@53067
   516
  assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
haftmann@53067
   517
  then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
haftmann@53067
   518
  have "0 < 2" by simp
haftmann@53067
   519
  with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
haftmann@53067
   520
  with `a mod 2 \<noteq> 0` have "0 < a mod 2" by simp
haftmann@53067
   521
  with discrete have "1 \<le> a mod 2" by simp
haftmann@53067
   522
  with `a mod 2 \<noteq> 1` have "1 < a mod 2" by simp
haftmann@53067
   523
  with discrete have "2 \<le> a mod 2" by simp
haftmann@53067
   524
  with `a mod 2 < 2` show False by simp
haftmann@53067
   525
qed
haftmann@53067
   526
haftmann@53067
   527
lemma divmod_digit_1:
haftmann@53067
   528
  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
haftmann@53067
   529
  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
haftmann@53067
   530
    and "a mod (2 * b) - b = a mod b" (is "?Q")
haftmann@53067
   531
proof -
haftmann@53067
   532
  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
haftmann@53067
   533
    by (auto intro: trans)
haftmann@53067
   534
  with `0 < b` have "0 < a div b" by (auto intro: div_positive)
haftmann@53067
   535
  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
haftmann@53067
   536
  with `0 < b` have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
haftmann@53067
   537
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   538
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   539
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   540
  from assms w_exhaust have "w = 1"
haftmann@53067
   541
    by (auto simp add: mod_w) (insert mod_less, auto)
haftmann@53067
   542
  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
haftmann@53067
   543
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   544
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067
   545
  with `w = 1` have div: "2 * (a div (2 * b)) = a div b - 1" by simp
haftmann@53067
   546
  then show ?P and ?Q
haftmann@53067
   547
    by (simp_all add: div mod diff_invert_add1 [symmetric] le_add_diff_inverse2)
haftmann@53067
   548
qed
haftmann@53067
   549
haftmann@53067
   550
lemma divmod_digit_0:
haftmann@53067
   551
  assumes "0 < b" and "a mod (2 * b) < b"
haftmann@53067
   552
  shows "2 * (a div (2 * b)) = a div b" (is "?P")
haftmann@53067
   553
    and "a mod (2 * b) = a mod b" (is "?Q")
haftmann@53067
   554
proof -
haftmann@53067
   555
  def w \<equiv> "a div b mod 2" with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
haftmann@53067
   556
  have mod_w: "a mod (2 * b) = a mod b + b * w"
haftmann@53067
   557
    by (simp add: w_def mod_mult2_eq ac_simps)
haftmann@53067
   558
  moreover have "b \<le> a mod b + b"
haftmann@53067
   559
  proof -
haftmann@53067
   560
    from `0 < b` pos_mod_sign have "0 \<le> a mod b" by blast
haftmann@53067
   561
    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
haftmann@53067
   562
    then show ?thesis by simp
haftmann@53067
   563
  qed
haftmann@53067
   564
  moreover note assms w_exhaust
haftmann@53067
   565
  ultimately have "w = 0" by auto
haftmann@53067
   566
  find_theorems "_ + ?b < _ + ?b"
haftmann@53067
   567
  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
haftmann@53067
   568
  have "2 * (a div (2 * b)) = a div b - w"
haftmann@53067
   569
    by (simp add: w_def div_mult2_eq mult_div_cancel ac_simps)
haftmann@53067
   570
  with `w = 0` have div: "2 * (a div (2 * b)) = a div b" by simp
haftmann@53067
   571
  then show ?P and ?Q
haftmann@53067
   572
    by (simp_all add: div mod)
haftmann@53067
   573
qed
haftmann@53067
   574
haftmann@53067
   575
definition divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
haftmann@53067
   576
where
haftmann@53067
   577
  "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@53067
   578
haftmann@53067
   579
lemma fst_divmod [simp]:
haftmann@53067
   580
  "fst (divmod m n) = numeral m div numeral n"
haftmann@53067
   581
  by (simp add: divmod_def)
haftmann@53067
   582
haftmann@53067
   583
lemma snd_divmod [simp]:
haftmann@53067
   584
  "snd (divmod m n) = numeral m mod numeral n"
haftmann@53067
   585
  by (simp add: divmod_def)
haftmann@53067
   586
haftmann@53067
   587
definition divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
haftmann@53067
   588
where
haftmann@53067
   589
  "divmod_step l qr = (let (q, r) = qr
haftmann@53067
   590
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@53067
   591
    else (2 * q, r))"
haftmann@53067
   592
haftmann@53067
   593
text {*
haftmann@53067
   594
  This is a formulation of one step (referring to one digit position)
haftmann@53067
   595
  in school-method division: compare the dividend at the current
haftmann@53067
   596
  digit position with the remained from previous division steps
haftmann@53067
   597
  and evaluate accordingly.
haftmann@53067
   598
*}
haftmann@53067
   599
haftmann@53067
   600
lemma divmod_step_eq [code]:
haftmann@53067
   601
  "divmod_step l (q, r) = (if numeral l \<le> r
haftmann@53067
   602
    then (2 * q + 1, r - numeral l) else (2 * q, r))"
haftmann@53067
   603
  by (simp add: divmod_step_def)
haftmann@53067
   604
haftmann@53067
   605
lemma divmod_step_simps [simp]:
haftmann@53067
   606
  "r < numeral l \<Longrightarrow> divmod_step l (q, r) = (2 * q, r)"
haftmann@53067
   607
  "numeral l \<le> r \<Longrightarrow> divmod_step l (q, r) = (2 * q + 1, r - numeral l)"
haftmann@53067
   608
  by (auto simp add: divmod_step_eq not_le)
haftmann@53067
   609
haftmann@53067
   610
text {*
haftmann@53067
   611
  This is a formulation of school-method division.
haftmann@53067
   612
  If the divisor is smaller than the dividend, terminate.
haftmann@53067
   613
  If not, shift the dividend to the right until termination
haftmann@53067
   614
  occurs and then reiterate single division steps in the
haftmann@53067
   615
  opposite direction.
haftmann@53067
   616
*}
haftmann@53067
   617
haftmann@53067
   618
lemma divmod_divmod_step [code]:
haftmann@53067
   619
  "divmod m n = (if m < n then (0, numeral m)
haftmann@53067
   620
    else divmod_step n (divmod m (Num.Bit0 n)))"
haftmann@53067
   621
proof (cases "m < n")
haftmann@53067
   622
  case True then have "numeral m < numeral n" by simp
haftmann@53067
   623
  then show ?thesis
haftmann@53067
   624
    by (simp add: prod_eq_iff div_less mod_less)
haftmann@53067
   625
next
haftmann@53067
   626
  case False
haftmann@53067
   627
  have "divmod m n =
haftmann@53067
   628
    divmod_step n (numeral m div (2 * numeral n),
haftmann@53067
   629
      numeral m mod (2 * numeral n))"
haftmann@53067
   630
  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
haftmann@53067
   631
    case True
haftmann@53067
   632
    with divmod_step_simps
haftmann@53067
   633
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   634
        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
haftmann@53067
   635
        by blast
haftmann@53067
   636
    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
haftmann@53067
   637
      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
haftmann@53067
   638
      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
haftmann@53067
   639
      by simp_all
haftmann@53067
   640
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   641
  next
haftmann@53067
   642
    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
haftmann@53067
   643
      by (simp add: not_le)
haftmann@53067
   644
    with divmod_step_simps
haftmann@53067
   645
      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
haftmann@53067
   646
        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
haftmann@53067
   647
        by blast
haftmann@53067
   648
    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
haftmann@53067
   649
      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
haftmann@53067
   650
      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
haftmann@53067
   651
      by (simp_all only: zero_less_numeral)
haftmann@53067
   652
    ultimately show ?thesis by (simp only: divmod_def)
haftmann@53067
   653
  qed
haftmann@53067
   654
  then have "divmod m n =
haftmann@53067
   655
    divmod_step n (numeral m div numeral (Num.Bit0 n),
haftmann@53067
   656
      numeral m mod numeral (Num.Bit0 n))"
haftmann@53067
   657
    by (simp only: numeral.simps distrib mult_1) 
haftmann@53067
   658
  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
haftmann@53067
   659
    by (simp add: divmod_def)
haftmann@53067
   660
  with False show ?thesis by simp
haftmann@53067
   661
qed
haftmann@53067
   662
haftmann@53067
   663
end
haftmann@53067
   664
haftmann@53067
   665
hide_fact (open) diff_invert_add1 le_add_diff_inverse2 diff_zero
haftmann@53067
   666
  -- {* restore simple accesses for more general variants of theorems *}
haftmann@53067
   667
haftmann@53067
   668
  
haftmann@26100
   669
subsection {* Division on @{typ nat} *}
haftmann@26100
   670
haftmann@26100
   671
text {*
haftmann@26100
   672
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   673
  of a characteristic relation with two input arguments
haftmann@26100
   674
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   675
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   676
*}
haftmann@26100
   677
haftmann@33340
   678
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   679
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   680
    m = fst qr * n + snd qr \<and>
haftmann@30923
   681
      (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
   682
haftmann@33340
   683
text {* @{const divmod_nat_rel} is total: *}
haftmann@26100
   684
haftmann@33340
   685
lemma divmod_nat_rel_ex:
haftmann@33340
   686
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   687
proof (cases "n = 0")
haftmann@30923
   688
  case True  with that show thesis
haftmann@33340
   689
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   690
next
haftmann@26100
   691
  case False
haftmann@26100
   692
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   693
  proof (induct m)
haftmann@26100
   694
    case 0 with `n \<noteq> 0`
haftmann@26100
   695
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   696
    then show ?case by blast
haftmann@26100
   697
  next
haftmann@26100
   698
    case (Suc m) then obtain q' r'
haftmann@26100
   699
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   700
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   701
      case True
haftmann@26100
   702
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   703
      with True show ?thesis by blast
haftmann@26100
   704
    next
haftmann@26100
   705
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   706
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   707
      ultimately have "n = Suc r'" by auto
haftmann@26100
   708
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   709
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   710
    qed
haftmann@26100
   711
  qed
haftmann@26100
   712
  with that show thesis
haftmann@33340
   713
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   714
qed
haftmann@26100
   715
haftmann@33340
   716
text {* @{const divmod_nat_rel} is injective: *}
haftmann@26100
   717
haftmann@33340
   718
lemma divmod_nat_rel_unique:
haftmann@33340
   719
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   720
    and "divmod_nat_rel m n qr'"
haftmann@30923
   721
  shows "qr = qr'"
haftmann@26100
   722
proof (cases "n = 0")
haftmann@26100
   723
  case True with assms show ?thesis
haftmann@30923
   724
    by (cases qr, cases qr')
haftmann@33340
   725
      (simp add: divmod_nat_rel_def)
haftmann@26100
   726
next
haftmann@26100
   727
  case False
haftmann@26100
   728
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   729
  apply (rule leI)
haftmann@26100
   730
  apply (subst less_iff_Suc_add)
haftmann@26100
   731
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   732
  done
haftmann@30923
   733
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
haftmann@33340
   734
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
haftmann@30923
   735
  moreover from this assms have "snd qr = snd qr'"
haftmann@33340
   736
    by (simp add: divmod_nat_rel_def)
haftmann@30923
   737
  ultimately show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   738
qed
haftmann@26100
   739
haftmann@26100
   740
text {*
haftmann@26100
   741
  We instantiate divisibility on the natural numbers by
haftmann@33340
   742
  means of @{const divmod_nat_rel}:
haftmann@26100
   743
*}
haftmann@25942
   744
haftmann@33340
   745
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@37767
   746
  "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   747
haftmann@33340
   748
lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   749
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   750
proof -
haftmann@33340
   751
  from divmod_nat_rel_ex
haftmann@33340
   752
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   753
  then show ?thesis
haftmann@33340
   754
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   755
qed
haftmann@30923
   756
huffman@47135
   757
lemma divmod_nat_unique:
haftmann@33340
   758
  assumes "divmod_nat_rel m n qr" 
haftmann@33340
   759
  shows "divmod_nat m n = qr"
haftmann@33340
   760
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   761
huffman@46551
   762
instantiation nat :: semiring_div
huffman@46551
   763
begin
huffman@46551
   764
haftmann@26100
   765
definition div_nat where
haftmann@33340
   766
  "m div n = fst (divmod_nat m n)"
haftmann@26100
   767
huffman@46551
   768
lemma fst_divmod_nat [simp]:
huffman@46551
   769
  "fst (divmod_nat m n) = m div n"
huffman@46551
   770
  by (simp add: div_nat_def)
huffman@46551
   771
haftmann@26100
   772
definition mod_nat where
haftmann@33340
   773
  "m mod n = snd (divmod_nat m n)"
haftmann@25571
   774
huffman@46551
   775
lemma snd_divmod_nat [simp]:
huffman@46551
   776
  "snd (divmod_nat m n) = m mod n"
huffman@46551
   777
  by (simp add: mod_nat_def)
huffman@46551
   778
haftmann@33340
   779
lemma divmod_nat_div_mod:
haftmann@33340
   780
  "divmod_nat m n = (m div n, m mod n)"
huffman@46551
   781
  by (simp add: prod_eq_iff)
haftmann@26100
   782
huffman@47135
   783
lemma div_nat_unique:
haftmann@33340
   784
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   785
  shows "m div n = q"
huffman@47135
   786
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   787
huffman@47135
   788
lemma mod_nat_unique:
haftmann@33340
   789
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   790
  shows "m mod n = r"
huffman@47135
   791
  using assms by (auto dest!: divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   792
haftmann@33340
   793
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
huffman@46551
   794
  using divmod_nat_rel_divmod_nat by (simp add: divmod_nat_div_mod)
paulson@14267
   795
huffman@47136
   796
lemma divmod_nat_zero: "divmod_nat m 0 = (0, m)"
huffman@47136
   797
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
huffman@47136
   798
huffman@47136
   799
lemma divmod_nat_zero_left: "divmod_nat 0 n = (0, 0)"
huffman@47136
   800
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   801
huffman@47137
   802
lemma divmod_nat_base: "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
huffman@47137
   803
  by (simp add: divmod_nat_unique divmod_nat_rel_def)
haftmann@25942
   804
haftmann@33340
   805
lemma divmod_nat_step:
haftmann@26100
   806
  assumes "0 < n" and "n \<le> m"
haftmann@33340
   807
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
huffman@47135
   808
proof (rule divmod_nat_unique)
huffman@47134
   809
  have "divmod_nat_rel (m - n) n ((m - n) div n, (m - n) mod n)"
huffman@47134
   810
    by (rule divmod_nat_rel)
huffman@47134
   811
  thus "divmod_nat_rel m n (Suc ((m - n) div n), (m - n) mod n)"
huffman@47134
   812
    unfolding divmod_nat_rel_def using assms by auto
haftmann@26100
   813
qed
haftmann@25942
   814
wenzelm@26300
   815
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   816
haftmann@26100
   817
lemma div_less [simp]:
haftmann@26100
   818
  fixes m n :: nat
haftmann@26100
   819
  assumes "m < n"
haftmann@26100
   820
  shows "m div n = 0"
huffman@46551
   821
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   822
haftmann@26100
   823
lemma le_div_geq:
haftmann@26100
   824
  fixes m n :: nat
haftmann@26100
   825
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   826
  shows "m div n = Suc ((m - n) div n)"
huffman@46551
   827
  using assms divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   828
haftmann@26100
   829
lemma mod_less [simp]:
haftmann@26100
   830
  fixes m n :: nat
haftmann@26100
   831
  assumes "m < n"
haftmann@26100
   832
  shows "m mod n = m"
huffman@46551
   833
  using assms divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   834
haftmann@26100
   835
lemma le_mod_geq:
haftmann@26100
   836
  fixes m n :: nat
haftmann@26100
   837
  assumes "n \<le> m"
haftmann@26100
   838
  shows "m mod n = (m - n) mod n"
huffman@46551
   839
  using assms divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   840
huffman@47136
   841
instance proof
huffman@47136
   842
  fix m n :: nat
huffman@47136
   843
  show "m div n * n + m mod n = m"
huffman@47136
   844
    using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
huffman@47136
   845
next
huffman@47136
   846
  fix m n q :: nat
huffman@47136
   847
  assume "n \<noteq> 0"
huffman@47136
   848
  then show "(q + m * n) div n = m + q div n"
huffman@47136
   849
    by (induct m) (simp_all add: le_div_geq)
huffman@47136
   850
next
huffman@47136
   851
  fix m n q :: nat
huffman@47136
   852
  assume "m \<noteq> 0"
huffman@47136
   853
  hence "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
huffman@47136
   854
    unfolding divmod_nat_rel_def
huffman@47136
   855
    by (auto split: split_if_asm, simp_all add: algebra_simps)
huffman@47136
   856
  moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
huffman@47136
   857
  ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
huffman@47136
   858
  thus "(m * n) div (m * q) = n div q" by (rule div_nat_unique)
huffman@47136
   859
next
huffman@47136
   860
  fix n :: nat show "n div 0 = 0"
haftmann@33340
   861
    by (simp add: div_nat_def divmod_nat_zero)
huffman@47136
   862
next
huffman@47136
   863
  fix n :: nat show "0 div n = 0"
huffman@47136
   864
    by (simp add: div_nat_def divmod_nat_zero_left)
haftmann@25942
   865
qed
haftmann@26100
   866
haftmann@25942
   867
end
paulson@14267
   868
haftmann@33361
   869
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@33361
   870
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
huffman@46551
   871
  by (simp add: prod_eq_iff prod_case_beta not_less le_div_geq le_mod_geq)
haftmann@33361
   872
haftmann@26100
   873
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   874
wenzelm@51299
   875
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299
   876
haftmann@30934
   877
ML {*
wenzelm@43594
   878
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
   879
(
haftmann@30934
   880
  val div_name = @{const_name div};
haftmann@30934
   881
  val mod_name = @{const_name mod};
haftmann@30934
   882
  val mk_binop = HOLogic.mk_binop;
huffman@48561
   883
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
   884
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
   885
  fun mk_sum [] = HOLogic.zero
huffman@48561
   886
    | mk_sum [t] = t
huffman@48561
   887
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
   888
  fun dest_sum tm =
huffman@48561
   889
    if HOLogic.is_zero tm then []
huffman@48561
   890
    else
huffman@48561
   891
      (case try HOLogic.dest_Suc tm of
huffman@48561
   892
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
   893
      | NONE =>
huffman@48561
   894
          (case try dest_plus tm of
huffman@48561
   895
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
   896
          | NONE => [tm]));
haftmann@25942
   897
haftmann@30934
   898
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
   899
haftmann@30934
   900
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@35050
   901
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
wenzelm@41550
   902
)
haftmann@25942
   903
*}
haftmann@25942
   904
wenzelm@43594
   905
simproc_setup cancel_div_mod_nat ("(m::nat) + n") = {* K Cancel_Div_Mod_Nat.proc *}
wenzelm@43594
   906
haftmann@26100
   907
haftmann@26100
   908
subsubsection {* Quotient *}
haftmann@26100
   909
haftmann@26100
   910
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   911
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   912
haftmann@26100
   913
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   914
by (simp add: div_geq)
haftmann@26100
   915
haftmann@26100
   916
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   917
by simp
haftmann@26100
   918
haftmann@26100
   919
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   920
by simp
haftmann@26100
   921
haftmann@53066
   922
lemma div_positive:
haftmann@53066
   923
  fixes m n :: nat
haftmann@53066
   924
  assumes "n > 0"
haftmann@53066
   925
  assumes "m \<ge> n"
haftmann@53066
   926
  shows "m div n > 0"
haftmann@53066
   927
proof -
haftmann@53066
   928
  from `m \<ge> n` obtain q where "m = n + q"
haftmann@53066
   929
    by (auto simp add: le_iff_add)
haftmann@53066
   930
  with `n > 0` show ?thesis by simp
haftmann@53066
   931
qed
haftmann@53066
   932
haftmann@25942
   933
haftmann@25942
   934
subsubsection {* Remainder *}
haftmann@25942
   935
haftmann@26100
   936
lemma mod_less_divisor [simp]:
haftmann@26100
   937
  fixes m n :: nat
haftmann@26100
   938
  assumes "n > 0"
haftmann@26100
   939
  shows "m mod n < (n::nat)"
haftmann@33340
   940
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
   941
haftmann@51173
   942
lemma mod_Suc_le_divisor [simp]:
haftmann@51173
   943
  "m mod Suc n \<le> n"
haftmann@51173
   944
  using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173
   945
haftmann@26100
   946
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   947
  fixes m n :: nat
haftmann@26100
   948
  shows "m mod n \<le> m"
haftmann@26100
   949
proof (rule add_leD2)
haftmann@26100
   950
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   951
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   952
qed
haftmann@26100
   953
haftmann@26100
   954
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
   955
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   956
haftmann@26100
   957
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
   958
by (simp add: le_mod_geq)
haftmann@26100
   959
paulson@14267
   960
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
   961
by (induct m) (simp_all add: mod_geq)
paulson@14267
   962
paulson@14267
   963
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   964
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
huffman@47138
   965
  using mod_div_equality2 [of n m] by arith
paulson@14267
   966
nipkow@15439
   967
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   968
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   969
  apply simp
wenzelm@22718
   970
  done
paulson@14267
   971
haftmann@26100
   972
subsubsection {* Quotient and Remainder *}
paulson@14267
   973
haftmann@33340
   974
lemma divmod_nat_rel_mult1_eq:
bulwahn@46552
   975
  "divmod_nat_rel b c (q, r)
haftmann@33340
   976
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
   977
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
   978
haftmann@30923
   979
lemma div_mult1_eq:
haftmann@30923
   980
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
huffman@47135
   981
by (blast intro: divmod_nat_rel_mult1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
   982
haftmann@33340
   983
lemma divmod_nat_rel_add1_eq:
bulwahn@46552
   984
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br)
haftmann@33340
   985
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
   986
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
   987
paulson@14267
   988
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   989
lemma div_add1_eq:
nipkow@25134
   990
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
huffman@47135
   991
by (blast intro: divmod_nat_rel_add1_eq [THEN div_nat_unique] divmod_nat_rel)
paulson@14267
   992
paulson@14267
   993
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   994
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   995
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   996
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   997
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   998
  done
paulson@10559
   999
haftmann@33340
  1000
lemma divmod_nat_rel_mult2_eq:
bulwahn@46552
  1001
  "divmod_nat_rel a b (q, r)
haftmann@33340
  1002
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@33340
  1003
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
  1004
paulson@14267
  1005
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
huffman@47135
  1006
by (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_nat_unique])
paulson@14267
  1007
paulson@14267
  1008
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
huffman@47135
  1009
by (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_nat_unique])
paulson@14267
  1010
paulson@14267
  1011
huffman@46551
  1012
subsubsection {* Further Facts about Quotient and Remainder *}
paulson@14267
  1013
paulson@14267
  1014
lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667
  1015
by (induct m) (simp_all add: div_geq)
paulson@14267
  1016
paulson@14267
  1017
(* Monotonicity of div in first argument *)
haftmann@30923
  1018
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
  1019
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
  1020
apply (case_tac "k=0", simp)
paulson@15251
  1021
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
  1022
apply (case_tac "n<k")
paulson@14267
  1023
(* 1  case n<k *)
paulson@14267
  1024
apply simp
paulson@14267
  1025
(* 2  case n >= k *)
paulson@14267
  1026
apply (case_tac "m<k")
paulson@14267
  1027
(* 2.1  case m<k *)
paulson@14267
  1028
apply simp
paulson@14267
  1029
(* 2.2  case m>=k *)
nipkow@15439
  1030
apply (simp add: div_geq diff_le_mono)
paulson@14267
  1031
done
paulson@14267
  1032
paulson@14267
  1033
(* Antimonotonicity of div in second argument *)
paulson@14267
  1034
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
  1035
apply (subgoal_tac "0<n")
wenzelm@22718
  1036
 prefer 2 apply simp
paulson@15251
  1037
apply (induct_tac k rule: nat_less_induct)
paulson@14267
  1038
apply (rename_tac "k")
paulson@14267
  1039
apply (case_tac "k<n", simp)
paulson@14267
  1040
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
  1041
 prefer 2 apply simp
paulson@14267
  1042
apply (simp add: div_geq)
paulson@15251
  1043
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
  1044
 prefer 2
paulson@14267
  1045
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
  1046
apply (rule le_trans, simp)
nipkow@15439
  1047
apply (simp)
paulson@14267
  1048
done
paulson@14267
  1049
paulson@14267
  1050
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
  1051
apply (case_tac "n=0", simp)
paulson@14267
  1052
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
  1053
apply (rule div_le_mono2)
paulson@14267
  1054
apply (simp_all (no_asm_simp))
paulson@14267
  1055
done
paulson@14267
  1056
wenzelm@22718
  1057
(* Similar for "less than" *)
huffman@47138
  1058
lemma div_less_dividend [simp]:
huffman@47138
  1059
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
  1060
apply (induct m rule: nat_less_induct)
paulson@14267
  1061
apply (rename_tac "m")
paulson@14267
  1062
apply (case_tac "m<n", simp)
paulson@14267
  1063
apply (subgoal_tac "0<n")
wenzelm@22718
  1064
 prefer 2 apply simp
paulson@14267
  1065
apply (simp add: div_geq)
paulson@14267
  1066
apply (case_tac "n<m")
paulson@15251
  1067
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
  1068
  apply (rule impI less_trans_Suc)+
paulson@14267
  1069
apply assumption
nipkow@15439
  1070
  apply (simp_all)
paulson@14267
  1071
done
paulson@14267
  1072
paulson@14267
  1073
text{*A fact for the mutilated chess board*}
paulson@14267
  1074
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
  1075
apply (case_tac "n=0", simp)
paulson@15251
  1076
apply (induct "m" rule: nat_less_induct)
paulson@14267
  1077
apply (case_tac "Suc (na) <n")
paulson@14267
  1078
(* case Suc(na) < n *)
paulson@14267
  1079
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
  1080
(* case n \<le> Suc(na) *)
paulson@16796
  1081
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
  1082
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
  1083
done
paulson@14267
  1084
paulson@14267
  1085
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
  1086
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
  1087
wenzelm@22718
  1088
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
  1089
paulson@14267
  1090
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
  1091
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
haftmann@27651
  1092
  apply (cut_tac a = m in mod_div_equality)
wenzelm@22718
  1093
  apply (simp only: add_ac)
wenzelm@22718
  1094
  apply (blast intro: sym)
wenzelm@22718
  1095
  done
paulson@14267
  1096
nipkow@13152
  1097
lemma split_div:
nipkow@13189
  1098
 "P(n div k :: nat) =
nipkow@13189
  1099
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
  1100
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1101
proof
nipkow@13189
  1102
  assume P: ?P
nipkow@13189
  1103
  show ?Q
nipkow@13189
  1104
  proof (cases)
nipkow@13189
  1105
    assume "k = 0"
haftmann@27651
  1106
    with P show ?Q by simp
nipkow@13189
  1107
  next
nipkow@13189
  1108
    assume not0: "k \<noteq> 0"
nipkow@13189
  1109
    thus ?Q
nipkow@13189
  1110
    proof (simp, intro allI impI)
nipkow@13189
  1111
      fix i j
nipkow@13189
  1112
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
  1113
      show "P i"
nipkow@13189
  1114
      proof (cases)
wenzelm@22718
  1115
        assume "i = 0"
wenzelm@22718
  1116
        with n j P show "P i" by simp
nipkow@13189
  1117
      next
wenzelm@22718
  1118
        assume "i \<noteq> 0"
wenzelm@22718
  1119
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
  1120
      qed
nipkow@13189
  1121
    qed
nipkow@13189
  1122
  qed
nipkow@13189
  1123
next
nipkow@13189
  1124
  assume Q: ?Q
nipkow@13189
  1125
  show ?P
nipkow@13189
  1126
  proof (cases)
nipkow@13189
  1127
    assume "k = 0"
haftmann@27651
  1128
    with Q show ?P by simp
nipkow@13189
  1129
  next
nipkow@13189
  1130
    assume not0: "k \<noteq> 0"
nipkow@13189
  1131
    with Q have R: ?R by simp
nipkow@13189
  1132
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1133
    show ?P by simp
nipkow@13189
  1134
  qed
nipkow@13189
  1135
qed
nipkow@13189
  1136
berghofe@13882
  1137
lemma split_div_lemma:
haftmann@26100
  1138
  assumes "0 < n"
haftmann@26100
  1139
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
  1140
proof
haftmann@26100
  1141
  assume ?rhs
haftmann@26100
  1142
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
  1143
  then have A: "n * q \<le> m" by simp
haftmann@26100
  1144
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
  1145
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
  1146
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
  1147
  with nq have "m < n + n * q" by simp
haftmann@26100
  1148
  then have B: "m < n * Suc q" by simp
haftmann@26100
  1149
  from A B show ?lhs ..
haftmann@26100
  1150
next
haftmann@26100
  1151
  assume P: ?lhs
haftmann@33340
  1152
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@33340
  1153
    unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
haftmann@33340
  1154
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
haftmann@30923
  1155
  have "(q, m - n * q) = (m div n, m mod n)" by auto
haftmann@30923
  1156
  then show ?rhs by simp
haftmann@26100
  1157
qed
berghofe@13882
  1158
berghofe@13882
  1159
theorem split_div':
berghofe@13882
  1160
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
  1161
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
  1162
  apply (case_tac "0 < n")
berghofe@13882
  1163
  apply (simp only: add: split_div_lemma)
haftmann@27651
  1164
  apply simp_all
berghofe@13882
  1165
  done
berghofe@13882
  1166
nipkow@13189
  1167
lemma split_mod:
nipkow@13189
  1168
 "P(n mod k :: nat) =
nipkow@13189
  1169
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
  1170
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1171
proof
nipkow@13189
  1172
  assume P: ?P
nipkow@13189
  1173
  show ?Q
nipkow@13189
  1174
  proof (cases)
nipkow@13189
  1175
    assume "k = 0"
haftmann@27651
  1176
    with P show ?Q by simp
nipkow@13189
  1177
  next
nipkow@13189
  1178
    assume not0: "k \<noteq> 0"
nipkow@13189
  1179
    thus ?Q
nipkow@13189
  1180
    proof (simp, intro allI impI)
nipkow@13189
  1181
      fix i j
nipkow@13189
  1182
      assume "n = k*i + j" "j < k"
nipkow@13189
  1183
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
  1184
    qed
nipkow@13189
  1185
  qed
nipkow@13189
  1186
next
nipkow@13189
  1187
  assume Q: ?Q
nipkow@13189
  1188
  show ?P
nipkow@13189
  1189
  proof (cases)
nipkow@13189
  1190
    assume "k = 0"
haftmann@27651
  1191
    with Q show ?P by simp
nipkow@13189
  1192
  next
nipkow@13189
  1193
    assume not0: "k \<noteq> 0"
nipkow@13189
  1194
    with Q have R: ?R by simp
nipkow@13189
  1195
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1196
    show ?P by simp
nipkow@13189
  1197
  qed
nipkow@13189
  1198
qed
nipkow@13189
  1199
berghofe@13882
  1200
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
huffman@47138
  1201
  using mod_div_equality [of m n] by arith
huffman@47138
  1202
huffman@47138
  1203
lemma div_mod_equality': "(m::nat) div n * n = m - m mod n"
huffman@47138
  1204
  using mod_div_equality [of m n] by arith
huffman@47138
  1205
(* FIXME: very similar to mult_div_cancel *)
haftmann@22800
  1206
noschinl@52398
  1207
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398
  1208
  apply rule
noschinl@52398
  1209
  apply (cases "b = 0")
noschinl@52398
  1210
  apply simp_all
noschinl@52398
  1211
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398
  1212
  done
noschinl@52398
  1213
haftmann@22800
  1214
huffman@46551
  1215
subsubsection {* An ``induction'' law for modulus arithmetic. *}
paulson@14640
  1216
paulson@14640
  1217
lemma mod_induct_0:
paulson@14640
  1218
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1219
  and base: "P i" and i: "i<p"
paulson@14640
  1220
  shows "P 0"
paulson@14640
  1221
proof (rule ccontr)
paulson@14640
  1222
  assume contra: "\<not>(P 0)"
paulson@14640
  1223
  from i have p: "0<p" by simp
paulson@14640
  1224
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1225
  proof
paulson@14640
  1226
    fix k
paulson@14640
  1227
    show "?A k"
paulson@14640
  1228
    proof (induct k)
paulson@14640
  1229
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1230
    next
paulson@14640
  1231
      fix n
paulson@14640
  1232
      assume ih: "?A n"
paulson@14640
  1233
      show "?A (Suc n)"
paulson@14640
  1234
      proof (clarsimp)
wenzelm@22718
  1235
        assume y: "P (p - Suc n)"
wenzelm@22718
  1236
        have n: "Suc n < p"
wenzelm@22718
  1237
        proof (rule ccontr)
wenzelm@22718
  1238
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1239
          hence "p - Suc n = 0"
wenzelm@22718
  1240
            by simp
wenzelm@22718
  1241
          with y contra show "False"
wenzelm@22718
  1242
            by simp
wenzelm@22718
  1243
        qed
wenzelm@22718
  1244
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1245
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1246
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1247
          by blast
wenzelm@22718
  1248
        show "False"
wenzelm@22718
  1249
        proof (cases "n=0")
wenzelm@22718
  1250
          case True
wenzelm@22718
  1251
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1252
        next
wenzelm@22718
  1253
          case False
wenzelm@22718
  1254
          with p have "p-n < p" by arith
wenzelm@22718
  1255
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1256
        qed
paulson@14640
  1257
      qed
paulson@14640
  1258
    qed
paulson@14640
  1259
  qed
paulson@14640
  1260
  moreover
paulson@14640
  1261
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1262
    by (blast dest: less_imp_add_positive)
paulson@14640
  1263
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1264
  moreover
paulson@14640
  1265
  note base
paulson@14640
  1266
  ultimately
paulson@14640
  1267
  show "False" by blast
paulson@14640
  1268
qed
paulson@14640
  1269
paulson@14640
  1270
lemma mod_induct:
paulson@14640
  1271
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1272
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1273
  shows "P j"
paulson@14640
  1274
proof -
paulson@14640
  1275
  have "\<forall>j<p. P j"
paulson@14640
  1276
  proof
paulson@14640
  1277
    fix j
paulson@14640
  1278
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1279
    proof (induct j)
paulson@14640
  1280
      from step base i show "?A 0"
wenzelm@22718
  1281
        by (auto elim: mod_induct_0)
paulson@14640
  1282
    next
paulson@14640
  1283
      fix k
paulson@14640
  1284
      assume ih: "?A k"
paulson@14640
  1285
      show "?A (Suc k)"
paulson@14640
  1286
      proof
wenzelm@22718
  1287
        assume suc: "Suc k < p"
wenzelm@22718
  1288
        hence k: "k<p" by simp
wenzelm@22718
  1289
        with ih have "P k" ..
wenzelm@22718
  1290
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1291
          by blast
wenzelm@22718
  1292
        moreover
wenzelm@22718
  1293
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1294
          by simp
wenzelm@22718
  1295
        ultimately
wenzelm@22718
  1296
        show "P (Suc k)" by simp
paulson@14640
  1297
      qed
paulson@14640
  1298
    qed
paulson@14640
  1299
  qed
paulson@14640
  1300
  with j show ?thesis by blast
paulson@14640
  1301
qed
paulson@14640
  1302
haftmann@33296
  1303
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1304
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1305
huffman@47138
  1306
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1307
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1308
haftmann@33296
  1309
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1310
by (simp add: mult_2 [symmetric])
haftmann@33296
  1311
haftmann@33296
  1312
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1313
proof -
boehmes@35815
  1314
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1315
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1316
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1317
  then show ?thesis by auto
haftmann@33296
  1318
qed
haftmann@33296
  1319
haftmann@33296
  1320
text{*These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1321
    at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@33296
  1322
    We already have some rules to simplify operands smaller than 3.*}
haftmann@33296
  1323
haftmann@33296
  1324
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1325
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1326
haftmann@33296
  1327
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1328
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1329
haftmann@33296
  1330
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1331
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1332
haftmann@33296
  1333
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1334
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1335
huffman@47108
  1336
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1337
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1338
haftmann@33361
  1339
haftmann@33361
  1340
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
haftmann@33361
  1341
apply (induct "m")
haftmann@33361
  1342
apply (simp_all add: mod_Suc)
haftmann@33361
  1343
done
haftmann@33361
  1344
huffman@47108
  1345
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1346
huffman@47138
  1347
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1348
by (simp add: div_le_mono)
haftmann@33361
  1349
haftmann@33361
  1350
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1351
by (cases n) simp_all
haftmann@33361
  1352
boehmes@35815
  1353
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1354
proof -
boehmes@35815
  1355
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
boehmes@35815
  1356
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
boehmes@35815
  1357
qed
haftmann@33361
  1358
haftmann@33361
  1359
  (* Potential use of algebra : Equality modulo n*)
haftmann@33361
  1360
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
haftmann@33361
  1361
by (simp add: mult_ac add_ac)
haftmann@33361
  1362
haftmann@33361
  1363
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1364
proof -
haftmann@33361
  1365
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
haftmann@33361
  1366
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
haftmann@33361
  1367
  finally show ?thesis .
haftmann@33361
  1368
qed
haftmann@33361
  1369
haftmann@33361
  1370
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
haftmann@33361
  1371
apply (subst mod_Suc [of m]) 
haftmann@33361
  1372
apply (subst mod_Suc [of "m mod n"], simp) 
haftmann@33361
  1373
done
haftmann@33361
  1374
huffman@47108
  1375
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1376
  fixes n :: nat
huffman@47108
  1377
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
huffman@47108
  1378
  by simp
huffman@47108
  1379
haftmann@53067
  1380
instance nat :: semiring_numeral_div
haftmann@53067
  1381
  by intro_classes (auto intro: div_positive simp add: mult_div_cancel mod_mult2_eq div_mult2_eq)
haftmann@53067
  1382
haftmann@33361
  1383
haftmann@33361
  1384
subsection {* Division on @{typ int} *}
haftmann@33361
  1385
haftmann@33361
  1386
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
haftmann@33361
  1387
    --{*definition of quotient and remainder*}
huffman@47139
  1388
  "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
huffman@47139
  1389
    (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
  1390
haftmann@53067
  1391
text {*
haftmann@53067
  1392
  The following algorithmic devlopment actually echos what has already
haftmann@53067
  1393
  been developed in class @{class semiring_numeral_div}.  In the long
haftmann@53067
  1394
  run it seems better to derive division on @{typ int} just from
haftmann@53067
  1395
  division on @{typ nat} and instantiate @{class semiring_numeral_div}
haftmann@53067
  1396
  accordingly.
haftmann@53067
  1397
*}
haftmann@53067
  1398
haftmann@33361
  1399
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1400
    --{*for the division algorithm*}
huffman@47108
  1401
    "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@33361
  1402
                         else (2 * q, r))"
haftmann@33361
  1403
haftmann@33361
  1404
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@33361
  1405
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1406
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@33361
  1407
     else adjust b (posDivAlg a (2 * b)))"
haftmann@33361
  1408
by auto
haftmann@33361
  1409
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
haftmann@33361
  1410
  (auto simp add: mult_2)
haftmann@33361
  1411
haftmann@33361
  1412
text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@33361
  1413
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1414
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@33361
  1415
     else adjust b (negDivAlg a (2 * b)))"
haftmann@33361
  1416
by auto
haftmann@33361
  1417
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
haftmann@33361
  1418
  (auto simp add: mult_2)
haftmann@33361
  1419
haftmann@33361
  1420
text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@33361
  1421
haftmann@33361
  1422
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1423
    --{*The full division algorithm considers all possible signs for a, b
haftmann@33361
  1424
       including the special case @{text "a=0, b<0"} because 
haftmann@33361
  1425
       @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@33361
  1426
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@33361
  1427
                  else if a = 0 then (0, 0)
huffman@46560
  1428
                       else apsnd uminus (negDivAlg (-a) (-b))
haftmann@33361
  1429
               else 
haftmann@33361
  1430
                  if 0 < b then negDivAlg a b
huffman@46560
  1431
                  else apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1432
haftmann@33361
  1433
instantiation int :: Divides.div
haftmann@33361
  1434
begin
haftmann@33361
  1435
huffman@46551
  1436
definition div_int where
haftmann@33361
  1437
  "a div b = fst (divmod_int a b)"
haftmann@33361
  1438
huffman@46551
  1439
lemma fst_divmod_int [simp]:
huffman@46551
  1440
  "fst (divmod_int a b) = a div b"
huffman@46551
  1441
  by (simp add: div_int_def)
huffman@46551
  1442
huffman@46551
  1443
definition mod_int where
huffman@46560
  1444
  "a mod b = snd (divmod_int a b)"
haftmann@33361
  1445
huffman@46551
  1446
lemma snd_divmod_int [simp]:
huffman@46551
  1447
  "snd (divmod_int a b) = a mod b"
huffman@46551
  1448
  by (simp add: mod_int_def)
huffman@46551
  1449
haftmann@33361
  1450
instance ..
haftmann@33361
  1451
paulson@3366
  1452
end
haftmann@33361
  1453
haftmann@33361
  1454
lemma divmod_int_mod_div:
haftmann@33361
  1455
  "divmod_int p q = (p div q, p mod q)"
huffman@46551
  1456
  by (simp add: prod_eq_iff)
haftmann@33361
  1457
haftmann@33361
  1458
text{*
haftmann@33361
  1459
Here is the division algorithm in ML:
haftmann@33361
  1460
haftmann@33361
  1461
\begin{verbatim}
haftmann@33361
  1462
    fun posDivAlg (a,b) =
haftmann@33361
  1463
      if a<b then (0,a)
haftmann@33361
  1464
      else let val (q,r) = posDivAlg(a, 2*b)
haftmann@33361
  1465
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1466
           end
haftmann@33361
  1467
haftmann@33361
  1468
    fun negDivAlg (a,b) =
haftmann@33361
  1469
      if 0\<le>a+b then (~1,a+b)
haftmann@33361
  1470
      else let val (q,r) = negDivAlg(a, 2*b)
haftmann@33361
  1471
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1472
           end;
haftmann@33361
  1473
haftmann@33361
  1474
    fun negateSnd (q,r:int) = (q,~r);
haftmann@33361
  1475
haftmann@33361
  1476
    fun divmod (a,b) = if 0\<le>a then 
haftmann@33361
  1477
                          if b>0 then posDivAlg (a,b) 
haftmann@33361
  1478
                           else if a=0 then (0,0)
haftmann@33361
  1479
                                else negateSnd (negDivAlg (~a,~b))
haftmann@33361
  1480
                       else 
haftmann@33361
  1481
                          if 0<b then negDivAlg (a,b)
haftmann@33361
  1482
                          else        negateSnd (posDivAlg (~a,~b));
haftmann@33361
  1483
\end{verbatim}
haftmann@33361
  1484
*}
haftmann@33361
  1485
haftmann@33361
  1486
huffman@46551
  1487
subsubsection {* Uniqueness and Monotonicity of Quotients and Remainders *}
haftmann@33361
  1488
haftmann@33361
  1489
lemma unique_quotient_lemma:
haftmann@33361
  1490
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
haftmann@33361
  1491
      ==> q' \<le> (q::int)"
haftmann@33361
  1492
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1493
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1494
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1495
apply (erule_tac [2] order_le_less_trans)
webertj@49962
  1496
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1497
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
  1498
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1499
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1500
done
haftmann@33361
  1501
haftmann@33361
  1502
lemma unique_quotient_lemma_neg:
haftmann@33361
  1503
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
haftmann@33361
  1504
      ==> q \<le> (q'::int)"
haftmann@33361
  1505
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
haftmann@33361
  1506
    auto)
haftmann@33361
  1507
haftmann@33361
  1508
lemma unique_quotient:
bulwahn@46552
  1509
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1510
      ==> q = q'"
haftmann@33361
  1511
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361
  1512
apply (blast intro: order_antisym
haftmann@33361
  1513
             dest: order_eq_refl [THEN unique_quotient_lemma] 
haftmann@33361
  1514
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1515
done
haftmann@33361
  1516
haftmann@33361
  1517
haftmann@33361
  1518
lemma unique_remainder:
bulwahn@46552
  1519
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r') |]  
haftmann@33361
  1520
      ==> r = r'"
haftmann@33361
  1521
apply (subgoal_tac "q = q'")
haftmann@33361
  1522
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1523
apply (blast intro: unique_quotient)
haftmann@33361
  1524
done
haftmann@33361
  1525
haftmann@33361
  1526
huffman@46551
  1527
subsubsection {* Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends *}
haftmann@33361
  1528
haftmann@33361
  1529
text{*And positive divisors*}
haftmann@33361
  1530
haftmann@33361
  1531
lemma adjust_eq [simp]:
huffman@47108
  1532
     "adjust b (q, r) = 
huffman@47108
  1533
      (let diff = r - b in  
huffman@47108
  1534
        if 0 \<le> diff then (2 * q + 1, diff)   
haftmann@33361
  1535
                     else (2*q, r))"
huffman@47108
  1536
  by (simp add: Let_def adjust_def)
haftmann@33361
  1537
haftmann@33361
  1538
declare posDivAlg.simps [simp del]
haftmann@33361
  1539
haftmann@33361
  1540
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1541
lemma posDivAlg_eqn:
haftmann@33361
  1542
     "0 < b ==>  
haftmann@33361
  1543
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
haftmann@33361
  1544
by (rule posDivAlg.simps [THEN trans], simp)
haftmann@33361
  1545
haftmann@33361
  1546
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
haftmann@33361
  1547
theorem posDivAlg_correct:
haftmann@33361
  1548
  assumes "0 \<le> a" and "0 < b"
haftmann@33361
  1549
  shows "divmod_int_rel a b (posDivAlg a b)"
wenzelm@41550
  1550
  using assms
wenzelm@41550
  1551
  apply (induct a b rule: posDivAlg.induct)
wenzelm@41550
  1552
  apply auto
wenzelm@41550
  1553
  apply (simp add: divmod_int_rel_def)
webertj@49962
  1554
  apply (subst posDivAlg_eqn, simp add: distrib_left)
wenzelm@41550
  1555
  apply (case_tac "a < b")
wenzelm@41550
  1556
  apply simp_all
wenzelm@41550
  1557
  apply (erule splitE)
webertj@49962
  1558
  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)
wenzelm@41550
  1559
  done
haftmann@33361
  1560
haftmann@33361
  1561
huffman@46551
  1562
subsubsection {* Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends *}
haftmann@33361
  1563
haftmann@33361
  1564
text{*And positive divisors*}
haftmann@33361
  1565
haftmann@33361
  1566
declare negDivAlg.simps [simp del]
haftmann@33361
  1567
haftmann@33361
  1568
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1569
lemma negDivAlg_eqn:
haftmann@33361
  1570
     "0 < b ==>  
haftmann@33361
  1571
      negDivAlg a b =       
haftmann@33361
  1572
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
haftmann@33361
  1573
by (rule negDivAlg.simps [THEN trans], simp)
haftmann@33361
  1574
haftmann@33361
  1575
(*Correctness of negDivAlg: it computes quotients correctly
haftmann@33361
  1576
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
haftmann@33361
  1577
lemma negDivAlg_correct:
haftmann@33361
  1578
  assumes "a < 0" and "b > 0"
haftmann@33361
  1579
  shows "divmod_int_rel a b (negDivAlg a b)"
wenzelm@41550
  1580
  using assms
wenzelm@41550
  1581
  apply (induct a b rule: negDivAlg.induct)
wenzelm@41550
  1582
  apply (auto simp add: linorder_not_le)
wenzelm@41550
  1583
  apply (simp add: divmod_int_rel_def)
wenzelm@41550
  1584
  apply (subst negDivAlg_eqn, assumption)
wenzelm@41550
  1585
  apply (case_tac "a + b < (0\<Colon>int)")
wenzelm@41550
  1586
  apply simp_all
wenzelm@41550
  1587
  apply (erule splitE)
webertj@49962
  1588
  apply (auto simp add: distrib_left Let_def mult_ac mult_2_right)
wenzelm@41550
  1589
  done
haftmann@33361
  1590
haftmann@33361
  1591
huffman@46551
  1592
subsubsection {* Existence Shown by Proving the Division Algorithm to be Correct *}
haftmann@33361
  1593
haftmann@33361
  1594
(*the case a=0*)
huffman@47139
  1595
lemma divmod_int_rel_0: "divmod_int_rel 0 b (0, 0)"
haftmann@33361
  1596
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1597
haftmann@33361
  1598
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
haftmann@33361
  1599
by (subst posDivAlg.simps, auto)
haftmann@33361
  1600
huffman@47139
  1601
lemma posDivAlg_0_right [simp]: "posDivAlg a 0 = (0, a)"
huffman@47139
  1602
by (subst posDivAlg.simps, auto)
huffman@47139
  1603
haftmann@33361
  1604
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
haftmann@33361
  1605
by (subst negDivAlg.simps, auto)
haftmann@33361
  1606
huffman@46560
  1607
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (apsnd uminus qr)"
huffman@47139
  1608
by (auto simp add: divmod_int_rel_def)
huffman@47139
  1609
huffman@47139
  1610
lemma divmod_int_correct: "divmod_int_rel a b (divmod_int a b)"
huffman@47139
  1611
apply (cases "b = 0", simp add: divmod_int_def divmod_int_rel_def)
haftmann@33361
  1612
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
haftmann@33361
  1613
                    posDivAlg_correct negDivAlg_correct)
haftmann@33361
  1614
huffman@47141
  1615
lemma divmod_int_unique:
huffman@47141
  1616
  assumes "divmod_int_rel a b qr" 
huffman@47141
  1617
  shows "divmod_int a b = qr"
huffman@47141
  1618
  using assms divmod_int_correct [of a b]
huffman@47141
  1619
  using unique_quotient [of a b] unique_remainder [of a b]
huffman@47141
  1620
  by (metis pair_collapse)
huffman@47141
  1621
huffman@47141
  1622
lemma divmod_int_rel_div_mod: "divmod_int_rel a b (a div b, a mod b)"
huffman@47141
  1623
  using divmod_int_correct by (simp add: divmod_int_mod_div)
huffman@47141
  1624
huffman@47141
  1625
lemma div_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a div b = q"
huffman@47141
  1626
  by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
huffman@47141
  1627
huffman@47141
  1628
lemma mod_int_unique: "divmod_int_rel a b (q, r) \<Longrightarrow> a mod b = r"
huffman@47141
  1629
  by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
huffman@47141
  1630
huffman@47141
  1631
instance int :: ring_div
huffman@47141
  1632
proof
huffman@47141
  1633
  fix a b :: int
huffman@47141
  1634
  show "a div b * b + a mod b = a"
huffman@47141
  1635
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1636
    unfolding divmod_int_rel_def by (simp add: mult_commute)
huffman@47141
  1637
next
huffman@47141
  1638
  fix a b c :: int
huffman@47141
  1639
  assume "b \<noteq> 0"
huffman@47141
  1640
  hence "divmod_int_rel (a + c * b) b (c + a div b, a mod b)"
huffman@47141
  1641
    using divmod_int_rel_div_mod [of a b]
huffman@47141
  1642
    unfolding divmod_int_rel_def by (auto simp: algebra_simps)
huffman@47141
  1643
  thus "(a + c * b) div b = c + a div b"
huffman@47141
  1644
    by (rule div_int_unique)
huffman@47141
  1645
next
huffman@47141
  1646
  fix a b c :: int
huffman@47141
  1647
  assume "c \<noteq> 0"
huffman@47141
  1648
  hence "\<And>q r. divmod_int_rel a b (q, r)
huffman@47141
  1649
    \<Longrightarrow> divmod_int_rel (c * a) (c * b) (q, c * r)"
huffman@47141
  1650
    unfolding divmod_int_rel_def
huffman@47141
  1651
    by - (rule linorder_cases [of 0 b], auto simp: algebra_simps
huffman@47141
  1652
      mult_less_0_iff zero_less_mult_iff mult_strict_right_mono
huffman@47141
  1653
      mult_strict_right_mono_neg zero_le_mult_iff mult_le_0_iff)
huffman@47141
  1654
  hence "divmod_int_rel (c * a) (c * b) (a div b, c * (a mod b))"
huffman@47141
  1655
    using divmod_int_rel_div_mod [of a b] .
huffman@47141
  1656
  thus "(c * a) div (c * b) = a div b"
huffman@47141
  1657
    by (rule div_int_unique)
huffman@47141
  1658
next
huffman@47141
  1659
  fix a :: int show "a div 0 = 0"
huffman@47141
  1660
    by (rule div_int_unique, simp add: divmod_int_rel_def)
huffman@47141
  1661
next
huffman@47141
  1662
  fix a :: int show "0 div a = 0"
huffman@47141
  1663
    by (rule div_int_unique, auto simp add: divmod_int_rel_def)
huffman@47141
  1664
qed
huffman@47141
  1665
haftmann@33361
  1666
text{*Basic laws about division and remainder*}
haftmann@33361
  1667
haftmann@33361
  1668
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
huffman@47141
  1669
  by (fact mod_div_equality2 [symmetric])
haftmann@33361
  1670
haftmann@33361
  1671
text {* Tool setup *}
haftmann@33361
  1672
huffman@47108
  1673
(* FIXME: Theorem list add_0s doesn't exist, because Numeral0 has gone. *)
huffman@47108
  1674
lemmas add_0s = add_0_left add_0_right
huffman@47108
  1675
haftmann@33361
  1676
ML {*
wenzelm@43594
  1677
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
wenzelm@41550
  1678
(
haftmann@33361
  1679
  val div_name = @{const_name div};
haftmann@33361
  1680
  val mod_name = @{const_name mod};
haftmann@33361
  1681
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1682
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1683
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1684
huffman@47165
  1685
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
haftmann@33361
  1686
haftmann@33361
  1687
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
haftmann@33361
  1688
    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
wenzelm@41550
  1689
)
haftmann@33361
  1690
*}
haftmann@33361
  1691
wenzelm@43594
  1692
simproc_setup cancel_div_mod_int ("(k::int) + l") = {* K Cancel_Div_Mod_Int.proc *}
wenzelm@43594
  1693
huffman@47141
  1694
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
huffman@47141
  1695
  using divmod_int_correct [of a b]
huffman@47141
  1696
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1697
wenzelm@45607
  1698
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1699
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1700
huffman@47141
  1701
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
huffman@47141
  1702
  using divmod_int_correct [of a b]
huffman@47141
  1703
  by (auto simp add: divmod_int_rel_def prod_eq_iff)
haftmann@33361
  1704
wenzelm@45607
  1705
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1706
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1707
haftmann@33361
  1708
huffman@46551
  1709
subsubsection {* General Properties of div and mod *}
haftmann@33361
  1710
haftmann@33361
  1711
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1712
apply (rule div_int_unique)
haftmann@33361
  1713
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1714
done
haftmann@33361
  1715
haftmann@33361
  1716
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1717
apply (rule div_int_unique)
haftmann@33361
  1718
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1719
done
haftmann@33361
  1720
haftmann@33361
  1721
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1722
apply (rule div_int_unique)
haftmann@33361
  1723
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1724
done
haftmann@33361
  1725
haftmann@33361
  1726
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1727
haftmann@33361
  1728
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1729
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1730
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1731
done
haftmann@33361
  1732
haftmann@33361
  1733
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1734
apply (rule_tac q = 0 in mod_int_unique)
haftmann@33361
  1735
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1736
done
haftmann@33361
  1737
haftmann@33361
  1738
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1739
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@33361
  1740
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1741
done
haftmann@33361
  1742
haftmann@33361
  1743
text{*There is no @{text mod_neg_pos_trivial}.*}
haftmann@33361
  1744
haftmann@33361
  1745
huffman@46551
  1746
subsubsection {* Laws for div and mod with Unary Minus *}
haftmann@33361
  1747
haftmann@33361
  1748
lemma zminus1_lemma:
huffman@47139
  1749
     "divmod_int_rel a b (q, r) ==> b \<noteq> 0
haftmann@33361
  1750
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@33361
  1751
                          if r=0 then 0 else b-r)"
haftmann@33361
  1752
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1753
haftmann@33361
  1754
haftmann@33361
  1755
lemma zdiv_zminus1_eq_if:
haftmann@33361
  1756
     "b \<noteq> (0::int)  
haftmann@33361
  1757
      ==> (-a) div b =  
haftmann@33361
  1758
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47140
  1759
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1760
haftmann@33361
  1761
lemma zmod_zminus1_eq_if:
haftmann@33361
  1762
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1763
apply (case_tac "b = 0", simp)
huffman@47140
  1764
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1765
done
haftmann@33361
  1766
haftmann@33361
  1767
lemma zmod_zminus1_not_zero:
haftmann@33361
  1768
  fixes k l :: int
haftmann@33361
  1769
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1770
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1771
haftmann@33361
  1772
lemma zdiv_zminus2_eq_if:
haftmann@33361
  1773
     "b \<noteq> (0::int)  
haftmann@33361
  1774
      ==> a div (-b) =  
haftmann@33361
  1775
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1776
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1777
haftmann@33361
  1778
lemma zmod_zminus2_eq_if:
haftmann@33361
  1779
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1780
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1781
haftmann@33361
  1782
lemma zmod_zminus2_not_zero:
haftmann@33361
  1783
  fixes k l :: int
haftmann@33361
  1784
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1785
  unfolding zmod_zminus2_eq_if by auto 
haftmann@33361
  1786
haftmann@33361
  1787
huffman@46551
  1788
subsubsection {* Computation of Division and Remainder *}
haftmann@33361
  1789
haftmann@33361
  1790
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@33361
  1791
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1792
haftmann@33361
  1793
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@33361
  1794
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1795
haftmann@33361
  1796
text{*a positive, b positive *}
haftmann@33361
  1797
haftmann@33361
  1798
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@33361
  1799
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1800
haftmann@33361
  1801
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@33361
  1802
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1803
haftmann@33361
  1804
text{*a negative, b positive *}
haftmann@33361
  1805
haftmann@33361
  1806
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@33361
  1807
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1808
haftmann@33361
  1809
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@33361
  1810
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1811
haftmann@33361
  1812
text{*a positive, b negative *}
haftmann@33361
  1813
haftmann@33361
  1814
lemma div_pos_neg:
huffman@46560
  1815
     "[| 0 < a;  b < 0 |] ==> a div b = fst (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1816
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1817
haftmann@33361
  1818
lemma mod_pos_neg:
huffman@46560
  1819
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (apsnd uminus (negDivAlg (-a) (-b)))"
haftmann@33361
  1820
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1821
haftmann@33361
  1822
text{*a negative, b negative *}
haftmann@33361
  1823
haftmann@33361
  1824
lemma div_neg_neg:
huffman@46560
  1825
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1826
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1827
haftmann@33361
  1828
lemma mod_neg_neg:
huffman@46560
  1829
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (apsnd uminus (posDivAlg (-a) (-b)))"
haftmann@33361
  1830
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1831
haftmann@33361
  1832
text {*Simplify expresions in which div and mod combine numerical constants*}
haftmann@33361
  1833
huffman@45530
  1834
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1835
  by (rule div_int_unique [of a b q r]) (simp add: divmod_int_rel_def)
huffman@45530
  1836
huffman@45530
  1837
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
huffman@47140
  1838
  by (rule div_int_unique [of a b q r],
bulwahn@46552
  1839
    simp add: divmod_int_rel_def)
huffman@45530
  1840
huffman@45530
  1841
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1842
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1843
    simp add: divmod_int_rel_def)
huffman@45530
  1844
huffman@45530
  1845
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
huffman@47140
  1846
  by (rule mod_int_unique [of a b q r],
bulwahn@46552
  1847
    simp add: divmod_int_rel_def)
huffman@45530
  1848
haftmann@33361
  1849
(* simprocs adapted from HOL/ex/Binary.thy *)
haftmann@33361
  1850
ML {*
haftmann@33361
  1851
local
huffman@45530
  1852
  val mk_number = HOLogic.mk_number HOLogic.intT
huffman@45530
  1853
  val plus = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1854
  val times = @{term "times :: int \<Rightarrow> int \<Rightarrow> int"}
huffman@45530
  1855
  val zero = @{term "0 :: int"}
huffman@45530
  1856
  val less = @{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1857
  val le = @{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"}
huffman@45530
  1858
  val simps = @{thms arith_simps} @ @{thms rel_simps} @
huffman@47108
  1859
    map (fn th => th RS sym) [@{thm numeral_1_eq_1}]
huffman@45530
  1860
  fun prove ctxt goal = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop goal)
wenzelm@51717
  1861
    (K (ALLGOALS (full_simp_tac (put_simpset HOL_basic_ss ctxt addsimps simps))));
wenzelm@51717
  1862
  fun binary_proc proc ctxt ct =
haftmann@33361
  1863
    (case Thm.term_of ct of
haftmann@33361
  1864
      _ $ t $ u =>
haftmann@33361
  1865
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
wenzelm@51717
  1866
        SOME args => proc ctxt args
haftmann@33361
  1867
      | NONE => NONE)
haftmann@33361
  1868
    | _ => NONE);
haftmann@33361
  1869
in
huffman@45530
  1870
  fun divmod_proc posrule negrule =
huffman@45530
  1871
    binary_proc (fn ctxt => fn ((a, t), (b, u)) =>
huffman@45530
  1872
      if b = 0 then NONE else let
huffman@45530
  1873
        val (q, r) = pairself mk_number (Integer.div_mod a b)
huffman@45530
  1874
        val goal1 = HOLogic.mk_eq (t, plus $ (times $ u $ q) $ r)
huffman@45530
  1875
        val (goal2, goal3, rule) = if b > 0
huffman@45530
  1876
          then (le $ zero $ r, less $ r $ u, posrule RS eq_reflection)
huffman@45530
  1877
          else (le $ r $ zero, less $ u $ r, negrule RS eq_reflection)
huffman@45530
  1878
      in SOME (rule OF map (prove ctxt) [goal1, goal2, goal3]) end)
haftmann@33361
  1879
end
haftmann@33361
  1880
*}
haftmann@33361
  1881
huffman@47108
  1882
simproc_setup binary_int_div
huffman@47108
  1883
  ("numeral m div numeral n :: int" |
huffman@47108
  1884
   "numeral m div neg_numeral n :: int" |
huffman@47108
  1885
   "neg_numeral m div numeral n :: int" |
huffman@47108
  1886
   "neg_numeral m div neg_numeral n :: int") =
huffman@45530
  1887
  {* K (divmod_proc @{thm int_div_pos_eq} @{thm int_div_neg_eq}) *}
haftmann@33361
  1888
huffman@47108
  1889
simproc_setup binary_int_mod
huffman@47108
  1890
  ("numeral m mod numeral n :: int" |
huffman@47108
  1891
   "numeral m mod neg_numeral n :: int" |
huffman@47108
  1892
   "neg_numeral m mod numeral n :: int" |
huffman@47108
  1893
   "neg_numeral m mod neg_numeral n :: int") =
huffman@45530
  1894
  {* K (divmod_proc @{thm int_mod_pos_eq} @{thm int_mod_neg_eq}) *}
haftmann@33361
  1895
huffman@47108
  1896
lemmas posDivAlg_eqn_numeral [simp] =
huffman@47108
  1897
    posDivAlg_eqn [of "numeral v" "numeral w", OF zero_less_numeral] for v w
huffman@47108
  1898
huffman@47108
  1899
lemmas negDivAlg_eqn_numeral [simp] =
huffman@47108
  1900
    negDivAlg_eqn [of "numeral v" "neg_numeral w", OF zero_less_numeral] for v w
haftmann@33361
  1901
haftmann@33361
  1902
haftmann@33361
  1903
text{*Special-case simplification *}
haftmann@33361
  1904
haftmann@33361
  1905
(** The last remaining special cases for constant arithmetic:
haftmann@33361
  1906
    1 div z and 1 mod z **)
haftmann@33361
  1907
huffman@47108
  1908
lemmas div_pos_pos_1_numeral [simp] =
huffman@47108
  1909
  div_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
huffman@47108
  1910
huffman@47108
  1911
lemmas div_pos_neg_1_numeral [simp] =
huffman@47108
  1912
  div_pos_neg [OF zero_less_one, of "neg_numeral w",
huffman@47108
  1913
  OF neg_numeral_less_zero] for w
huffman@47108
  1914
huffman@47108
  1915
lemmas mod_pos_pos_1_numeral [simp] =
huffman@47108
  1916
  mod_pos_pos [OF zero_less_one, of "numeral w", OF zero_le_numeral] for w
huffman@47108
  1917
huffman@47108
  1918
lemmas mod_pos_neg_1_numeral [simp] =
huffman@47108
  1919
  mod_pos_neg [OF zero_less_one, of "neg_numeral w",
huffman@47108
  1920
  OF neg_numeral_less_zero] for w
huffman@47108
  1921
huffman@47108
  1922
lemmas posDivAlg_eqn_1_numeral [simp] =
huffman@47108
  1923
    posDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
huffman@47108
  1924
huffman@47108
  1925
lemmas negDivAlg_eqn_1_numeral [simp] =
huffman@47108
  1926
    negDivAlg_eqn [of concl: 1 "numeral w", OF zero_less_numeral] for w
haftmann@33361
  1927
haftmann@33361
  1928
huffman@46551
  1929
subsubsection {* Monotonicity in the First Argument (Dividend) *}
haftmann@33361
  1930
haftmann@33361
  1931
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361
  1932
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1933
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1934
apply (rule unique_quotient_lemma)
haftmann@33361
  1935
apply (erule subst)
haftmann@33361
  1936
apply (erule subst, simp_all)
haftmann@33361
  1937
done
haftmann@33361
  1938
haftmann@33361
  1939
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361
  1940
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1941
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1942
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1943
apply (erule subst)
haftmann@33361
  1944
apply (erule subst, simp_all)
haftmann@33361
  1945
done
haftmann@33361
  1946
haftmann@33361
  1947
huffman@46551
  1948
subsubsection {* Monotonicity in the Second Argument (Divisor) *}
haftmann@33361
  1949
haftmann@33361
  1950
lemma q_pos_lemma:
haftmann@33361
  1951
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1952
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1953
 apply (simp add: zero_less_mult_iff)
webertj@49962
  1954
apply (simp add: distrib_left)
haftmann@33361
  1955
done
haftmann@33361
  1956
haftmann@33361
  1957
lemma zdiv_mono2_lemma:
haftmann@33361
  1958
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
haftmann@33361
  1959
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
haftmann@33361
  1960
      ==> q \<le> (q'::int)"
haftmann@33361
  1961
apply (frule q_pos_lemma, assumption+) 
haftmann@33361
  1962
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  1963
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1964
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  1965
 prefer 2 apply simp
webertj@49962
  1966
apply (simp (no_asm_simp) add: distrib_left)
huffman@44766
  1967
apply (subst add_commute, rule add_less_le_mono, arith)
haftmann@33361
  1968
apply (rule mult_right_mono, auto)
haftmann@33361
  1969
done
haftmann@33361
  1970
haftmann@33361
  1971
lemma zdiv_mono2:
haftmann@33361
  1972
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  1973
apply (subgoal_tac "b \<noteq> 0")
haftmann@33361
  1974
 prefer 2 apply arith
haftmann@33361
  1975
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1976
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1977
apply (rule zdiv_mono2_lemma)
haftmann@33361
  1978
apply (erule subst)
haftmann@33361
  1979
apply (erule subst, simp_all)
haftmann@33361
  1980
done
haftmann@33361
  1981
haftmann@33361
  1982
lemma q_neg_lemma:
haftmann@33361
  1983
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  1984
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  1985
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  1986
done
haftmann@33361
  1987
haftmann@33361
  1988
lemma zdiv_mono2_neg_lemma:
haftmann@33361
  1989
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
haftmann@33361
  1990
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
haftmann@33361
  1991
      ==> q' \<le> (q::int)"
haftmann@33361
  1992
apply (frule q_neg_lemma, assumption+) 
haftmann@33361
  1993
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  1994
 apply (simp add: mult_less_cancel_left)
webertj@49962
  1995
apply (simp add: distrib_left)
haftmann@33361
  1996
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  1997
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  1998
done
haftmann@33361
  1999
haftmann@33361
  2000
lemma zdiv_mono2_neg:
haftmann@33361
  2001
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361
  2002
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  2003
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  2004
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  2005
apply (erule subst)
haftmann@33361
  2006
apply (erule subst, simp_all)
haftmann@33361
  2007
done
haftmann@33361
  2008
haftmann@33361
  2009
huffman@46551
  2010
subsubsection {* More Algebraic Laws for div and mod *}
haftmann@33361
  2011
haftmann@33361
  2012
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
haftmann@33361
  2013
haftmann@33361
  2014
lemma zmult1_lemma:
bulwahn@46552
  2015
     "[| divmod_int_rel b c (q, r) |]  
haftmann@33361
  2016
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
webertj@49962
  2017
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left mult_ac)
haftmann@33361
  2018
haftmann@33361
  2019
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  2020
apply (case_tac "c = 0", simp)
huffman@47140
  2021
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  2022
done
haftmann@33361
  2023
haftmann@33361
  2024
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@33361
  2025
haftmann@33361
  2026
lemma zadd1_lemma:
bulwahn@46552
  2027
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br) |]  
haftmann@33361
  2028
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
webertj@49962
  2029
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff distrib_left)
haftmann@33361
  2030
haftmann@33361
  2031
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  2032
lemma zdiv_zadd1_eq:
haftmann@33361
  2033
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  2034
apply (case_tac "c = 0", simp)
huffman@47140
  2035
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] div_int_unique)
haftmann@33361
  2036
done
haftmann@33361
  2037
haftmann@33361
  2038
lemma posDivAlg_div_mod:
haftmann@33361
  2039
  assumes "k \<ge> 0"
haftmann@33361
  2040
  and "l \<ge> 0"
haftmann@33361
  2041
  shows "posDivAlg k l = (k div l, k mod l)"
haftmann@33361
  2042
proof (cases "l = 0")
haftmann@33361
  2043
  case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@33361
  2044
next
haftmann@33361
  2045
  case False with assms posDivAlg_correct
haftmann@33361
  2046
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@33361
  2047
    by simp
huffman@47140
  2048
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  2049
  show ?thesis by simp
haftmann@33361
  2050
qed
haftmann@33361
  2051
haftmann@33361
  2052
lemma negDivAlg_div_mod:
haftmann@33361
  2053
  assumes "k < 0"
haftmann@33361
  2054
  and "l > 0"
haftmann@33361
  2055
  shows "negDivAlg k l = (k div l, k mod l)"
haftmann@33361
  2056
proof -
haftmann@33361
  2057
  from assms have "l \<noteq> 0" by simp
haftmann@33361
  2058
  from assms negDivAlg_correct
haftmann@33361
  2059
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@33361
  2060
    by simp
huffman@47140
  2061
  from div_int_unique [OF this] mod_int_unique [OF this]
haftmann@33361
  2062
  show ?thesis by simp
haftmann@33361
  2063
qed
haftmann@33361
  2064
haftmann@33361
  2065
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  2066
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  2067
haftmann@33361
  2068
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  2069
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  2070
huffman@47108
  2071
lemma zmod_zdiv_equality':
huffman@47108
  2072
  "(m\<Colon>int) mod n = m - (m div n) * n"
huffman@47141
  2073
  using mod_div_equality [of m n] by arith
huffman@47108
  2074
haftmann@33361
  2075
huffman@46551
  2076
subsubsection {* Proving  @{term "a div (b*c) = (a div b) div c"} *}
haftmann@33361
  2077
haftmann@33361
  2078
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  2079
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  2080
  to cause particular problems.*)
haftmann@33361
  2081
haftmann@33361
  2082
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
haftmann@33361
  2083
haftmann@33361
  2084
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
haftmann@33361
  2085
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  2086
 apply (simp add: algebra_simps)
haftmann@33361
  2087
apply (rule order_le_less_trans)
haftmann@33361
  2088
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  2089
 apply (rule mult_left_mono_neg)
huffman@35216
  2090
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  2091
 apply (simp)
haftmann@33361
  2092
apply (simp)
haftmann@33361
  2093
done
haftmann@33361
  2094
haftmann@33361
  2095
lemma zmult2_lemma_aux2:
haftmann@33361
  2096
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  2097
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  2098
 apply arith
haftmann@33361
  2099
apply (simp add: mult_le_0_iff)
haftmann@33361
  2100
done
haftmann@33361
  2101
haftmann@33361
  2102
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  2103
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  2104
apply arith
haftmann@33361
  2105
apply (simp add: zero_le_mult_iff)
haftmann@33361
  2106
done
haftmann@33361
  2107
haftmann@33361
  2108
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  2109
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  2110
 apply (simp add: right_diff_distrib)
haftmann@33361
  2111
apply (rule order_less_le_trans)
haftmann@33361
  2112
 apply (erule mult_strict_right_mono)
haftmann@33361
  2113
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  2114
  apply simp
huffman@35216
  2115
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  2116
apply simp
haftmann@33361
  2117
done
haftmann@33361
  2118
bulwahn@46552
  2119
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r); 0 < c |]  
haftmann@33361
  2120
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@33361
  2121
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
webertj@49962
  2122
                   zero_less_mult_iff distrib_left [symmetric] 
huffman@47139
  2123
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: split_if_asm)
haftmann@33361
  2124
haftmann@53068
  2125
lemma zdiv_zmult2_eq:
haftmann@53068
  2126
  fixes a b c :: int
haftmann@53068
  2127
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
  2128
apply (case_tac "b = 0", simp)
haftmann@53068
  2129
apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  2130
done
haftmann@33361
  2131
haftmann@33361
  2132
lemma zmod_zmult2_eq:
haftmann@53068
  2133
  fixes a b c :: int
haftmann@53068
  2134
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
  2135
apply (case_tac "b = 0", simp)
haftmann@53068
  2136
apply (force simp add: le_less divmod_int_rel_div_mod [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  2137
done
haftmann@33361
  2138
huffman@47108
  2139
lemma div_pos_geq:
huffman@47108
  2140
  fixes k l :: int
huffman@47108
  2141
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2142
  shows "k div l = (k - l) div l + 1"
huffman@47108
  2143
proof -
huffman@47108
  2144
  have "k = (k - l) + l" by simp
huffman@47108
  2145
  then obtain j where k: "k = j + l" ..
huffman@47108
  2146
  with assms show ?thesis by simp
huffman@47108
  2147
qed
huffman@47108
  2148
huffman@47108
  2149
lemma mod_pos_geq:
huffman@47108
  2150
  fixes k l :: int
huffman@47108
  2151
  assumes "0 < l" and "l \<le> k"
huffman@47108
  2152
  shows "k mod l = (k - l) mod l"
huffman@47108
  2153
proof -
huffman@47108
  2154
  have "k = (k - l) + l" by simp
huffman@47108
  2155
  then obtain j where k: "k = j + l" ..
huffman@47108
  2156
  with assms show ?thesis by simp
huffman@47108
  2157
qed
huffman@47108
  2158
haftmann@33361
  2159
huffman@46551
  2160
subsubsection {* Splitting Rules for div and mod *}
haftmann@33361
  2161
haftmann@33361
  2162
text{*The proofs of the two lemmas below are essentially identical*}
haftmann@33361
  2163
haftmann@33361
  2164
lemma split_pos_lemma:
haftmann@33361
  2165
 "0<k ==> 
haftmann@33361
  2166
    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
  2167
apply (rule iffI, clarify)
haftmann@33361
  2168
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2169
 apply (subst mod_add_eq) 
haftmann@33361
  2170
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2171
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
haftmann@33361
  2172
txt{*converse direction*}
haftmann@33361
  2173
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2174
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2175
done
haftmann@33361
  2176
haftmann@33361
  2177
lemma split_neg_lemma:
haftmann@33361
  2178
 "k<0 ==>
haftmann@33361
  2179
    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
  2180
apply (rule iffI, clarify)
haftmann@33361
  2181
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2182
 apply (subst mod_add_eq) 
haftmann@33361
  2183
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2184
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
haftmann@33361
  2185
txt{*converse direction*}
haftmann@33361
  2186
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2187
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2188
done
haftmann@33361
  2189
haftmann@33361
  2190
lemma split_zdiv:
haftmann@33361
  2191
 "P(n div k :: int) =
haftmann@33361
  2192
  ((k = 0 --> P 0) & 
haftmann@33361
  2193
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
haftmann@33361
  2194
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  2195
apply (case_tac "k=0", simp)
haftmann@33361
  2196
apply (simp only: linorder_neq_iff)
haftmann@33361
  2197
apply (erule disjE) 
haftmann@33361
  2198
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
haftmann@33361
  2199
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  2200
done
haftmann@33361
  2201
haftmann@33361
  2202
lemma split_zmod:
haftmann@33361
  2203
 "P(n mod k :: int) =
haftmann@33361
  2204
  ((k = 0 --> P n) & 
haftmann@33361
  2205
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
haftmann@33361
  2206
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  2207
apply (case_tac "k=0", simp)
haftmann@33361
  2208
apply (simp only: linorder_neq_iff)
haftmann@33361
  2209
apply (erule disjE) 
haftmann@33361
  2210
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
haftmann@33361
  2211
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  2212
done
haftmann@33361
  2213
webertj@33730
  2214
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
webertj@33730
  2215
  when these are applied to some constant that is of the form
huffman@47108
  2216
  @{term "numeral k"}: *}
huffman@47108
  2217
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  2218
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  2219
haftmann@33361
  2220
huffman@47166
  2221
subsubsection {* Computing @{text "div"} and @{text "mod"} with shifting *}
huffman@47166
  2222
huffman@47166
  2223
lemma pos_divmod_int_rel_mult_2:
huffman@47166
  2224
  assumes "0 \<le> b"
huffman@47166
  2225
  assumes "divmod_int_rel a b (q, r)"
huffman@47166
  2226
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 1 + 2*r)"
huffman@47166
  2227
  using assms unfolding divmod_int_rel_def by auto
huffman@47166
  2228
huffman@47166
  2229
lemma neg_divmod_int_rel_mult_2:
huffman@47166
  2230
  assumes "b \<le> 0"
huffman@47166
  2231
  assumes "divmod_int_rel (a + 1) b (q, r)"
huffman@47166
  2232
  shows "divmod_int_rel (1 + 2*a) (2*b) (q, 2*r - 1)"
huffman@47166
  2233
  using assms unfolding divmod_int_rel_def by auto
haftmann@33361
  2234
haftmann@33361
  2235
text{*computing div by shifting *}
haftmann@33361
  2236
haftmann@33361
  2237
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
huffman@47166
  2238
  using pos_divmod_int_rel_mult_2 [OF _ divmod_int_rel_div_mod]
huffman@47166
  2239
  by (rule div_int_unique)
haftmann@33361
  2240
boehmes@35815
  2241
lemma neg_zdiv_mult_2: 
boehmes@35815
  2242
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
huffman@47166
  2243
  using neg_divmod_int_rel_mult_2 [OF A divmod_int_rel_div_mod]
huffman@47166
  2244
  by (rule div_int_unique)
haftmann@33361
  2245
huffman@47108
  2246
(* FIXME: add rules for negative numerals *)
huffman@47108
  2247
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  2248
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  2249
    numeral v div (numeral w :: int)"
huffman@47108
  2250
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  2251
  by (rule div_mult_mult1, simp)
huffman@47108
  2252
huffman@47108
  2253
lemma zdiv_numeral_Bit1 [simp]:
huffman@47108
  2254
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =  
huffman@47108
  2255
    (numeral v div (numeral w :: int))"
huffman@47108
  2256
  unfolding numeral.simps
huffman@47108
  2257
  unfolding mult_2 [symmetric] add_commute [of _ 1]
huffman@47108
  2258
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  2259
haftmann@33361
  2260
lemma pos_zmod_mult_2:
haftmann@33361
  2261
  fixes a b :: int
haftmann@33361
  2262
  assumes "0 \<le> a"
haftmann@33361
  2263
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
huffman@47166
  2264
  using pos_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2265
  by (rule mod_int_unique)
haftmann@33361
  2266
haftmann@33361
  2267
lemma neg_zmod_mult_2:
haftmann@33361
  2268
  fixes a b :: int
haftmann@33361
  2269
  assumes "a \<le> 0"
haftmann@33361
  2270
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
huffman@47166
  2271
  using neg_divmod_int_rel_mult_2 [OF assms divmod_int_rel_div_mod]
huffman@47166
  2272
  by (rule mod_int_unique)
haftmann@33361
  2273
huffman@47108
  2274
(* FIXME: add rules for negative numerals *)
huffman@47108
  2275
lemma zmod_numeral_Bit0 [simp]:
huffman@47108
  2276
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =  
huffman@47108
  2277
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  2278
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  2279
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  2280
huffman@47108
  2281
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  2282
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  2283
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  2284
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
huffman@47108
  2285
  unfolding mult_2 [symmetric] add_commute [of _ 1]
huffman@47108
  2286
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  2287
nipkow@39489
  2288
lemma zdiv_eq_0_iff:
nipkow@39489
  2289
 "(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
  2290
proof
nipkow@39489
  2291
  assume ?L
nipkow@39489
  2292
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
nipkow@39489
  2293
  with `?L` show ?R by blast
nipkow@39489
  2294
next
nipkow@39489
  2295
  assume ?R thus ?L
nipkow@39489
  2296
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  2297
qed
nipkow@39489
  2298
nipkow@39489
  2299
huffman@46551
  2300
subsubsection {* Quotients of Signs *}
haftmann@33361
  2301
haftmann@33361
  2302
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2303
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2304
apply (rule order_trans)
haftmann@33361
  2305
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2306
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2307
done
haftmann@33361
  2308
haftmann@33361
  2309
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2310
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2311
haftmann@33361
  2312
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2313
by (drule zdiv_mono1, auto)
haftmann@33361
  2314
nipkow@33804
  2315
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
nipkow@33804
  2316
conditional upon the sign of @{text a} or @{text b}. There are many more.
nipkow@33804
  2317
They should all be simp rules unless that causes too much search. *}
nipkow@33804
  2318
haftmann@33361
  2319
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2320
apply auto
haftmann@33361
  2321
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2322
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2323
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2324
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2325
done
haftmann@33361
  2326
haftmann@33361
  2327
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2328
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2329
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2330
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2331
done
haftmann@33361
  2332
haftmann@33361
  2333
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2334
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2335
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2336
nipkow@39489
  2337
lemma pos_imp_zdiv_pos_iff:
nipkow@39489
  2338
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
nipkow@39489
  2339
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
nipkow@39489
  2340
by arith
nipkow@39489
  2341
haftmann@33361
  2342
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2343
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2344
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2345
nipkow@33804
  2346
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2347
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2348
apply rule
nipkow@33804
  2349
 apply rule
nipkow@33804
  2350
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2351
 apply(cases "b=0")apply simp
nipkow@33804
  2352
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2353
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2354
done
nipkow@33804
  2355
nipkow@39489
  2356
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2357
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2358
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2359
done
nipkow@39489
  2360
nipkow@39489
  2361
haftmann@33361
  2362
subsubsection {* The Divides Relation *}
haftmann@33361
  2363
huffman@47268
  2364
lemma dvd_neg_numeral_left [simp]:
huffman@47268
  2365
  fixes y :: "'a::comm_ring_1"
huffman@47268
  2366
  shows "(neg_numeral k) dvd y \<longleftrightarrow> (numeral k) dvd y"
huffman@47268
  2367
  unfolding neg_numeral_def minus_dvd_iff ..
huffman@47268
  2368
huffman@47268
  2369
lemma dvd_neg_numeral_right [simp]:
huffman@47268
  2370
  fixes x :: "'a::comm_ring_1"
huffman@47268
  2371
  shows "x dvd (neg_numeral k) \<longleftrightarrow> x dvd (numeral k)"
huffman@47268
  2372
  unfolding neg_numeral_def dvd_minus_iff ..
haftmann@33361
  2373
huffman@47108
  2374
lemmas dvd_eq_mod_eq_0_numeral [simp] =
huffman@47108
  2375
  dvd_eq_mod_eq_0 [of "numeral x" "numeral y"] for x y
huffman@47108
  2376
huffman@47108
  2377
huffman@47108
  2378
subsubsection {* Further properties *}
huffman@47108
  2379
haftmann@33361
  2380
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
haftmann@33361
  2381
  using zmod_zdiv_equality[where a="m" and b="n"]
huffman@47142
  2382
  by (simp add: algebra_simps) (* FIXME: generalize *)
haftmann@33361
  2383
haftmann@33361
  2384
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
haftmann@33361
  2385
apply (subst split_div, auto)
haftmann@33361
  2386
apply (subst split_zdiv, auto)
haftmann@33361
  2387
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
  2388
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2389
done
haftmann@33361
  2390
haftmann@33361
  2391
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
haftmann@33361
  2392
apply (subst split_mod, auto)
haftmann@33361
  2393
apply (subst split_zmod, auto)
haftmann@33361
  2394
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
haftmann@33361
  2395
       in unique_remainder)
haftmann@33361
  2396
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2397
done
haftmann@33361
  2398
haftmann@33361
  2399
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
haftmann@33361
  2400
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2401
haftmann@33361
  2402
text{*Suggested by Matthias Daum*}
haftmann@33361
  2403
lemma int_power_div_base:
haftmann@33361
  2404
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2405
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2406
 apply (erule ssubst)
haftmann@33361
  2407
 apply (simp only: power_add)
haftmann@33361
  2408
 apply simp_all
haftmann@33361
  2409
done
haftmann@33361
  2410
haftmann@33361
  2411
text {* by Brian Huffman *}
haftmann@33361
  2412
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2413
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2414
haftmann@33361
  2415
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2416
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2417
haftmann@33361
  2418
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2419
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2420
haftmann@33361
  2421
lemmas zmod_simps =
haftmann@33361
  2422
  mod_add_left_eq  [symmetric]
haftmann@33361
  2423
  mod_add_right_eq [symmetric]
huffman@47142
  2424
  mod_mult_right_eq[symmetric]
haftmann@33361
  2425
  mod_mult_left_eq [symmetric]
huffman@47164
  2426
  power_mod
haftmann@33361
  2427
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2428
haftmann@33361
  2429
text {* Distributive laws for function @{text nat}. *}
haftmann@33361
  2430
haftmann@33361
  2431
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2432
apply (rule linorder_cases [of y 0])
haftmann@33361
  2433
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2434
apply simp
haftmann@33361
  2435
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2436
done
haftmann@33361
  2437
haftmann@33361
  2438
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2439
lemma nat_mod_distrib:
haftmann@33361
  2440
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2441
apply (case_tac "y = 0", simp)
haftmann@33361
  2442
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2443
done
haftmann@33361
  2444
haftmann@33361
  2445
text  {* transfer setup *}
haftmann@33361
  2446
haftmann@33361
  2447
lemma transfer_nat_int_functions:
haftmann@33361
  2448
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2449
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2450
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2451
haftmann@33361
  2452
lemma transfer_nat_int_function_closures:
haftmann@33361
  2453
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2454
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2455
  apply (cases "y = 0")
haftmann@33361
  2456
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2457
  apply (cases "y = 0")
haftmann@33361
  2458
  apply auto
haftmann@33361
  2459
done
haftmann@33361
  2460
haftmann@35644
  2461
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2462
  transfer_nat_int_functions
haftmann@33361
  2463
  transfer_nat_int_function_closures
haftmann@33361
  2464
]
haftmann@33361
  2465
haftmann@33361
  2466
lemma transfer_int_nat_functions:
haftmann@33361
  2467
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2468
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2469
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2470
haftmann@33361
  2471
lemma transfer_int_nat_function_closures:
haftmann@33361
  2472
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  2473
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  2474
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  2475
haftmann@35644
  2476
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  2477
  transfer_int_nat_functions
haftmann@33361
  2478
  transfer_int_nat_function_closures
haftmann@33361
  2479
]
haftmann@33361
  2480
haftmann@33361
  2481
text{*Suggested by Matthias Daum*}
haftmann@33361
  2482
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  2483
apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225
  2484
 apply (simp add: nat_div_distrib [symmetric])
haftmann@33361
  2485
apply (rule Divides.div_less_dividend, simp_all)
haftmann@33361
  2486
done
haftmann@33361
  2487
haftmann@33361
  2488
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
haftmann@33361
  2489
proof
haftmann@33361
  2490
  assume H: "x mod n = y mod n"
haftmann@33361
  2491
  hence "x mod n - y mod n = 0" by simp
haftmann@33361
  2492
  hence "(x mod n - y mod n) mod n = 0" by simp 
haftmann@33361
  2493
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
haftmann@33361
  2494
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
haftmann@33361
  2495
next
haftmann@33361
  2496
  assume H: "n dvd x - y"
haftmann@33361
  2497
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
haftmann@33361
  2498
  hence "x = n*k + y" by simp
haftmann@33361
  2499
  hence "x mod n = (n*k + y) mod n" by simp
haftmann@33361
  2500
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
haftmann@33361
  2501
qed
haftmann@33361
  2502
haftmann@33361
  2503
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
haftmann@33361
  2504
  shows "\<exists>q. x = y + n * q"
haftmann@33361
  2505
proof-
haftmann@33361
  2506
  from xy have th: "int x - int y = int (x - y)" by simp 
haftmann@33361
  2507
  from xyn have "int x mod int n = int y mod int n" 
huffman@46551
  2508
    by (simp add: zmod_int [symmetric])
haftmann@33361
  2509
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
haftmann@33361
  2510
  hence "n dvd x - y" by (simp add: th zdvd_int)
haftmann@33361
  2511
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
haftmann@33361
  2512
qed
haftmann@33361
  2513
haftmann@33361
  2514
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
haftmann@33361
  2515
  (is "?lhs = ?rhs")
haftmann@33361
  2516
proof
haftmann@33361
  2517
  assume H: "x mod n = y mod n"
haftmann@33361
  2518
  {assume xy: "x \<le> y"
haftmann@33361
  2519
    from H have th: "y mod n = x mod n" by simp
haftmann@33361
  2520
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
haftmann@33361
  2521
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361
  2522
  moreover
haftmann@33361
  2523
  {assume xy: "y \<le> x"
haftmann@33361