src/HOL/Divides.thy
author boehmes
Wed Mar 17 19:55:07 2010 +0100 (2010-03-17)
changeset 35815 10e723e54076
parent 35673 178caf872f95
child 36634 f9b43d197d16
permissions -rw-r--r--
tuned proofs (to avoid linarith error message caused by bootstrapping of HOL)
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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_Numeral Nat_Transfer
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uses "~~/src/Provers/Arith/cancel_div_mod.ML"
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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 left_distrib)
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  then have "a div b * b + (a + c * b) mod b = a div b * b + a mod b"
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    by (simp add: mod_div_equality)
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  then show ?thesis by simp
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qed
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lemma mod_mult_self2 [simp]: "(a + b * c) mod b = a mod b"
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  by (simp add: mult_commute [of b])
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lemma div_mult_self1_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> b * a div b = a"
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  using div_mult_self2 [of b 0 a] by simp
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lemma div_mult_self2_is_id [simp]: "b \<noteq> 0 \<Longrightarrow> a * b div b = a"
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  using div_mult_self1 [of b 0 a] by simp
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lemma mod_mult_self1_is_0 [simp]: "b * a mod b = 0"
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  using mod_mult_self2 [of 0 b a] by simp
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lemma mod_mult_self2_is_0 [simp]: "a * b mod b = 0"
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  using mod_mult_self1 [of 0 a b] by simp
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lemma div_by_1 [simp]: "a div 1 = a"
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  using div_mult_self2_is_id [of 1 a] zero_neq_one by simp
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lemma mod_by_1 [simp]: "a mod 1 = 0"
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proof -
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  from mod_div_equality [of a one] div_by_1 have "a + a mod 1 = a" by simp
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  then have "a + a mod 1 = a + 0" by simp
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  then show ?thesis by (rule add_left_imp_eq)
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qed
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lemma mod_self [simp]: "a mod a = 0"
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  using mod_mult_self2_is_0 [of 1] by simp
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lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
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  using div_mult_self2_is_id [of _ 1] by simp
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lemma div_add_self1 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(b + a) div b = a div b + 1"
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  using assms div_mult_self1 [of b a 1] by (simp add: add_commute)
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lemma div_add_self2 [simp]:
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  assumes "b \<noteq> 0"
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  shows "(a + b) div b = a div b + 1"
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  using assms div_add_self1 [of b a] by (simp add: add_commute)
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lemma mod_add_self1 [simp]:
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  "(b + a) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by (simp add: add_commute)
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lemma mod_add_self2 [simp]:
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  "(a + b) mod b = a mod b"
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  using mod_mult_self1 [of a 1 b] by simp
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lemma mod_div_decomp:
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  fixes a b
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  obtains q r where "q = a div b" and "r = a mod b"
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    and "a = q * b + r"
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proof -
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  from mod_div_equality have "a = a div b * b + a mod b" by simp
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  moreover have "a div b = a div b" ..
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  moreover have "a mod b = a mod b" ..
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  note that ultimately show thesis by blast
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qed
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lemma dvd_eq_mod_eq_0 [code, code_unfold, code_inline del]: "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(fastsimp 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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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)"
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  apply (cases "y = 0", simp)
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  apply (cases "z = 0", simp)
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  apply (auto elim!: dvdE simp add: algebra_simps)
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  apply (subst mult_assoc [symmetric])
nipkow@30476
   309
  apply (simp add: no_zero_divisors)
haftmann@30930
   310
  done
haftmann@30930
   311
haftmann@35367
   312
lemma div_mult_swap:
haftmann@35367
   313
  assumes "c dvd b"
haftmann@35367
   314
  shows "a * (b div c) = (a * b) div c"
haftmann@35367
   315
proof -
haftmann@35367
   316
  from assms have "b div c * (a div 1) = b * a div (c * 1)"
haftmann@35367
   317
    by (simp only: div_mult_div_if_dvd one_dvd)
haftmann@35367
   318
  then show ?thesis by (simp add: mult_commute)
haftmann@35367
   319
qed
haftmann@35367
   320
   
haftmann@30930
   321
lemma div_mult_mult2 [simp]:
haftmann@30930
   322
  "c \<noteq> 0 \<Longrightarrow> (a * c) div (b * c) = a div b"
haftmann@30930
   323
  by (drule div_mult_mult1) (simp add: mult_commute)
haftmann@30930
   324
haftmann@30930
   325
lemma div_mult_mult1_if [simp]:
haftmann@30930
   326
  "(c * a) div (c * b) = (if c = 0 then 0 else a div b)"
haftmann@30930
   327
  by simp_all
nipkow@30476
   328
haftmann@30930
   329
lemma mod_mult_mult1:
haftmann@30930
   330
  "(c * a) mod (c * b) = c * (a mod b)"
haftmann@30930
   331
proof (cases "c = 0")
haftmann@30930
   332
  case True then show ?thesis by simp
haftmann@30930
   333
next
haftmann@30930
   334
  case False
haftmann@30930
   335
  from mod_div_equality
haftmann@30930
   336
  have "((c * a) div (c * b)) * (c * b) + (c * a) mod (c * b) = c * a" .
haftmann@30930
   337
  with False have "c * ((a div b) * b + a mod b) + (c * a) mod (c * b)
haftmann@30930
   338
    = c * a + c * (a mod b)" by (simp add: algebra_simps)
haftmann@30930
   339
  with mod_div_equality show ?thesis by simp 
haftmann@30930
   340
qed
haftmann@30930
   341
  
haftmann@30930
   342
lemma mod_mult_mult2:
haftmann@30930
   343
  "(a * c) mod (b * c) = (a mod b) * c"
haftmann@30930
   344
  using mod_mult_mult1 [of c a b] by (simp add: mult_commute)
haftmann@30930
   345
huffman@31662
   346
lemma dvd_mod: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m mod n)"
huffman@31662
   347
  unfolding dvd_def by (auto simp add: mod_mult_mult1)
huffman@31662
   348
huffman@31662
   349
lemma dvd_mod_iff: "k dvd n \<Longrightarrow> k dvd (m mod n) \<longleftrightarrow> k dvd m"
huffman@31662
   350
by (blast intro: dvd_mod_imp_dvd dvd_mod)
huffman@31662
   351
haftmann@31009
   352
lemma div_power:
huffman@31661
   353
  "y dvd x \<Longrightarrow> (x div y) ^ n = x ^ n div y ^ n"
nipkow@30476
   354
apply (induct n)
nipkow@30476
   355
 apply simp
nipkow@30476
   356
apply(simp add: div_mult_div_if_dvd dvd_power_same)
nipkow@30476
   357
done
nipkow@30476
   358
haftmann@35367
   359
lemma dvd_div_eq_mult:
haftmann@35367
   360
  assumes "a \<noteq> 0" and "a dvd b"  
haftmann@35367
   361
  shows "b div a = c \<longleftrightarrow> b = c * a"
haftmann@35367
   362
proof
haftmann@35367
   363
  assume "b = c * a"
haftmann@35367
   364
  then show "b div a = c" by (simp add: assms)
haftmann@35367
   365
next
haftmann@35367
   366
  assume "b div a = c"
haftmann@35367
   367
  then have "b div a * a = c * a" by simp
haftmann@35367
   368
  moreover from `a dvd b` have "b div a * a = b" by (simp add: dvd_div_mult_self)
haftmann@35367
   369
  ultimately show "b = c * a" by simp
haftmann@35367
   370
qed
haftmann@35367
   371
   
haftmann@35367
   372
lemma dvd_div_div_eq_mult:
haftmann@35367
   373
  assumes "a \<noteq> 0" "c \<noteq> 0" and "a dvd b" "c dvd d"
haftmann@35367
   374
  shows "b div a = d div c \<longleftrightarrow> b * c = a * d"
haftmann@35367
   375
  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
   376
huffman@31661
   377
end
huffman@31661
   378
haftmann@35673
   379
class ring_div = semiring_div + comm_ring_1
huffman@29405
   380
begin
huffman@29405
   381
huffman@29405
   382
text {* Negation respects modular equivalence. *}
huffman@29405
   383
huffman@29405
   384
lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
huffman@29405
   385
proof -
huffman@29405
   386
  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
huffman@29405
   387
    by (simp only: mod_div_equality)
huffman@29405
   388
  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
huffman@29405
   389
    by (simp only: minus_add_distrib minus_mult_left add_ac)
huffman@29405
   390
  also have "\<dots> = (- (a mod b)) mod b"
huffman@29405
   391
    by (rule mod_mult_self1)
huffman@29405
   392
  finally show ?thesis .
huffman@29405
   393
qed
huffman@29405
   394
huffman@29405
   395
lemma mod_minus_cong:
huffman@29405
   396
  assumes "a mod b = a' mod b"
huffman@29405
   397
  shows "(- a) mod b = (- a') mod b"
huffman@29405
   398
proof -
huffman@29405
   399
  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
huffman@29405
   400
    unfolding assms ..
huffman@29405
   401
  thus ?thesis
huffman@29405
   402
    by (simp only: mod_minus_eq [symmetric])
huffman@29405
   403
qed
huffman@29405
   404
huffman@29405
   405
text {* Subtraction respects modular equivalence. *}
huffman@29405
   406
huffman@29405
   407
lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
huffman@29405
   408
  unfolding diff_minus
huffman@29405
   409
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   410
huffman@29405
   411
lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
huffman@29405
   412
  unfolding diff_minus
huffman@29405
   413
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   414
huffman@29405
   415
lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
huffman@29405
   416
  unfolding diff_minus
huffman@29405
   417
  by (intro mod_add_cong mod_minus_cong) simp_all
huffman@29405
   418
huffman@29405
   419
lemma mod_diff_cong:
huffman@29405
   420
  assumes "a mod c = a' mod c"
huffman@29405
   421
  assumes "b mod c = b' mod c"
huffman@29405
   422
  shows "(a - b) mod c = (a' - b') mod c"
huffman@29405
   423
  unfolding diff_minus using assms
huffman@29405
   424
  by (intro mod_add_cong mod_minus_cong)
huffman@29405
   425
nipkow@30180
   426
lemma dvd_neg_div: "y dvd x \<Longrightarrow> -x div y = - (x div y)"
nipkow@30180
   427
apply (case_tac "y = 0") apply simp
nipkow@30180
   428
apply (auto simp add: dvd_def)
nipkow@30180
   429
apply (subgoal_tac "-(y * k) = y * - k")
nipkow@30180
   430
 apply (erule ssubst)
nipkow@30180
   431
 apply (erule div_mult_self1_is_id)
nipkow@30180
   432
apply simp
nipkow@30180
   433
done
nipkow@30180
   434
nipkow@30180
   435
lemma dvd_div_neg: "y dvd x \<Longrightarrow> x div -y = - (x div y)"
nipkow@30180
   436
apply (case_tac "y = 0") apply simp
nipkow@30180
   437
apply (auto simp add: dvd_def)
nipkow@30180
   438
apply (subgoal_tac "y * k = -y * -k")
nipkow@30180
   439
 apply (erule ssubst)
nipkow@30180
   440
 apply (rule div_mult_self1_is_id)
nipkow@30180
   441
 apply simp
nipkow@30180
   442
apply simp
nipkow@30180
   443
done
nipkow@30180
   444
huffman@29405
   445
end
huffman@29405
   446
haftmann@25942
   447
haftmann@26100
   448
subsection {* Division on @{typ nat} *}
haftmann@26100
   449
haftmann@26100
   450
text {*
haftmann@26100
   451
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   452
  of a characteristic relation with two input arguments
haftmann@26100
   453
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   454
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   455
*}
haftmann@26100
   456
haftmann@33340
   457
definition divmod_nat_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
haftmann@33340
   458
  "divmod_nat_rel m n qr \<longleftrightarrow>
haftmann@30923
   459
    m = fst qr * n + snd qr \<and>
haftmann@30923
   460
      (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
   461
haftmann@33340
   462
text {* @{const divmod_nat_rel} is total: *}
haftmann@26100
   463
haftmann@33340
   464
lemma divmod_nat_rel_ex:
haftmann@33340
   465
  obtains q r where "divmod_nat_rel m n (q, r)"
haftmann@26100
   466
proof (cases "n = 0")
haftmann@30923
   467
  case True  with that show thesis
haftmann@33340
   468
    by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   469
next
haftmann@26100
   470
  case False
haftmann@26100
   471
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   472
  proof (induct m)
haftmann@26100
   473
    case 0 with `n \<noteq> 0`
haftmann@26100
   474
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   475
    then show ?case by blast
haftmann@26100
   476
  next
haftmann@26100
   477
    case (Suc m) then obtain q' r'
haftmann@26100
   478
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   479
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   480
      case True
haftmann@26100
   481
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   482
      with True show ?thesis by blast
haftmann@26100
   483
    next
haftmann@26100
   484
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   485
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   486
      ultimately have "n = Suc r'" by auto
haftmann@26100
   487
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   488
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   489
    qed
haftmann@26100
   490
  qed
haftmann@26100
   491
  with that show thesis
haftmann@33340
   492
    using `n \<noteq> 0` by (auto simp add: divmod_nat_rel_def)
haftmann@26100
   493
qed
haftmann@26100
   494
haftmann@33340
   495
text {* @{const divmod_nat_rel} is injective: *}
haftmann@26100
   496
haftmann@33340
   497
lemma divmod_nat_rel_unique:
haftmann@33340
   498
  assumes "divmod_nat_rel m n qr"
haftmann@33340
   499
    and "divmod_nat_rel m n qr'"
haftmann@30923
   500
  shows "qr = qr'"
haftmann@26100
   501
proof (cases "n = 0")
haftmann@26100
   502
  case True with assms show ?thesis
haftmann@30923
   503
    by (cases qr, cases qr')
haftmann@33340
   504
      (simp add: divmod_nat_rel_def)
haftmann@26100
   505
next
haftmann@26100
   506
  case False
haftmann@26100
   507
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   508
  apply (rule leI)
haftmann@26100
   509
  apply (subst less_iff_Suc_add)
haftmann@26100
   510
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   511
  done
haftmann@30923
   512
  from `n \<noteq> 0` assms have "fst qr = fst qr'"
haftmann@33340
   513
    by (auto simp add: divmod_nat_rel_def intro: order_antisym dest: aux sym)
haftmann@30923
   514
  moreover from this assms have "snd qr = snd qr'"
haftmann@33340
   515
    by (simp add: divmod_nat_rel_def)
haftmann@30923
   516
  ultimately show ?thesis by (cases qr, cases qr') simp
haftmann@26100
   517
qed
haftmann@26100
   518
haftmann@26100
   519
text {*
haftmann@26100
   520
  We instantiate divisibility on the natural numbers by
haftmann@33340
   521
  means of @{const divmod_nat_rel}:
haftmann@26100
   522
*}
haftmann@25942
   523
haftmann@25942
   524
instantiation nat :: semiring_div
haftmann@25571
   525
begin
haftmann@25571
   526
haftmann@33340
   527
definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@33340
   528
  [code del]: "divmod_nat m n = (THE qr. divmod_nat_rel m n qr)"
haftmann@30923
   529
haftmann@33340
   530
lemma divmod_nat_rel_divmod_nat:
haftmann@33340
   531
  "divmod_nat_rel m n (divmod_nat m n)"
haftmann@30923
   532
proof -
haftmann@33340
   533
  from divmod_nat_rel_ex
haftmann@33340
   534
    obtain qr where rel: "divmod_nat_rel m n qr" .
haftmann@30923
   535
  then show ?thesis
haftmann@33340
   536
  by (auto simp add: divmod_nat_def intro: theI elim: divmod_nat_rel_unique)
haftmann@30923
   537
qed
haftmann@30923
   538
haftmann@33340
   539
lemma divmod_nat_eq:
haftmann@33340
   540
  assumes "divmod_nat_rel m n qr" 
haftmann@33340
   541
  shows "divmod_nat m n = qr"
haftmann@33340
   542
  using assms by (auto intro: divmod_nat_rel_unique divmod_nat_rel_divmod_nat)
haftmann@26100
   543
haftmann@26100
   544
definition div_nat where
haftmann@33340
   545
  "m div n = fst (divmod_nat m n)"
haftmann@26100
   546
haftmann@26100
   547
definition mod_nat where
haftmann@33340
   548
  "m mod n = snd (divmod_nat m n)"
haftmann@25571
   549
haftmann@33340
   550
lemma divmod_nat_div_mod:
haftmann@33340
   551
  "divmod_nat m n = (m div n, m mod n)"
haftmann@26100
   552
  unfolding div_nat_def mod_nat_def by simp
haftmann@26100
   553
haftmann@26100
   554
lemma div_eq:
haftmann@33340
   555
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   556
  shows "m div n = q"
haftmann@33340
   557
  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
haftmann@26100
   558
haftmann@26100
   559
lemma mod_eq:
haftmann@33340
   560
  assumes "divmod_nat_rel m n (q, r)" 
haftmann@26100
   561
  shows "m mod n = r"
haftmann@33340
   562
  using assms by (auto dest: divmod_nat_eq simp add: divmod_nat_div_mod)
haftmann@25571
   563
haftmann@33340
   564
lemma divmod_nat_rel: "divmod_nat_rel m n (m div n, m mod n)"
haftmann@33340
   565
  by (simp add: div_nat_def mod_nat_def divmod_nat_rel_divmod_nat)
paulson@14267
   566
haftmann@33340
   567
lemma divmod_nat_zero:
haftmann@33340
   568
  "divmod_nat m 0 = (0, m)"
haftmann@26100
   569
proof -
haftmann@33340
   570
  from divmod_nat_rel [of m 0] show ?thesis
haftmann@33340
   571
    unfolding divmod_nat_div_mod divmod_nat_rel_def by simp
haftmann@26100
   572
qed
haftmann@25942
   573
haftmann@33340
   574
lemma divmod_nat_base:
haftmann@26100
   575
  assumes "m < n"
haftmann@33340
   576
  shows "divmod_nat m n = (0, m)"
haftmann@26100
   577
proof -
haftmann@33340
   578
  from divmod_nat_rel [of m n] show ?thesis
haftmann@33340
   579
    unfolding divmod_nat_div_mod divmod_nat_rel_def
haftmann@26100
   580
    using assms by (cases "m div n = 0")
haftmann@26100
   581
      (auto simp add: gr0_conv_Suc [of "m div n"])
haftmann@26100
   582
qed
haftmann@25942
   583
haftmann@33340
   584
lemma divmod_nat_step:
haftmann@26100
   585
  assumes "0 < n" and "n \<le> m"
haftmann@33340
   586
  shows "divmod_nat m n = (Suc ((m - n) div n), (m - n) mod n)"
haftmann@26100
   587
proof -
haftmann@33340
   588
  from divmod_nat_rel have divmod_nat_m_n: "divmod_nat_rel m n (m div n, m mod n)" .
haftmann@26100
   589
  with assms have m_div_n: "m div n \<ge> 1"
haftmann@33340
   590
    by (cases "m div n") (auto simp add: divmod_nat_rel_def)
boehmes@35815
   591
  have "divmod_nat_rel (m - n) n (m div n - Suc 0, m mod n)"
boehmes@35815
   592
  proof -
boehmes@35815
   593
    from assms have
boehmes@35815
   594
      "n \<noteq> 0"
boehmes@35815
   595
      "\<And>k. m = Suc k * n + m mod n ==> m - n = (Suc k - Suc 0) * n + m mod n"
boehmes@35815
   596
      by simp_all
boehmes@35815
   597
    then show ?thesis using assms divmod_nat_m_n 
boehmes@35815
   598
      by (cases "m div n")
boehmes@35815
   599
         (simp_all only: divmod_nat_rel_def fst_conv snd_conv, simp_all)
boehmes@35815
   600
  qed
haftmann@33340
   601
  with divmod_nat_eq have "divmod_nat (m - n) n = (m div n - Suc 0, m mod n)" by simp
haftmann@33340
   602
  moreover from divmod_nat_div_mod have "divmod_nat (m - n) n = ((m - n) div n, (m - n) mod n)" .
haftmann@26100
   603
  ultimately have "m div n = Suc ((m - n) div n)"
haftmann@26100
   604
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
haftmann@33340
   605
  then show ?thesis using divmod_nat_div_mod by simp
haftmann@26100
   606
qed
haftmann@25942
   607
wenzelm@26300
   608
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   609
haftmann@26100
   610
lemma div_less [simp]:
haftmann@26100
   611
  fixes m n :: nat
haftmann@26100
   612
  assumes "m < n"
haftmann@26100
   613
  shows "m div n = 0"
haftmann@33340
   614
  using assms divmod_nat_base divmod_nat_div_mod by simp
haftmann@25942
   615
haftmann@26100
   616
lemma le_div_geq:
haftmann@26100
   617
  fixes m n :: nat
haftmann@26100
   618
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   619
  shows "m div n = Suc ((m - n) div n)"
haftmann@33340
   620
  using assms divmod_nat_step divmod_nat_div_mod by simp
paulson@14267
   621
haftmann@26100
   622
lemma mod_less [simp]:
haftmann@26100
   623
  fixes m n :: nat
haftmann@26100
   624
  assumes "m < n"
haftmann@26100
   625
  shows "m mod n = m"
haftmann@33340
   626
  using assms divmod_nat_base divmod_nat_div_mod by simp
haftmann@26100
   627
haftmann@26100
   628
lemma le_mod_geq:
haftmann@26100
   629
  fixes m n :: nat
haftmann@26100
   630
  assumes "n \<le> m"
haftmann@26100
   631
  shows "m mod n = (m - n) mod n"
haftmann@33340
   632
  using assms divmod_nat_step divmod_nat_div_mod by (cases "n = 0") simp_all
paulson@14267
   633
haftmann@30930
   634
instance proof -
haftmann@30930
   635
  have [simp]: "\<And>n::nat. n div 0 = 0"
haftmann@33340
   636
    by (simp add: div_nat_def divmod_nat_zero)
haftmann@30930
   637
  have [simp]: "\<And>n::nat. 0 div n = 0"
haftmann@30930
   638
  proof -
haftmann@30930
   639
    fix n :: nat
haftmann@30930
   640
    show "0 div n = 0"
haftmann@30930
   641
      by (cases "n = 0") simp_all
haftmann@30930
   642
  qed
haftmann@30930
   643
  show "OFCLASS(nat, semiring_div_class)" proof
haftmann@30930
   644
    fix m n :: nat
haftmann@30930
   645
    show "m div n * n + m mod n = m"
haftmann@33340
   646
      using divmod_nat_rel [of m n] by (simp add: divmod_nat_rel_def)
haftmann@30930
   647
  next
haftmann@30930
   648
    fix m n q :: nat
haftmann@30930
   649
    assume "n \<noteq> 0"
haftmann@30930
   650
    then show "(q + m * n) div n = m + q div n"
haftmann@30930
   651
      by (induct m) (simp_all add: le_div_geq)
haftmann@30930
   652
  next
haftmann@30930
   653
    fix m n q :: nat
haftmann@30930
   654
    assume "m \<noteq> 0"
haftmann@30930
   655
    then show "(m * n) div (m * q) = n div q"
haftmann@30930
   656
    proof (cases "n \<noteq> 0 \<and> q \<noteq> 0")
haftmann@30930
   657
      case False then show ?thesis by auto
haftmann@30930
   658
    next
haftmann@30930
   659
      case True with `m \<noteq> 0`
haftmann@30930
   660
        have "m > 0" and "n > 0" and "q > 0" by auto
haftmann@33340
   661
      then have "\<And>a b. divmod_nat_rel n q (a, b) \<Longrightarrow> divmod_nat_rel (m * n) (m * q) (a, m * b)"
haftmann@33340
   662
        by (auto simp add: divmod_nat_rel_def) (simp_all add: algebra_simps)
haftmann@33340
   663
      moreover from divmod_nat_rel have "divmod_nat_rel n q (n div q, n mod q)" .
haftmann@33340
   664
      ultimately have "divmod_nat_rel (m * n) (m * q) (n div q, m * (n mod q))" .
haftmann@30930
   665
      then show ?thesis by (simp add: div_eq)
haftmann@30930
   666
    qed
haftmann@30930
   667
  qed simp_all
haftmann@25942
   668
qed
haftmann@26100
   669
haftmann@25942
   670
end
paulson@14267
   671
haftmann@33361
   672
lemma divmod_nat_if [code]: "divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@33361
   673
  let (q, r) = divmod_nat (m - n) n in (Suc q, r))"
haftmann@33361
   674
by (simp add: divmod_nat_zero divmod_nat_base divmod_nat_step)
haftmann@33361
   675
    (simp add: divmod_nat_div_mod)
haftmann@33361
   676
haftmann@26100
   677
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   678
haftmann@30934
   679
ML {*
haftmann@30934
   680
local
haftmann@30934
   681
haftmann@30934
   682
structure CancelDivMod = CancelDivModFun(struct
haftmann@25942
   683
haftmann@30934
   684
  val div_name = @{const_name div};
haftmann@30934
   685
  val mod_name = @{const_name mod};
haftmann@30934
   686
  val mk_binop = HOLogic.mk_binop;
haftmann@30934
   687
  val mk_sum = Nat_Arith.mk_sum;
haftmann@30934
   688
  val dest_sum = Nat_Arith.dest_sum;
haftmann@25942
   689
haftmann@30934
   690
  val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}];
paulson@14267
   691
haftmann@30934
   692
  val trans = trans;
haftmann@25942
   693
haftmann@30934
   694
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@35050
   695
    (@{thm add_0_left} :: @{thm add_0_right} :: @{thms add_ac}))
haftmann@25942
   696
haftmann@30934
   697
end)
haftmann@25942
   698
haftmann@30934
   699
in
haftmann@25942
   700
wenzelm@32010
   701
val cancel_div_mod_nat_proc = Simplifier.simproc @{theory}
haftmann@26100
   702
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
haftmann@25942
   703
haftmann@30934
   704
val _ = Addsimprocs [cancel_div_mod_nat_proc];
haftmann@30934
   705
haftmann@30934
   706
end
haftmann@25942
   707
*}
haftmann@25942
   708
haftmann@26100
   709
haftmann@26100
   710
subsubsection {* Quotient *}
haftmann@26100
   711
haftmann@26100
   712
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   713
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   714
haftmann@26100
   715
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   716
by (simp add: div_geq)
haftmann@26100
   717
haftmann@26100
   718
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   719
by simp
haftmann@26100
   720
haftmann@26100
   721
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   722
by simp
haftmann@26100
   723
haftmann@25942
   724
haftmann@25942
   725
subsubsection {* Remainder *}
haftmann@25942
   726
haftmann@26100
   727
lemma mod_less_divisor [simp]:
haftmann@26100
   728
  fixes m n :: nat
haftmann@26100
   729
  assumes "n > 0"
haftmann@26100
   730
  shows "m mod n < (n::nat)"
haftmann@33340
   731
  using assms divmod_nat_rel [of m n] unfolding divmod_nat_rel_def by auto
paulson@14267
   732
haftmann@26100
   733
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   734
  fixes m n :: nat
haftmann@26100
   735
  shows "m mod n \<le> m"
haftmann@26100
   736
proof (rule add_leD2)
haftmann@26100
   737
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   738
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   739
qed
haftmann@26100
   740
haftmann@26100
   741
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
   742
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   743
haftmann@26100
   744
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
   745
by (simp add: le_mod_geq)
haftmann@26100
   746
paulson@14267
   747
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
   748
by (induct m) (simp_all add: mod_geq)
paulson@14267
   749
haftmann@26100
   750
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
wenzelm@22718
   751
  apply (cases "n = 0", simp)
wenzelm@22718
   752
  apply (cases "k = 0", simp)
wenzelm@22718
   753
  apply (induct m rule: nat_less_induct)
wenzelm@22718
   754
  apply (subst mod_if, simp)
wenzelm@22718
   755
  apply (simp add: mod_geq diff_mult_distrib)
wenzelm@22718
   756
  done
paulson@14267
   757
paulson@14267
   758
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
nipkow@29667
   759
by (simp add: mult_commute [of k] mod_mult_distrib)
paulson@14267
   760
paulson@14267
   761
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   762
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
nipkow@29667
   763
by (cut_tac a = m and b = n in mod_div_equality2, arith)
paulson@14267
   764
nipkow@15439
   765
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   766
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   767
  apply simp
wenzelm@22718
   768
  done
paulson@14267
   769
haftmann@26100
   770
subsubsection {* Quotient and Remainder *}
paulson@14267
   771
haftmann@33340
   772
lemma divmod_nat_rel_mult1_eq:
haftmann@33340
   773
  "divmod_nat_rel b c (q, r) \<Longrightarrow> c > 0
haftmann@33340
   774
   \<Longrightarrow> divmod_nat_rel (a * b) c (a * q + a * r div c, a * r mod c)"
haftmann@33340
   775
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
   776
haftmann@30923
   777
lemma div_mult1_eq:
haftmann@30923
   778
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
nipkow@25134
   779
apply (cases "c = 0", simp)
haftmann@33340
   780
apply (blast intro: divmod_nat_rel [THEN divmod_nat_rel_mult1_eq, THEN div_eq])
nipkow@25134
   781
done
paulson@14267
   782
haftmann@33340
   783
lemma divmod_nat_rel_add1_eq:
haftmann@33340
   784
  "divmod_nat_rel a c (aq, ar) \<Longrightarrow> divmod_nat_rel b c (bq, br) \<Longrightarrow>  c > 0
haftmann@33340
   785
   \<Longrightarrow> divmod_nat_rel (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@33340
   786
by (auto simp add: split_ifs divmod_nat_rel_def algebra_simps)
paulson@14267
   787
paulson@14267
   788
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   789
lemma div_add1_eq:
nipkow@25134
   790
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
nipkow@25134
   791
apply (cases "c = 0", simp)
haftmann@33340
   792
apply (blast intro: divmod_nat_rel_add1_eq [THEN div_eq] divmod_nat_rel)
nipkow@25134
   793
done
paulson@14267
   794
paulson@14267
   795
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   796
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   797
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   798
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   799
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   800
  done
paulson@10559
   801
haftmann@33340
   802
lemma divmod_nat_rel_mult2_eq:
haftmann@33340
   803
  "divmod_nat_rel a b (q, r) \<Longrightarrow> 0 < b \<Longrightarrow> 0 < c
haftmann@33340
   804
   \<Longrightarrow> divmod_nat_rel a (b * c) (q div c, b *(q mod c) + r)"
haftmann@33340
   805
by (auto simp add: mult_ac divmod_nat_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   806
paulson@14267
   807
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
wenzelm@22718
   808
  apply (cases "b = 0", simp)
wenzelm@22718
   809
  apply (cases "c = 0", simp)
haftmann@33340
   810
  apply (force simp add: divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN div_eq])
wenzelm@22718
   811
  done
paulson@14267
   812
paulson@14267
   813
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
wenzelm@22718
   814
  apply (cases "b = 0", simp)
wenzelm@22718
   815
  apply (cases "c = 0", simp)
haftmann@33340
   816
  apply (auto simp add: mult_commute divmod_nat_rel [THEN divmod_nat_rel_mult2_eq, THEN mod_eq])
wenzelm@22718
   817
  done
paulson@14267
   818
paulson@14267
   819
haftmann@25942
   820
subsubsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   821
paulson@14267
   822
lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667
   823
by (induct m) (simp_all add: div_geq)
paulson@14267
   824
paulson@14267
   825
paulson@14267
   826
(* Monotonicity of div in first argument *)
haftmann@30923
   827
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   828
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   829
apply (case_tac "k=0", simp)
paulson@15251
   830
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   831
apply (case_tac "n<k")
paulson@14267
   832
(* 1  case n<k *)
paulson@14267
   833
apply simp
paulson@14267
   834
(* 2  case n >= k *)
paulson@14267
   835
apply (case_tac "m<k")
paulson@14267
   836
(* 2.1  case m<k *)
paulson@14267
   837
apply simp
paulson@14267
   838
(* 2.2  case m>=k *)
nipkow@15439
   839
apply (simp add: div_geq diff_le_mono)
paulson@14267
   840
done
paulson@14267
   841
paulson@14267
   842
(* Antimonotonicity of div in second argument *)
paulson@14267
   843
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   844
apply (subgoal_tac "0<n")
wenzelm@22718
   845
 prefer 2 apply simp
paulson@15251
   846
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   847
apply (rename_tac "k")
paulson@14267
   848
apply (case_tac "k<n", simp)
paulson@14267
   849
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   850
 prefer 2 apply simp
paulson@14267
   851
apply (simp add: div_geq)
paulson@15251
   852
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   853
 prefer 2
paulson@14267
   854
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   855
apply (rule le_trans, simp)
nipkow@15439
   856
apply (simp)
paulson@14267
   857
done
paulson@14267
   858
paulson@14267
   859
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   860
apply (case_tac "n=0", simp)
paulson@14267
   861
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   862
apply (rule div_le_mono2)
paulson@14267
   863
apply (simp_all (no_asm_simp))
paulson@14267
   864
done
paulson@14267
   865
wenzelm@22718
   866
(* Similar for "less than" *)
paulson@17085
   867
lemma div_less_dividend [rule_format]:
paulson@14267
   868
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   869
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   870
apply (rename_tac "m")
paulson@14267
   871
apply (case_tac "m<n", simp)
paulson@14267
   872
apply (subgoal_tac "0<n")
wenzelm@22718
   873
 prefer 2 apply simp
paulson@14267
   874
apply (simp add: div_geq)
paulson@14267
   875
apply (case_tac "n<m")
paulson@15251
   876
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   877
  apply (rule impI less_trans_Suc)+
paulson@14267
   878
apply assumption
nipkow@15439
   879
  apply (simp_all)
paulson@14267
   880
done
paulson@14267
   881
paulson@17085
   882
declare div_less_dividend [simp]
paulson@17085
   883
paulson@14267
   884
text{*A fact for the mutilated chess board*}
paulson@14267
   885
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   886
apply (case_tac "n=0", simp)
paulson@15251
   887
apply (induct "m" rule: nat_less_induct)
paulson@14267
   888
apply (case_tac "Suc (na) <n")
paulson@14267
   889
(* case Suc(na) < n *)
paulson@14267
   890
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   891
(* case n \<le> Suc(na) *)
paulson@16796
   892
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   893
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   894
done
paulson@14267
   895
paulson@14267
   896
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
   897
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   898
wenzelm@22718
   899
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   900
paulson@14267
   901
(*Loses information, namely we also have r<d provided d is nonzero*)
paulson@14267
   902
lemma mod_eqD: "(m mod d = r) ==> \<exists>q::nat. m = r + q*d"
haftmann@27651
   903
  apply (cut_tac a = m in mod_div_equality)
wenzelm@22718
   904
  apply (simp only: add_ac)
wenzelm@22718
   905
  apply (blast intro: sym)
wenzelm@22718
   906
  done
paulson@14267
   907
nipkow@13152
   908
lemma split_div:
nipkow@13189
   909
 "P(n div k :: nat) =
nipkow@13189
   910
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   911
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   912
proof
nipkow@13189
   913
  assume P: ?P
nipkow@13189
   914
  show ?Q
nipkow@13189
   915
  proof (cases)
nipkow@13189
   916
    assume "k = 0"
haftmann@27651
   917
    with P show ?Q by simp
nipkow@13189
   918
  next
nipkow@13189
   919
    assume not0: "k \<noteq> 0"
nipkow@13189
   920
    thus ?Q
nipkow@13189
   921
    proof (simp, intro allI impI)
nipkow@13189
   922
      fix i j
nipkow@13189
   923
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   924
      show "P i"
nipkow@13189
   925
      proof (cases)
wenzelm@22718
   926
        assume "i = 0"
wenzelm@22718
   927
        with n j P show "P i" by simp
nipkow@13189
   928
      next
wenzelm@22718
   929
        assume "i \<noteq> 0"
wenzelm@22718
   930
        with not0 n j P show "P i" by(simp add:add_ac)
nipkow@13189
   931
      qed
nipkow@13189
   932
    qed
nipkow@13189
   933
  qed
nipkow@13189
   934
next
nipkow@13189
   935
  assume Q: ?Q
nipkow@13189
   936
  show ?P
nipkow@13189
   937
  proof (cases)
nipkow@13189
   938
    assume "k = 0"
haftmann@27651
   939
    with Q show ?P by simp
nipkow@13189
   940
  next
nipkow@13189
   941
    assume not0: "k \<noteq> 0"
nipkow@13189
   942
    with Q have R: ?R by simp
nipkow@13189
   943
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   944
    show ?P by simp
nipkow@13189
   945
  qed
nipkow@13189
   946
qed
nipkow@13189
   947
berghofe@13882
   948
lemma split_div_lemma:
haftmann@26100
   949
  assumes "0 < n"
haftmann@26100
   950
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m\<Colon>nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
   951
proof
haftmann@26100
   952
  assume ?rhs
haftmann@26100
   953
  with mult_div_cancel have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
   954
  then have A: "n * q \<le> m" by simp
haftmann@26100
   955
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
   956
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
   957
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
   958
  with nq have "m < n + n * q" by simp
haftmann@26100
   959
  then have B: "m < n * Suc q" by simp
haftmann@26100
   960
  from A B show ?lhs ..
haftmann@26100
   961
next
haftmann@26100
   962
  assume P: ?lhs
haftmann@33340
   963
  then have "divmod_nat_rel m n (q, m - n * q)"
haftmann@33340
   964
    unfolding divmod_nat_rel_def by (auto simp add: mult_ac)
haftmann@33340
   965
  with divmod_nat_rel_unique divmod_nat_rel [of m n]
haftmann@30923
   966
  have "(q, m - n * q) = (m div n, m mod n)" by auto
haftmann@30923
   967
  then show ?rhs by simp
haftmann@26100
   968
qed
berghofe@13882
   969
berghofe@13882
   970
theorem split_div':
berghofe@13882
   971
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   972
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   973
  apply (case_tac "0 < n")
berghofe@13882
   974
  apply (simp only: add: split_div_lemma)
haftmann@27651
   975
  apply simp_all
berghofe@13882
   976
  done
berghofe@13882
   977
nipkow@13189
   978
lemma split_mod:
nipkow@13189
   979
 "P(n mod k :: nat) =
nipkow@13189
   980
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   981
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   982
proof
nipkow@13189
   983
  assume P: ?P
nipkow@13189
   984
  show ?Q
nipkow@13189
   985
  proof (cases)
nipkow@13189
   986
    assume "k = 0"
haftmann@27651
   987
    with P show ?Q by simp
nipkow@13189
   988
  next
nipkow@13189
   989
    assume not0: "k \<noteq> 0"
nipkow@13189
   990
    thus ?Q
nipkow@13189
   991
    proof (simp, intro allI impI)
nipkow@13189
   992
      fix i j
nipkow@13189
   993
      assume "n = k*i + j" "j < k"
nipkow@13189
   994
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   995
    qed
nipkow@13189
   996
  qed
nipkow@13189
   997
next
nipkow@13189
   998
  assume Q: ?Q
nipkow@13189
   999
  show ?P
nipkow@13189
  1000
  proof (cases)
nipkow@13189
  1001
    assume "k = 0"
haftmann@27651
  1002
    with Q show ?P by simp
nipkow@13189
  1003
  next
nipkow@13189
  1004
    assume not0: "k \<noteq> 0"
nipkow@13189
  1005
    with Q have R: ?R by simp
nipkow@13189
  1006
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1007
    show ?P by simp
nipkow@13189
  1008
  qed
nipkow@13189
  1009
qed
nipkow@13189
  1010
berghofe@13882
  1011
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
  1012
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
  1013
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
  1014
  apply arith
berghofe@13882
  1015
  done
berghofe@13882
  1016
haftmann@22800
  1017
lemma div_mod_equality':
haftmann@22800
  1018
  fixes m n :: nat
haftmann@22800
  1019
  shows "m div n * n = m - m mod n"
haftmann@22800
  1020
proof -
haftmann@22800
  1021
  have "m mod n \<le> m mod n" ..
haftmann@22800
  1022
  from div_mod_equality have 
haftmann@22800
  1023
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
haftmann@22800
  1024
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
haftmann@22800
  1025
    "m div n * n + (m mod n - m mod n) = m - m mod n"
haftmann@22800
  1026
    by simp
haftmann@22800
  1027
  then show ?thesis by simp
haftmann@22800
  1028
qed
haftmann@22800
  1029
haftmann@22800
  1030
haftmann@25942
  1031
subsubsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
  1032
paulson@14640
  1033
lemma mod_induct_0:
paulson@14640
  1034
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1035
  and base: "P i" and i: "i<p"
paulson@14640
  1036
  shows "P 0"
paulson@14640
  1037
proof (rule ccontr)
paulson@14640
  1038
  assume contra: "\<not>(P 0)"
paulson@14640
  1039
  from i have p: "0<p" by simp
paulson@14640
  1040
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1041
  proof
paulson@14640
  1042
    fix k
paulson@14640
  1043
    show "?A k"
paulson@14640
  1044
    proof (induct k)
paulson@14640
  1045
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1046
    next
paulson@14640
  1047
      fix n
paulson@14640
  1048
      assume ih: "?A n"
paulson@14640
  1049
      show "?A (Suc n)"
paulson@14640
  1050
      proof (clarsimp)
wenzelm@22718
  1051
        assume y: "P (p - Suc n)"
wenzelm@22718
  1052
        have n: "Suc n < p"
wenzelm@22718
  1053
        proof (rule ccontr)
wenzelm@22718
  1054
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1055
          hence "p - Suc n = 0"
wenzelm@22718
  1056
            by simp
wenzelm@22718
  1057
          with y contra show "False"
wenzelm@22718
  1058
            by simp
wenzelm@22718
  1059
        qed
wenzelm@22718
  1060
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1061
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1062
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1063
          by blast
wenzelm@22718
  1064
        show "False"
wenzelm@22718
  1065
        proof (cases "n=0")
wenzelm@22718
  1066
          case True
wenzelm@22718
  1067
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1068
        next
wenzelm@22718
  1069
          case False
wenzelm@22718
  1070
          with p have "p-n < p" by arith
wenzelm@22718
  1071
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1072
        qed
paulson@14640
  1073
      qed
paulson@14640
  1074
    qed
paulson@14640
  1075
  qed
paulson@14640
  1076
  moreover
paulson@14640
  1077
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1078
    by (blast dest: less_imp_add_positive)
paulson@14640
  1079
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1080
  moreover
paulson@14640
  1081
  note base
paulson@14640
  1082
  ultimately
paulson@14640
  1083
  show "False" by blast
paulson@14640
  1084
qed
paulson@14640
  1085
paulson@14640
  1086
lemma mod_induct:
paulson@14640
  1087
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1088
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1089
  shows "P j"
paulson@14640
  1090
proof -
paulson@14640
  1091
  have "\<forall>j<p. P j"
paulson@14640
  1092
  proof
paulson@14640
  1093
    fix j
paulson@14640
  1094
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1095
    proof (induct j)
paulson@14640
  1096
      from step base i show "?A 0"
wenzelm@22718
  1097
        by (auto elim: mod_induct_0)
paulson@14640
  1098
    next
paulson@14640
  1099
      fix k
paulson@14640
  1100
      assume ih: "?A k"
paulson@14640
  1101
      show "?A (Suc k)"
paulson@14640
  1102
      proof
wenzelm@22718
  1103
        assume suc: "Suc k < p"
wenzelm@22718
  1104
        hence k: "k<p" by simp
wenzelm@22718
  1105
        with ih have "P k" ..
wenzelm@22718
  1106
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1107
          by blast
wenzelm@22718
  1108
        moreover
wenzelm@22718
  1109
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1110
          by simp
wenzelm@22718
  1111
        ultimately
wenzelm@22718
  1112
        show "P (Suc k)" by simp
paulson@14640
  1113
      qed
paulson@14640
  1114
    qed
paulson@14640
  1115
  qed
paulson@14640
  1116
  with j show ?thesis by blast
paulson@14640
  1117
qed
paulson@14640
  1118
haftmann@33296
  1119
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
haftmann@33296
  1120
by (auto simp add: numeral_2_eq_2 le_div_geq)
haftmann@33296
  1121
haftmann@33296
  1122
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
haftmann@33296
  1123
by (simp add: nat_mult_2 [symmetric])
haftmann@33296
  1124
haftmann@33296
  1125
lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2"
haftmann@33296
  1126
apply (subgoal_tac "m mod 2 < 2")
haftmann@33296
  1127
apply (erule less_2_cases [THEN disjE])
huffman@35216
  1128
apply (simp_all (no_asm_simp) add: Let_def mod_Suc)
haftmann@33296
  1129
done
haftmann@33296
  1130
haftmann@33296
  1131
lemma mod2_gr_0 [simp]: "0 < (m\<Colon>nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1132
proof -
boehmes@35815
  1133
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1134
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1135
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1136
  then show ?thesis by auto
haftmann@33296
  1137
qed
haftmann@33296
  1138
haftmann@33296
  1139
text{*These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1140
    at least three Sucs, since two and fewer are rewritten back to Suc again!
haftmann@33296
  1141
    We already have some rules to simplify operands smaller than 3.*}
haftmann@33296
  1142
haftmann@33296
  1143
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1144
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1145
haftmann@33296
  1146
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1147
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1148
haftmann@33296
  1149
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1150
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1151
haftmann@33296
  1152
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1153
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1154
haftmann@33296
  1155
lemmas Suc_div_eq_add3_div_number_of =
haftmann@33296
  1156
    Suc_div_eq_add3_div [of _ "number_of v", standard]
haftmann@33296
  1157
declare Suc_div_eq_add3_div_number_of [simp]
haftmann@33296
  1158
haftmann@33296
  1159
lemmas Suc_mod_eq_add3_mod_number_of =
haftmann@33296
  1160
    Suc_mod_eq_add3_mod [of _ "number_of v", standard]
haftmann@33296
  1161
declare Suc_mod_eq_add3_mod_number_of [simp]
haftmann@33296
  1162
haftmann@33361
  1163
haftmann@33361
  1164
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1" 
haftmann@33361
  1165
apply (induct "m")
haftmann@33361
  1166
apply (simp_all add: mod_Suc)
haftmann@33361
  1167
done
haftmann@33361
  1168
haftmann@33361
  1169
declare Suc_times_mod_eq [of "number_of w", standard, simp]
haftmann@33361
  1170
haftmann@33361
  1171
lemma [simp]: "n div k \<le> (Suc n) div k"
haftmann@33361
  1172
by (simp add: div_le_mono) 
haftmann@33361
  1173
haftmann@33361
  1174
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1175
by (cases n) simp_all
haftmann@33361
  1176
boehmes@35815
  1177
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1178
proof -
boehmes@35815
  1179
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
boehmes@35815
  1180
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp 
boehmes@35815
  1181
qed
haftmann@33361
  1182
haftmann@33361
  1183
  (* Potential use of algebra : Equality modulo n*)
haftmann@33361
  1184
lemma mod_mult_self3 [simp]: "(k*n + m) mod n = m mod (n::nat)"
haftmann@33361
  1185
by (simp add: mult_ac add_ac)
haftmann@33361
  1186
haftmann@33361
  1187
lemma mod_mult_self4 [simp]: "Suc (k*n + m) mod n = Suc m mod n"
haftmann@33361
  1188
proof -
haftmann@33361
  1189
  have "Suc (k * n + m) mod n = (k * n + Suc m) mod n" by simp
haftmann@33361
  1190
  also have "... = Suc m mod n" by (rule mod_mult_self3) 
haftmann@33361
  1191
  finally show ?thesis .
haftmann@33361
  1192
qed
haftmann@33361
  1193
haftmann@33361
  1194
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
haftmann@33361
  1195
apply (subst mod_Suc [of m]) 
haftmann@33361
  1196
apply (subst mod_Suc [of "m mod n"], simp) 
haftmann@33361
  1197
done
haftmann@33361
  1198
haftmann@33361
  1199
haftmann@33361
  1200
subsection {* Division on @{typ int} *}
haftmann@33361
  1201
haftmann@33361
  1202
definition divmod_int_rel :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool" where
haftmann@33361
  1203
    --{*definition of quotient and remainder*}
haftmann@33361
  1204
    [code]: "divmod_int_rel a b = (\<lambda>(q, r). a = b * q + r \<and>
haftmann@33361
  1205
               (if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0))"
haftmann@33361
  1206
haftmann@33361
  1207
definition adjust :: "int \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1208
    --{*for the division algorithm*}
haftmann@33361
  1209
    [code]: "adjust b = (\<lambda>(q, r). if 0 \<le> r - b then (2 * q + 1, r - b)
haftmann@33361
  1210
                         else (2 * q, r))"
haftmann@33361
  1211
haftmann@33361
  1212
text{*algorithm for the case @{text "a\<ge>0, b>0"}*}
haftmann@33361
  1213
function posDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1214
  "posDivAlg a b = (if a < b \<or>  b \<le> 0 then (0, a)
haftmann@33361
  1215
     else adjust b (posDivAlg a (2 * b)))"
haftmann@33361
  1216
by auto
haftmann@33361
  1217
termination by (relation "measure (\<lambda>(a, b). nat (a - b + 1))")
haftmann@33361
  1218
  (auto simp add: mult_2)
haftmann@33361
  1219
haftmann@33361
  1220
text{*algorithm for the case @{text "a<0, b>0"}*}
haftmann@33361
  1221
function negDivAlg :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1222
  "negDivAlg a b = (if 0 \<le>a + b \<or> b \<le> 0  then (-1, a + b)
haftmann@33361
  1223
     else adjust b (negDivAlg a (2 * b)))"
haftmann@33361
  1224
by auto
haftmann@33361
  1225
termination by (relation "measure (\<lambda>(a, b). nat (- a - b))")
haftmann@33361
  1226
  (auto simp add: mult_2)
haftmann@33361
  1227
haftmann@33361
  1228
text{*algorithm for the general case @{term "b\<noteq>0"}*}
haftmann@33361
  1229
definition negateSnd :: "int \<times> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1230
  [code_unfold]: "negateSnd = apsnd uminus"
haftmann@33361
  1231
haftmann@33361
  1232
definition divmod_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  1233
    --{*The full division algorithm considers all possible signs for a, b
haftmann@33361
  1234
       including the special case @{text "a=0, b<0"} because 
haftmann@33361
  1235
       @{term negDivAlg} requires @{term "a<0"}.*}
haftmann@33361
  1236
  "divmod_int a b = (if 0 \<le> a then if 0 \<le> b then posDivAlg a b
haftmann@33361
  1237
                  else if a = 0 then (0, 0)
haftmann@33361
  1238
                       else negateSnd (negDivAlg (-a) (-b))
haftmann@33361
  1239
               else 
haftmann@33361
  1240
                  if 0 < b then negDivAlg a b
haftmann@33361
  1241
                  else negateSnd (posDivAlg (-a) (-b)))"
haftmann@33361
  1242
haftmann@33361
  1243
instantiation int :: Divides.div
haftmann@33361
  1244
begin
haftmann@33361
  1245
haftmann@33361
  1246
definition
haftmann@33361
  1247
  "a div b = fst (divmod_int a b)"
haftmann@33361
  1248
haftmann@33361
  1249
definition
haftmann@33361
  1250
 "a mod b = snd (divmod_int a b)"
haftmann@33361
  1251
haftmann@33361
  1252
instance ..
haftmann@33361
  1253
paulson@3366
  1254
end
haftmann@33361
  1255
haftmann@33361
  1256
lemma divmod_int_mod_div:
haftmann@33361
  1257
  "divmod_int p q = (p div q, p mod q)"
haftmann@33361
  1258
  by (auto simp add: div_int_def mod_int_def)
haftmann@33361
  1259
haftmann@33361
  1260
text{*
haftmann@33361
  1261
Here is the division algorithm in ML:
haftmann@33361
  1262
haftmann@33361
  1263
\begin{verbatim}
haftmann@33361
  1264
    fun posDivAlg (a,b) =
haftmann@33361
  1265
      if a<b then (0,a)
haftmann@33361
  1266
      else let val (q,r) = posDivAlg(a, 2*b)
haftmann@33361
  1267
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1268
           end
haftmann@33361
  1269
haftmann@33361
  1270
    fun negDivAlg (a,b) =
haftmann@33361
  1271
      if 0\<le>a+b then (~1,a+b)
haftmann@33361
  1272
      else let val (q,r) = negDivAlg(a, 2*b)
haftmann@33361
  1273
               in  if 0\<le>r-b then (2*q+1, r-b) else (2*q, r)
haftmann@33361
  1274
           end;
haftmann@33361
  1275
haftmann@33361
  1276
    fun negateSnd (q,r:int) = (q,~r);
haftmann@33361
  1277
haftmann@33361
  1278
    fun divmod (a,b) = if 0\<le>a then 
haftmann@33361
  1279
                          if b>0 then posDivAlg (a,b) 
haftmann@33361
  1280
                           else if a=0 then (0,0)
haftmann@33361
  1281
                                else negateSnd (negDivAlg (~a,~b))
haftmann@33361
  1282
                       else 
haftmann@33361
  1283
                          if 0<b then negDivAlg (a,b)
haftmann@33361
  1284
                          else        negateSnd (posDivAlg (~a,~b));
haftmann@33361
  1285
\end{verbatim}
haftmann@33361
  1286
*}
haftmann@33361
  1287
haftmann@33361
  1288
haftmann@33361
  1289
haftmann@33361
  1290
subsubsection{*Uniqueness and Monotonicity of Quotients and Remainders*}
haftmann@33361
  1291
haftmann@33361
  1292
lemma unique_quotient_lemma:
haftmann@33361
  1293
     "[| b*q' + r'  \<le> b*q + r;  0 \<le> r';  r' < b;  r < b |]  
haftmann@33361
  1294
      ==> q' \<le> (q::int)"
haftmann@33361
  1295
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1296
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1297
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1298
apply (erule_tac [2] order_le_less_trans)
haftmann@33361
  1299
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
haftmann@33361
  1300
apply (subgoal_tac "b * q' < b * (1 + q) ")
haftmann@33361
  1301
 prefer 2 apply (simp add: right_diff_distrib right_distrib)
haftmann@33361
  1302
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1303
done
haftmann@33361
  1304
haftmann@33361
  1305
lemma unique_quotient_lemma_neg:
haftmann@33361
  1306
     "[| b*q' + r' \<le> b*q + r;  r \<le> 0;  b < r;  b < r' |]  
haftmann@33361
  1307
      ==> q \<le> (q'::int)"
haftmann@33361
  1308
by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma, 
haftmann@33361
  1309
    auto)
haftmann@33361
  1310
haftmann@33361
  1311
lemma unique_quotient:
haftmann@33361
  1312
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
haftmann@33361
  1313
      ==> q = q'"
haftmann@33361
  1314
apply (simp add: divmod_int_rel_def linorder_neq_iff split: split_if_asm)
haftmann@33361
  1315
apply (blast intro: order_antisym
haftmann@33361
  1316
             dest: order_eq_refl [THEN unique_quotient_lemma] 
haftmann@33361
  1317
             order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@33361
  1318
done
haftmann@33361
  1319
haftmann@33361
  1320
haftmann@33361
  1321
lemma unique_remainder:
haftmann@33361
  1322
     "[| divmod_int_rel a b (q, r); divmod_int_rel a b (q', r');  b \<noteq> 0 |]  
haftmann@33361
  1323
      ==> r = r'"
haftmann@33361
  1324
apply (subgoal_tac "q = q'")
haftmann@33361
  1325
 apply (simp add: divmod_int_rel_def)
haftmann@33361
  1326
apply (blast intro: unique_quotient)
haftmann@33361
  1327
done
haftmann@33361
  1328
haftmann@33361
  1329
haftmann@33361
  1330
subsubsection{*Correctness of @{term posDivAlg}, the Algorithm for Non-Negative Dividends*}
haftmann@33361
  1331
haftmann@33361
  1332
text{*And positive divisors*}
haftmann@33361
  1333
haftmann@33361
  1334
lemma adjust_eq [simp]:
haftmann@33361
  1335
     "adjust b (q,r) = 
haftmann@33361
  1336
      (let diff = r-b in  
haftmann@33361
  1337
        if 0 \<le> diff then (2*q + 1, diff)   
haftmann@33361
  1338
                     else (2*q, r))"
haftmann@33361
  1339
by (simp add: Let_def adjust_def)
haftmann@33361
  1340
haftmann@33361
  1341
declare posDivAlg.simps [simp del]
haftmann@33361
  1342
haftmann@33361
  1343
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1344
lemma posDivAlg_eqn:
haftmann@33361
  1345
     "0 < b ==>  
haftmann@33361
  1346
      posDivAlg a b = (if a<b then (0,a) else adjust b (posDivAlg a (2*b)))"
haftmann@33361
  1347
by (rule posDivAlg.simps [THEN trans], simp)
haftmann@33361
  1348
haftmann@33361
  1349
text{*Correctness of @{term posDivAlg}: it computes quotients correctly*}
haftmann@33361
  1350
theorem posDivAlg_correct:
haftmann@33361
  1351
  assumes "0 \<le> a" and "0 < b"
haftmann@33361
  1352
  shows "divmod_int_rel a b (posDivAlg a b)"
haftmann@33361
  1353
using prems apply (induct a b rule: posDivAlg.induct)
haftmann@33361
  1354
apply auto
haftmann@33361
  1355
apply (simp add: divmod_int_rel_def)
haftmann@33361
  1356
apply (subst posDivAlg_eqn, simp add: right_distrib)
haftmann@33361
  1357
apply (case_tac "a < b")
haftmann@33361
  1358
apply simp_all
haftmann@33361
  1359
apply (erule splitE)
haftmann@33361
  1360
apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
haftmann@33361
  1361
done
haftmann@33361
  1362
haftmann@33361
  1363
haftmann@33361
  1364
subsubsection{*Correctness of @{term negDivAlg}, the Algorithm for Negative Dividends*}
haftmann@33361
  1365
haftmann@33361
  1366
text{*And positive divisors*}
haftmann@33361
  1367
haftmann@33361
  1368
declare negDivAlg.simps [simp del]
haftmann@33361
  1369
haftmann@33361
  1370
text{*use with a simproc to avoid repeatedly proving the premise*}
haftmann@33361
  1371
lemma negDivAlg_eqn:
haftmann@33361
  1372
     "0 < b ==>  
haftmann@33361
  1373
      negDivAlg a b =       
haftmann@33361
  1374
       (if 0\<le>a+b then (-1,a+b) else adjust b (negDivAlg a (2*b)))"
haftmann@33361
  1375
by (rule negDivAlg.simps [THEN trans], simp)
haftmann@33361
  1376
haftmann@33361
  1377
(*Correctness of negDivAlg: it computes quotients correctly
haftmann@33361
  1378
  It doesn't work if a=0 because the 0/b equals 0, not -1*)
haftmann@33361
  1379
lemma negDivAlg_correct:
haftmann@33361
  1380
  assumes "a < 0" and "b > 0"
haftmann@33361
  1381
  shows "divmod_int_rel a b (negDivAlg a b)"
haftmann@33361
  1382
using prems apply (induct a b rule: negDivAlg.induct)
haftmann@33361
  1383
apply (auto simp add: linorder_not_le)
haftmann@33361
  1384
apply (simp add: divmod_int_rel_def)
haftmann@33361
  1385
apply (subst negDivAlg_eqn, assumption)
haftmann@33361
  1386
apply (case_tac "a + b < (0\<Colon>int)")
haftmann@33361
  1387
apply simp_all
haftmann@33361
  1388
apply (erule splitE)
haftmann@33361
  1389
apply (auto simp add: right_distrib Let_def mult_ac mult_2_right)
haftmann@33361
  1390
done
haftmann@33361
  1391
haftmann@33361
  1392
haftmann@33361
  1393
subsubsection{*Existence Shown by Proving the Division Algorithm to be Correct*}
haftmann@33361
  1394
haftmann@33361
  1395
(*the case a=0*)
haftmann@33361
  1396
lemma divmod_int_rel_0: "b \<noteq> 0 ==> divmod_int_rel 0 b (0, 0)"
haftmann@33361
  1397
by (auto simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1398
haftmann@33361
  1399
lemma posDivAlg_0 [simp]: "posDivAlg 0 b = (0, 0)"
haftmann@33361
  1400
by (subst posDivAlg.simps, auto)
haftmann@33361
  1401
haftmann@33361
  1402
lemma negDivAlg_minus1 [simp]: "negDivAlg -1 b = (-1, b - 1)"
haftmann@33361
  1403
by (subst negDivAlg.simps, auto)
haftmann@33361
  1404
haftmann@33361
  1405
lemma negateSnd_eq [simp]: "negateSnd(q,r) = (q,-r)"
haftmann@33361
  1406
by (simp add: negateSnd_def)
haftmann@33361
  1407
haftmann@33361
  1408
lemma divmod_int_rel_neg: "divmod_int_rel (-a) (-b) qr ==> divmod_int_rel a b (negateSnd qr)"
haftmann@33361
  1409
by (auto simp add: split_ifs divmod_int_rel_def)
haftmann@33361
  1410
haftmann@33361
  1411
lemma divmod_int_correct: "b \<noteq> 0 ==> divmod_int_rel a b (divmod_int a b)"
haftmann@33361
  1412
by (force simp add: linorder_neq_iff divmod_int_rel_0 divmod_int_def divmod_int_rel_neg
haftmann@33361
  1413
                    posDivAlg_correct negDivAlg_correct)
haftmann@33361
  1414
haftmann@33361
  1415
text{*Arbitrary definitions for division by zero.  Useful to simplify 
haftmann@33361
  1416
    certain equations.*}
haftmann@33361
  1417
haftmann@33361
  1418
lemma DIVISION_BY_ZERO [simp]: "a div (0::int) = 0 & a mod (0::int) = a"
haftmann@33361
  1419
by (simp add: div_int_def mod_int_def divmod_int_def posDivAlg.simps)  
haftmann@33361
  1420
haftmann@33361
  1421
haftmann@33361
  1422
text{*Basic laws about division and remainder*}
haftmann@33361
  1423
haftmann@33361
  1424
lemma zmod_zdiv_equality: "(a::int) = b * (a div b) + (a mod b)"
haftmann@33361
  1425
apply (case_tac "b = 0", simp)
haftmann@33361
  1426
apply (cut_tac a = a and b = b in divmod_int_correct)
haftmann@33361
  1427
apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
haftmann@33361
  1428
done
haftmann@33361
  1429
haftmann@33361
  1430
lemma zdiv_zmod_equality: "(b * (a div b) + (a mod b)) + k = (a::int)+k"
haftmann@33361
  1431
by(simp add: zmod_zdiv_equality[symmetric])
haftmann@33361
  1432
haftmann@33361
  1433
lemma zdiv_zmod_equality2: "((a div b) * b + (a mod b)) + k = (a::int)+k"
haftmann@33361
  1434
by(simp add: mult_commute zmod_zdiv_equality[symmetric])
haftmann@33361
  1435
haftmann@33361
  1436
text {* Tool setup *}
haftmann@33361
  1437
haftmann@33361
  1438
ML {*
haftmann@33361
  1439
local
haftmann@33361
  1440
haftmann@33361
  1441
structure CancelDivMod = CancelDivModFun(struct
haftmann@33361
  1442
haftmann@33361
  1443
  val div_name = @{const_name div};
haftmann@33361
  1444
  val mod_name = @{const_name mod};
haftmann@33361
  1445
  val mk_binop = HOLogic.mk_binop;
haftmann@33361
  1446
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@33361
  1447
  val dest_sum = Arith_Data.dest_sum;
haftmann@33361
  1448
haftmann@33361
  1449
  val div_mod_eqs = map mk_meta_eq [@{thm zdiv_zmod_equality}, @{thm zdiv_zmod_equality2}];
haftmann@33361
  1450
haftmann@33361
  1451
  val trans = trans;
haftmann@33361
  1452
haftmann@33361
  1453
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac 
haftmann@33361
  1454
    (@{thm diff_minus} :: @{thms add_0s} @ @{thms add_ac}))
haftmann@33361
  1455
haftmann@33361
  1456
end)
haftmann@33361
  1457
haftmann@33361
  1458
in
haftmann@33361
  1459
haftmann@33361
  1460
val cancel_div_mod_int_proc = Simplifier.simproc @{theory}
haftmann@33361
  1461
  "cancel_zdiv_zmod" ["(k::int) + l"] (K CancelDivMod.proc);
haftmann@33361
  1462
haftmann@33361
  1463
val _ = Addsimprocs [cancel_div_mod_int_proc];
haftmann@33361
  1464
haftmann@33361
  1465
end
haftmann@33361
  1466
*}
haftmann@33361
  1467
haftmann@33361
  1468
lemma pos_mod_conj : "(0::int) < b ==> 0 \<le> a mod b & a mod b < b"
haftmann@33361
  1469
apply (cut_tac a = a and b = b in divmod_int_correct)
haftmann@33361
  1470
apply (auto simp add: divmod_int_rel_def mod_int_def)
haftmann@33361
  1471
done
haftmann@33361
  1472
haftmann@33361
  1473
lemmas pos_mod_sign  [simp] = pos_mod_conj [THEN conjunct1, standard]
haftmann@33361
  1474
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2, standard]
haftmann@33361
  1475
haftmann@33361
  1476
lemma neg_mod_conj : "b < (0::int) ==> a mod b \<le> 0 & b < a mod b"
haftmann@33361
  1477
apply (cut_tac a = a and b = b in divmod_int_correct)
haftmann@33361
  1478
apply (auto simp add: divmod_int_rel_def div_int_def mod_int_def)
haftmann@33361
  1479
done
haftmann@33361
  1480
haftmann@33361
  1481
lemmas neg_mod_sign  [simp] = neg_mod_conj [THEN conjunct1, standard]
haftmann@33361
  1482
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2, standard]
haftmann@33361
  1483
haftmann@33361
  1484
haftmann@33361
  1485
haftmann@33361
  1486
subsubsection{*General Properties of div and mod*}
haftmann@33361
  1487
haftmann@33361
  1488
lemma divmod_int_rel_div_mod: "b \<noteq> 0 ==> divmod_int_rel a b (a div b, a mod b)"
haftmann@33361
  1489
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1490
apply (force simp add: divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1491
done
haftmann@33361
  1492
haftmann@33361
  1493
lemma divmod_int_rel_div: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a div b = q"
haftmann@33361
  1494
by (simp add: divmod_int_rel_div_mod [THEN unique_quotient])
haftmann@33361
  1495
haftmann@33361
  1496
lemma divmod_int_rel_mod: "[| divmod_int_rel a b (q, r);  b \<noteq> 0 |] ==> a mod b = r"
haftmann@33361
  1497
by (simp add: divmod_int_rel_div_mod [THEN unique_remainder])
haftmann@33361
  1498
haftmann@33361
  1499
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
haftmann@33361
  1500
apply (rule divmod_int_rel_div)
haftmann@33361
  1501
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1502
done
haftmann@33361
  1503
haftmann@33361
  1504
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
haftmann@33361
  1505
apply (rule divmod_int_rel_div)
haftmann@33361
  1506
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1507
done
haftmann@33361
  1508
haftmann@33361
  1509
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
haftmann@33361
  1510
apply (rule divmod_int_rel_div)
haftmann@33361
  1511
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1512
done
haftmann@33361
  1513
haftmann@33361
  1514
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1515
haftmann@33361
  1516
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
haftmann@33361
  1517
apply (rule_tac q = 0 in divmod_int_rel_mod)
haftmann@33361
  1518
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1519
done
haftmann@33361
  1520
haftmann@33361
  1521
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
haftmann@33361
  1522
apply (rule_tac q = 0 in divmod_int_rel_mod)
haftmann@33361
  1523
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1524
done
haftmann@33361
  1525
haftmann@33361
  1526
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
haftmann@33361
  1527
apply (rule_tac q = "-1" in divmod_int_rel_mod)
haftmann@33361
  1528
apply (auto simp add: divmod_int_rel_def)
haftmann@33361
  1529
done
haftmann@33361
  1530
haftmann@33361
  1531
text{*There is no @{text mod_neg_pos_trivial}.*}
haftmann@33361
  1532
haftmann@33361
  1533
haftmann@33361
  1534
(*Simpler laws such as -a div b = -(a div b) FAIL, but see just below*)
haftmann@33361
  1535
lemma zdiv_zminus_zminus [simp]: "(-a) div (-b) = a div (b::int)"
haftmann@33361
  1536
apply (case_tac "b = 0", simp)
haftmann@33361
  1537
apply (simp add: divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, 
haftmann@33361
  1538
                                 THEN divmod_int_rel_div, THEN sym])
haftmann@33361
  1539
haftmann@33361
  1540
done
haftmann@33361
  1541
haftmann@33361
  1542
(*Simpler laws such as -a mod b = -(a mod b) FAIL, but see just below*)
haftmann@33361
  1543
lemma zmod_zminus_zminus [simp]: "(-a) mod (-b) = - (a mod (b::int))"
haftmann@33361
  1544
apply (case_tac "b = 0", simp)
haftmann@33361
  1545
apply (subst divmod_int_rel_div_mod [THEN divmod_int_rel_neg, simplified, THEN divmod_int_rel_mod],
haftmann@33361
  1546
       auto)
haftmann@33361
  1547
done
haftmann@33361
  1548
haftmann@33361
  1549
haftmann@33361
  1550
subsubsection{*Laws for div and mod with Unary Minus*}
haftmann@33361
  1551
haftmann@33361
  1552
lemma zminus1_lemma:
haftmann@33361
  1553
     "divmod_int_rel a b (q, r)
haftmann@33361
  1554
      ==> divmod_int_rel (-a) b (if r=0 then -q else -q - 1,  
haftmann@33361
  1555
                          if r=0 then 0 else b-r)"
haftmann@33361
  1556
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_diff_distrib)
haftmann@33361
  1557
haftmann@33361
  1558
haftmann@33361
  1559
lemma zdiv_zminus1_eq_if:
haftmann@33361
  1560
     "b \<noteq> (0::int)  
haftmann@33361
  1561
      ==> (-a) div b =  
haftmann@33361
  1562
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@33361
  1563
by (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_div])
haftmann@33361
  1564
haftmann@33361
  1565
lemma zmod_zminus1_eq_if:
haftmann@33361
  1566
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1567
apply (case_tac "b = 0", simp)
haftmann@33361
  1568
apply (blast intro: divmod_int_rel_div_mod [THEN zminus1_lemma, THEN divmod_int_rel_mod])
haftmann@33361
  1569
done
haftmann@33361
  1570
haftmann@33361
  1571
lemma zmod_zminus1_not_zero:
haftmann@33361
  1572
  fixes k l :: int
haftmann@33361
  1573
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1574
  unfolding zmod_zminus1_eq_if by auto
haftmann@33361
  1575
haftmann@33361
  1576
lemma zdiv_zminus2: "a div (-b) = (-a::int) div b"
haftmann@33361
  1577
by (cut_tac a = "-a" in zdiv_zminus_zminus, auto)
haftmann@33361
  1578
haftmann@33361
  1579
lemma zmod_zminus2: "a mod (-b) = - ((-a::int) mod b)"
haftmann@33361
  1580
by (cut_tac a = "-a" and b = b in zmod_zminus_zminus, auto)
haftmann@33361
  1581
haftmann@33361
  1582
lemma zdiv_zminus2_eq_if:
haftmann@33361
  1583
     "b \<noteq> (0::int)  
haftmann@33361
  1584
      ==> a div (-b) =  
haftmann@33361
  1585
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@33361
  1586
by (simp add: zdiv_zminus1_eq_if zdiv_zminus2)
haftmann@33361
  1587
haftmann@33361
  1588
lemma zmod_zminus2_eq_if:
haftmann@33361
  1589
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
haftmann@33361
  1590
by (simp add: zmod_zminus1_eq_if zmod_zminus2)
haftmann@33361
  1591
haftmann@33361
  1592
lemma zmod_zminus2_not_zero:
haftmann@33361
  1593
  fixes k l :: int
haftmann@33361
  1594
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@33361
  1595
  unfolding zmod_zminus2_eq_if by auto 
haftmann@33361
  1596
haftmann@33361
  1597
haftmann@33361
  1598
subsubsection{*Division of a Number by Itself*}
haftmann@33361
  1599
haftmann@33361
  1600
lemma self_quotient_aux1: "[| (0::int) < a; a = r + a*q; r < a |] ==> 1 \<le> q"
haftmann@33361
  1601
apply (subgoal_tac "0 < a*q")
haftmann@33361
  1602
 apply (simp add: zero_less_mult_iff, arith)
haftmann@33361
  1603
done
haftmann@33361
  1604
haftmann@33361
  1605
lemma self_quotient_aux2: "[| (0::int) < a; a = r + a*q; 0 \<le> r |] ==> q \<le> 1"
haftmann@33361
  1606
apply (subgoal_tac "0 \<le> a* (1-q) ")
haftmann@33361
  1607
 apply (simp add: zero_le_mult_iff)
haftmann@33361
  1608
apply (simp add: right_diff_distrib)
haftmann@33361
  1609
done
haftmann@33361
  1610
haftmann@33361
  1611
lemma self_quotient: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> q = 1"
haftmann@33361
  1612
apply (simp add: split_ifs divmod_int_rel_def linorder_neq_iff)
haftmann@33361
  1613
apply (rule order_antisym, safe, simp_all)
haftmann@33361
  1614
apply (rule_tac [3] a = "-a" and r = "-r" in self_quotient_aux1)
haftmann@33361
  1615
apply (rule_tac a = "-a" and r = "-r" in self_quotient_aux2)
haftmann@33361
  1616
apply (force intro: self_quotient_aux1 self_quotient_aux2 simp add: add_commute)+
haftmann@33361
  1617
done
haftmann@33361
  1618
haftmann@33361
  1619
lemma self_remainder: "[| divmod_int_rel a a (q, r);  a \<noteq> (0::int) |] ==> r = 0"
haftmann@33361
  1620
apply (frule self_quotient, assumption)
haftmann@33361
  1621
apply (simp add: divmod_int_rel_def)
haftmann@33361
  1622
done
haftmann@33361
  1623
haftmann@33361
  1624
lemma zdiv_self [simp]: "a \<noteq> 0 ==> a div a = (1::int)"
haftmann@33361
  1625
by (simp add: divmod_int_rel_div_mod [THEN self_quotient])
haftmann@33361
  1626
haftmann@33361
  1627
(*Here we have 0 mod 0 = 0, also assumed by Knuth (who puts m mod 0 = 0) *)
haftmann@33361
  1628
lemma zmod_self [simp]: "a mod a = (0::int)"
haftmann@33361
  1629
apply (case_tac "a = 0", simp)
haftmann@33361
  1630
apply (simp add: divmod_int_rel_div_mod [THEN self_remainder])
haftmann@33361
  1631
done
haftmann@33361
  1632
haftmann@33361
  1633
haftmann@33361
  1634
subsubsection{*Computation of Division and Remainder*}
haftmann@33361
  1635
haftmann@33361
  1636
lemma zdiv_zero [simp]: "(0::int) div b = 0"
haftmann@33361
  1637
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1638
haftmann@33361
  1639
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@33361
  1640
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1641
haftmann@33361
  1642
lemma zmod_zero [simp]: "(0::int) mod b = 0"
haftmann@33361
  1643
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1644
haftmann@33361
  1645
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@33361
  1646
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1647
haftmann@33361
  1648
text{*a positive, b positive *}
haftmann@33361
  1649
haftmann@33361
  1650
lemma div_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a div b = fst (posDivAlg a b)"
haftmann@33361
  1651
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1652
haftmann@33361
  1653
lemma mod_pos_pos: "[| 0 < a;  0 \<le> b |] ==> a mod b = snd (posDivAlg a b)"
haftmann@33361
  1654
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1655
haftmann@33361
  1656
text{*a negative, b positive *}
haftmann@33361
  1657
haftmann@33361
  1658
lemma div_neg_pos: "[| a < 0;  0 < b |] ==> a div b = fst (negDivAlg a b)"
haftmann@33361
  1659
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1660
haftmann@33361
  1661
lemma mod_neg_pos: "[| a < 0;  0 < b |] ==> a mod b = snd (negDivAlg a b)"
haftmann@33361
  1662
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1663
haftmann@33361
  1664
text{*a positive, b negative *}
haftmann@33361
  1665
haftmann@33361
  1666
lemma div_pos_neg:
haftmann@33361
  1667
     "[| 0 < a;  b < 0 |] ==> a div b = fst (negateSnd (negDivAlg (-a) (-b)))"
haftmann@33361
  1668
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1669
haftmann@33361
  1670
lemma mod_pos_neg:
haftmann@33361
  1671
     "[| 0 < a;  b < 0 |] ==> a mod b = snd (negateSnd (negDivAlg (-a) (-b)))"
haftmann@33361
  1672
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1673
haftmann@33361
  1674
text{*a negative, b negative *}
haftmann@33361
  1675
haftmann@33361
  1676
lemma div_neg_neg:
haftmann@33361
  1677
     "[| a < 0;  b \<le> 0 |] ==> a div b = fst (negateSnd (posDivAlg (-a) (-b)))"
haftmann@33361
  1678
by (simp add: div_int_def divmod_int_def)
haftmann@33361
  1679
haftmann@33361
  1680
lemma mod_neg_neg:
haftmann@33361
  1681
     "[| a < 0;  b \<le> 0 |] ==> a mod b = snd (negateSnd (posDivAlg (-a) (-b)))"
haftmann@33361
  1682
by (simp add: mod_int_def divmod_int_def)
haftmann@33361
  1683
haftmann@33361
  1684
text {*Simplify expresions in which div and mod combine numerical constants*}
haftmann@33361
  1685
haftmann@33361
  1686
lemma divmod_int_relI:
haftmann@33361
  1687
  "\<lbrakk>a == b * q + r; if 0 < b then 0 \<le> r \<and> r < b else b < r \<and> r \<le> 0\<rbrakk>
haftmann@33361
  1688
    \<Longrightarrow> divmod_int_rel a b (q, r)"
haftmann@33361
  1689
  unfolding divmod_int_rel_def by simp
haftmann@33361
  1690
haftmann@33361
  1691
lemmas divmod_int_rel_div_eq = divmod_int_relI [THEN divmod_int_rel_div, THEN eq_reflection]
haftmann@33361
  1692
lemmas divmod_int_rel_mod_eq = divmod_int_relI [THEN divmod_int_rel_mod, THEN eq_reflection]
haftmann@33361
  1693
lemmas arithmetic_simps =
haftmann@33361
  1694
  arith_simps
haftmann@33361
  1695
  add_special
haftmann@35050
  1696
  add_0_left
haftmann@35050
  1697
  add_0_right
haftmann@33361
  1698
  mult_zero_left
haftmann@33361
  1699
  mult_zero_right
haftmann@33361
  1700
  mult_1_left
haftmann@33361
  1701
  mult_1_right
haftmann@33361
  1702
haftmann@33361
  1703
(* simprocs adapted from HOL/ex/Binary.thy *)
haftmann@33361
  1704
ML {*
haftmann@33361
  1705
local
haftmann@33361
  1706
  val mk_number = HOLogic.mk_number HOLogic.intT;
haftmann@33361
  1707
  fun mk_cert u k l = @{term "plus :: int \<Rightarrow> int \<Rightarrow> int"} $
haftmann@33361
  1708
    (@{term "times :: int \<Rightarrow> int \<Rightarrow> int"} $ u $ mk_number k) $
haftmann@33361
  1709
      mk_number l;
haftmann@33361
  1710
  fun prove ctxt prop = Goal.prove ctxt [] [] prop
haftmann@33361
  1711
    (K (ALLGOALS (full_simp_tac (HOL_basic_ss addsimps @{thms arithmetic_simps}))));
haftmann@33361
  1712
  fun binary_proc proc ss ct =
haftmann@33361
  1713
    (case Thm.term_of ct of
haftmann@33361
  1714
      _ $ t $ u =>
haftmann@33361
  1715
      (case try (pairself (`(snd o HOLogic.dest_number))) (t, u) of
haftmann@33361
  1716
        SOME args => proc (Simplifier.the_context ss) args
haftmann@33361
  1717
      | NONE => NONE)
haftmann@33361
  1718
    | _ => NONE);
haftmann@33361
  1719
in
haftmann@33361
  1720
  fun divmod_proc rule = binary_proc (fn ctxt => fn ((m, t), (n, u)) =>
haftmann@33361
  1721
    if n = 0 then NONE
haftmann@33361
  1722
    else let val (k, l) = Integer.div_mod m n;
haftmann@33361
  1723
    in SOME (rule OF [prove ctxt (Logic.mk_equals (t, mk_cert u k l))]) end);
haftmann@33361
  1724
end
haftmann@33361
  1725
*}
haftmann@33361
  1726
haftmann@33361
  1727
simproc_setup binary_int_div ("number_of m div number_of n :: int") =
haftmann@33361
  1728
  {* K (divmod_proc (@{thm divmod_int_rel_div_eq})) *}
haftmann@33361
  1729
haftmann@33361
  1730
simproc_setup binary_int_mod ("number_of m mod number_of n :: int") =
haftmann@33361
  1731
  {* K (divmod_proc (@{thm divmod_int_rel_mod_eq})) *}
haftmann@33361
  1732
haftmann@33361
  1733
lemmas posDivAlg_eqn_number_of [simp] =
haftmann@33361
  1734
    posDivAlg_eqn [of "number_of v" "number_of w", standard]
haftmann@33361
  1735
haftmann@33361
  1736
lemmas negDivAlg_eqn_number_of [simp] =
haftmann@33361
  1737
    negDivAlg_eqn [of "number_of v" "number_of w", standard]
haftmann@33361
  1738
haftmann@33361
  1739
haftmann@33361
  1740
text{*Special-case simplification *}
haftmann@33361
  1741
haftmann@33361
  1742
lemma zmod_minus1_right [simp]: "a mod (-1::int) = 0"
haftmann@33361
  1743
apply (cut_tac a = a and b = "-1" in neg_mod_sign)
haftmann@33361
  1744
apply (cut_tac [2] a = a and b = "-1" in neg_mod_bound)
haftmann@33361
  1745
apply (auto simp del: neg_mod_sign neg_mod_bound)
haftmann@33361
  1746
done
haftmann@33361
  1747
haftmann@33361
  1748
lemma zdiv_minus1_right [simp]: "a div (-1::int) = -a"
haftmann@33361
  1749
by (cut_tac a = a and b = "-1" in zmod_zdiv_equality, auto)
haftmann@33361
  1750
haftmann@33361
  1751
(** The last remaining special cases for constant arithmetic:
haftmann@33361
  1752
    1 div z and 1 mod z **)
haftmann@33361
  1753
haftmann@33361
  1754
lemmas div_pos_pos_1_number_of [simp] =
haftmann@33361
  1755
    div_pos_pos [OF int_0_less_1, of "number_of w", standard]
haftmann@33361
  1756
haftmann@33361
  1757
lemmas div_pos_neg_1_number_of [simp] =
haftmann@33361
  1758
    div_pos_neg [OF int_0_less_1, of "number_of w", standard]
haftmann@33361
  1759
haftmann@33361
  1760
lemmas mod_pos_pos_1_number_of [simp] =
haftmann@33361
  1761
    mod_pos_pos [OF int_0_less_1, of "number_of w", standard]
haftmann@33361
  1762
haftmann@33361
  1763
lemmas mod_pos_neg_1_number_of [simp] =
haftmann@33361
  1764
    mod_pos_neg [OF int_0_less_1, of "number_of w", standard]
haftmann@33361
  1765
haftmann@33361
  1766
haftmann@33361
  1767
lemmas posDivAlg_eqn_1_number_of [simp] =
haftmann@33361
  1768
    posDivAlg_eqn [of concl: 1 "number_of w", standard]
haftmann@33361
  1769
haftmann@33361
  1770
lemmas negDivAlg_eqn_1_number_of [simp] =
haftmann@33361
  1771
    negDivAlg_eqn [of concl: 1 "number_of w", standard]
haftmann@33361
  1772
haftmann@33361
  1773
haftmann@33361
  1774
haftmann@33361
  1775
subsubsection{*Monotonicity in the First Argument (Dividend)*}
haftmann@33361
  1776
haftmann@33361
  1777
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@33361
  1778
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1779
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1780
apply (rule unique_quotient_lemma)
haftmann@33361
  1781
apply (erule subst)
haftmann@33361
  1782
apply (erule subst, simp_all)
haftmann@33361
  1783
done
haftmann@33361
  1784
haftmann@33361
  1785
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@33361
  1786
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1787
apply (cut_tac a = a' and b = b in zmod_zdiv_equality)
haftmann@33361
  1788
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1789
apply (erule subst)
haftmann@33361
  1790
apply (erule subst, simp_all)
haftmann@33361
  1791
done
haftmann@33361
  1792
haftmann@33361
  1793
haftmann@33361
  1794
subsubsection{*Monotonicity in the Second Argument (Divisor)*}
haftmann@33361
  1795
haftmann@33361
  1796
lemma q_pos_lemma:
haftmann@33361
  1797
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1798
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1799
 apply (simp add: zero_less_mult_iff)
haftmann@33361
  1800
apply (simp add: right_distrib)
haftmann@33361
  1801
done
haftmann@33361
  1802
haftmann@33361
  1803
lemma zdiv_mono2_lemma:
haftmann@33361
  1804
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';   
haftmann@33361
  1805
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]   
haftmann@33361
  1806
      ==> q \<le> (q'::int)"
haftmann@33361
  1807
apply (frule q_pos_lemma, assumption+) 
haftmann@33361
  1808
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  1809
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1810
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  1811
 prefer 2 apply simp
haftmann@33361
  1812
apply (simp (no_asm_simp) add: right_distrib)
haftmann@33361
  1813
apply (subst add_commute, rule zadd_zless_mono, arith)
haftmann@33361
  1814
apply (rule mult_right_mono, auto)
haftmann@33361
  1815
done
haftmann@33361
  1816
haftmann@33361
  1817
lemma zdiv_mono2:
haftmann@33361
  1818
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  1819
apply (subgoal_tac "b \<noteq> 0")
haftmann@33361
  1820
 prefer 2 apply arith
haftmann@33361
  1821
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1822
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1823
apply (rule zdiv_mono2_lemma)
haftmann@33361
  1824
apply (erule subst)
haftmann@33361
  1825
apply (erule subst, simp_all)
haftmann@33361
  1826
done
haftmann@33361
  1827
haftmann@33361
  1828
lemma q_neg_lemma:
haftmann@33361
  1829
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  1830
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  1831
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  1832
done
haftmann@33361
  1833
haftmann@33361
  1834
lemma zdiv_mono2_neg_lemma:
haftmann@33361
  1835
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;   
haftmann@33361
  1836
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]   
haftmann@33361
  1837
      ==> q' \<le> (q::int)"
haftmann@33361
  1838
apply (frule q_neg_lemma, assumption+) 
haftmann@33361
  1839
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  1840
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1841
apply (simp add: right_distrib)
haftmann@33361
  1842
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  1843
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  1844
done
haftmann@33361
  1845
haftmann@33361
  1846
lemma zdiv_mono2_neg:
haftmann@33361
  1847
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@33361
  1848
apply (cut_tac a = a and b = b in zmod_zdiv_equality)
haftmann@33361
  1849
apply (cut_tac a = a and b = b' in zmod_zdiv_equality)
haftmann@33361
  1850
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  1851
apply (erule subst)
haftmann@33361
  1852
apply (erule subst, simp_all)
haftmann@33361
  1853
done
haftmann@33361
  1854
haftmann@33361
  1855
haftmann@33361
  1856
subsubsection{*More Algebraic Laws for div and mod*}
haftmann@33361
  1857
haftmann@33361
  1858
text{*proving (a*b) div c = a * (b div c) + a * (b mod c) *}
haftmann@33361
  1859
haftmann@33361
  1860
lemma zmult1_lemma:
haftmann@33361
  1861
     "[| divmod_int_rel b c (q, r);  c \<noteq> 0 |]  
haftmann@33361
  1862
      ==> divmod_int_rel (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@33361
  1863
by (auto simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib mult_ac)
haftmann@33361
  1864
haftmann@33361
  1865
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  1866
apply (case_tac "c = 0", simp)
haftmann@33361
  1867
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_div])
haftmann@33361
  1868
done
haftmann@33361
  1869
haftmann@33361
  1870
lemma zmod_zmult1_eq: "(a*b) mod c = a*(b mod c) mod (c::int)"
haftmann@33361
  1871
apply (case_tac "c = 0", simp)
haftmann@33361
  1872
apply (blast intro: divmod_int_rel_div_mod [THEN zmult1_lemma, THEN divmod_int_rel_mod])
haftmann@33361
  1873
done
haftmann@33361
  1874
haftmann@33361
  1875
lemma zmod_zdiv_trivial: "(a mod b) div b = (0::int)"
haftmann@33361
  1876
apply (case_tac "b = 0", simp)
haftmann@33361
  1877
apply (auto simp add: linorder_neq_iff div_pos_pos_trivial div_neg_neg_trivial)
haftmann@33361
  1878
done
haftmann@33361
  1879
haftmann@33361
  1880
text{*proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c) *}
haftmann@33361
  1881
haftmann@33361
  1882
lemma zadd1_lemma:
haftmann@33361
  1883
     "[| divmod_int_rel a c (aq, ar);  divmod_int_rel b c (bq, br);  c \<noteq> 0 |]  
haftmann@33361
  1884
      ==> divmod_int_rel (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
haftmann@33361
  1885
by (force simp add: split_ifs divmod_int_rel_def linorder_neq_iff right_distrib)
haftmann@33361
  1886
haftmann@33361
  1887
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  1888
lemma zdiv_zadd1_eq:
haftmann@33361
  1889
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  1890
apply (case_tac "c = 0", simp)
haftmann@33361
  1891
apply (blast intro: zadd1_lemma [OF divmod_int_rel_div_mod divmod_int_rel_div_mod] divmod_int_rel_div)
haftmann@33361
  1892
done
haftmann@33361
  1893
haftmann@33361
  1894
instance int :: ring_div
haftmann@33361
  1895
proof
haftmann@33361
  1896
  fix a b c :: int
haftmann@33361
  1897
  assume not0: "b \<noteq> 0"
haftmann@33361
  1898
  show "(a + c * b) div b = c + a div b"
haftmann@33361
  1899
    unfolding zdiv_zadd1_eq [of a "c * b"] using not0 
haftmann@33361
  1900
      by (simp add: zmod_zmult1_eq zmod_zdiv_trivial zdiv_zmult1_eq)
haftmann@33361
  1901
next
haftmann@33361
  1902
  fix a b c :: int
haftmann@33361
  1903
  assume "a \<noteq> 0"
haftmann@33361
  1904
  then show "(a * b) div (a * c) = b div c"
haftmann@33361
  1905
  proof (cases "b \<noteq> 0 \<and> c \<noteq> 0")
haftmann@33361
  1906
    case False then show ?thesis by auto
haftmann@33361
  1907
  next
haftmann@33361
  1908
    case True then have "b \<noteq> 0" and "c \<noteq> 0" by auto
haftmann@33361
  1909
    with `a \<noteq> 0`
haftmann@33361
  1910
    have "\<And>q r. divmod_int_rel b c (q, r) \<Longrightarrow> divmod_int_rel (a * b) (a * c) (q, a * r)"
haftmann@33361
  1911
      apply (auto simp add: divmod_int_rel_def) 
haftmann@33361
  1912
      apply (auto simp add: algebra_simps)
haftmann@33361
  1913
      apply (auto simp add: zero_less_mult_iff zero_le_mult_iff mult_le_0_iff mult_commute [of a] mult_less_cancel_right)
haftmann@33361
  1914
      done
haftmann@33361
  1915
    moreover with `c \<noteq> 0` divmod_int_rel_div_mod have "divmod_int_rel b c (b div c, b mod c)" by auto
haftmann@33361
  1916
    ultimately have "divmod_int_rel (a * b) (a * c) (b div c, a * (b mod c))" .
haftmann@33361
  1917
    moreover from  `a \<noteq> 0` `c \<noteq> 0` have "a * c \<noteq> 0" by simp
haftmann@33361
  1918
    ultimately show ?thesis by (rule divmod_int_rel_div)
haftmann@33361
  1919
  qed
haftmann@33361
  1920
qed auto
haftmann@33361
  1921
haftmann@33361
  1922
lemma posDivAlg_div_mod:
haftmann@33361
  1923
  assumes "k \<ge> 0"
haftmann@33361
  1924
  and "l \<ge> 0"
haftmann@33361
  1925
  shows "posDivAlg k l = (k div l, k mod l)"
haftmann@33361
  1926
proof (cases "l = 0")
haftmann@33361
  1927
  case True then show ?thesis by (simp add: posDivAlg.simps)
haftmann@33361
  1928
next
haftmann@33361
  1929
  case False with assms posDivAlg_correct
haftmann@33361
  1930
    have "divmod_int_rel k l (fst (posDivAlg k l), snd (posDivAlg k l))"
haftmann@33361
  1931
    by simp
haftmann@33361
  1932
  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
haftmann@33361
  1933
  show ?thesis by simp
haftmann@33361
  1934
qed
haftmann@33361
  1935
haftmann@33361
  1936
lemma negDivAlg_div_mod:
haftmann@33361
  1937
  assumes "k < 0"
haftmann@33361
  1938
  and "l > 0"
haftmann@33361
  1939
  shows "negDivAlg k l = (k div l, k mod l)"
haftmann@33361
  1940
proof -
haftmann@33361
  1941
  from assms have "l \<noteq> 0" by simp
haftmann@33361
  1942
  from assms negDivAlg_correct
haftmann@33361
  1943
    have "divmod_int_rel k l (fst (negDivAlg k l), snd (negDivAlg k l))"
haftmann@33361
  1944
    by simp
haftmann@33361
  1945
  from divmod_int_rel_div [OF this `l \<noteq> 0`] divmod_int_rel_mod [OF this `l \<noteq> 0`]
haftmann@33361
  1946
  show ?thesis by simp
haftmann@33361
  1947
qed
haftmann@33361
  1948
haftmann@33361
  1949
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  1950
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  1951
haftmann@33361
  1952
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  1953
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  1954
haftmann@33361
  1955
haftmann@33361
  1956
subsubsection{*Proving  @{term "a div (b*c) = (a div b) div c"} *}
haftmann@33361
  1957
haftmann@33361
  1958
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  1959
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  1960
  to cause particular problems.*)
haftmann@33361
  1961
haftmann@33361
  1962
text{*first, four lemmas to bound the remainder for the cases b<0 and b>0 *}
haftmann@33361
  1963
haftmann@33361
  1964
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b*c < b*(q mod c) + r"
haftmann@33361
  1965
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  1966
 apply (simp add: algebra_simps)
haftmann@33361
  1967
apply (rule order_le_less_trans)
haftmann@33361
  1968
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  1969
 apply (rule mult_left_mono_neg)
huffman@35216
  1970
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  1971
 apply (simp)
haftmann@33361
  1972
apply (simp)
haftmann@33361
  1973
done
haftmann@33361
  1974
haftmann@33361
  1975
lemma zmult2_lemma_aux2:
haftmann@33361
  1976
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  1977
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  1978
 apply arith
haftmann@33361
  1979
apply (simp add: mult_le_0_iff)
haftmann@33361
  1980
done
haftmann@33361
  1981
haftmann@33361
  1982
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  1983
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  1984
apply arith
haftmann@33361
  1985
apply (simp add: zero_le_mult_iff)
haftmann@33361
  1986
done
haftmann@33361
  1987
haftmann@33361
  1988
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  1989
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  1990
 apply (simp add: right_diff_distrib)
haftmann@33361
  1991
apply (rule order_less_le_trans)
haftmann@33361
  1992
 apply (erule mult_strict_right_mono)
haftmann@33361
  1993
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  1994
  apply simp
huffman@35216
  1995
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  1996
apply simp
haftmann@33361
  1997
done
haftmann@33361
  1998
haftmann@33361
  1999
lemma zmult2_lemma: "[| divmod_int_rel a b (q, r);  b \<noteq> 0;  0 < c |]  
haftmann@33361
  2000
      ==> divmod_int_rel a (b * c) (q div c, b*(q mod c) + r)"
haftmann@33361
  2001
by (auto simp add: mult_ac divmod_int_rel_def linorder_neq_iff
haftmann@33361
  2002
                   zero_less_mult_iff right_distrib [symmetric] 
haftmann@33361
  2003
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4)
haftmann@33361
  2004
haftmann@33361
  2005
lemma zdiv_zmult2_eq: "(0::int) < c ==> a div (b*c) = (a div b) div c"
haftmann@33361
  2006
apply (case_tac "b = 0", simp)
haftmann@33361
  2007
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_div])
haftmann@33361
  2008
done
haftmann@33361
  2009
haftmann@33361
  2010
lemma zmod_zmult2_eq:
haftmann@33361
  2011
     "(0::int) < c ==> a mod (b*c) = b*(a div b mod c) + a mod b"
haftmann@33361
  2012
apply (case_tac "b = 0", simp)
haftmann@33361
  2013
apply (force simp add: divmod_int_rel_div_mod [THEN zmult2_lemma, THEN divmod_int_rel_mod])
haftmann@33361
  2014
done
haftmann@33361
  2015
haftmann@33361
  2016
haftmann@33361
  2017
subsubsection {*Splitting Rules for div and mod*}
haftmann@33361
  2018
haftmann@33361
  2019
text{*The proofs of the two lemmas below are essentially identical*}
haftmann@33361
  2020
haftmann@33361
  2021
lemma split_pos_lemma:
haftmann@33361
  2022
 "0<k ==> 
haftmann@33361
  2023
    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
  2024
apply (rule iffI, clarify)
haftmann@33361
  2025
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2026
 apply (subst mod_add_eq) 
haftmann@33361
  2027
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2028
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)  
haftmann@33361
  2029
txt{*converse direction*}
haftmann@33361
  2030
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2031
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2032
done
haftmann@33361
  2033
haftmann@33361
  2034
lemma split_neg_lemma:
haftmann@33361
  2035
 "k<0 ==>
haftmann@33361
  2036
    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
  2037
apply (rule iffI, clarify)
haftmann@33361
  2038
 apply (erule_tac P="P ?x ?y" in rev_mp)  
haftmann@33361
  2039
 apply (subst mod_add_eq) 
haftmann@33361
  2040
 apply (subst zdiv_zadd1_eq) 
haftmann@33361
  2041
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)  
haftmann@33361
  2042
txt{*converse direction*}
haftmann@33361
  2043
apply (drule_tac x = "n div k" in spec) 
haftmann@33361
  2044
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  2045
done
haftmann@33361
  2046
haftmann@33361
  2047
lemma split_zdiv:
haftmann@33361
  2048
 "P(n div k :: int) =
haftmann@33361
  2049
  ((k = 0 --> P 0) & 
haftmann@33361
  2050
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) & 
haftmann@33361
  2051
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  2052
apply (case_tac "k=0", simp)
haftmann@33361
  2053
apply (simp only: linorder_neq_iff)
haftmann@33361
  2054
apply (erule disjE) 
haftmann@33361
  2055
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"] 
haftmann@33361
  2056
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  2057
done
haftmann@33361
  2058
haftmann@33361
  2059
lemma split_zmod:
haftmann@33361
  2060
 "P(n mod k :: int) =
haftmann@33361
  2061
  ((k = 0 --> P n) & 
haftmann@33361
  2062
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) & 
haftmann@33361
  2063
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  2064
apply (case_tac "k=0", simp)
haftmann@33361
  2065
apply (simp only: linorder_neq_iff)
haftmann@33361
  2066
apply (erule disjE) 
haftmann@33361
  2067
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"] 
haftmann@33361
  2068
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  2069
done
haftmann@33361
  2070
webertj@33730
  2071
text {* Enable (lin)arith to deal with @{const div} and @{const mod}
webertj@33730
  2072
  when these are applied to some constant that is of the form
webertj@33730
  2073
  @{term "number_of k"}: *}
webertj@33728
  2074
declare split_zdiv [of _ _ "number_of k", standard, arith_split]
webertj@33728
  2075
declare split_zmod [of _ _ "number_of k", standard, arith_split]
haftmann@33361
  2076
haftmann@33361
  2077
haftmann@33361
  2078
subsubsection{*Speeding up the Division Algorithm with Shifting*}
haftmann@33361
  2079
haftmann@33361
  2080
text{*computing div by shifting *}
haftmann@33361
  2081
haftmann@33361
  2082
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
haftmann@33361
  2083
proof cases
haftmann@33361
  2084
  assume "a=0"
haftmann@33361
  2085
    thus ?thesis by simp
haftmann@33361
  2086
next
haftmann@33361
  2087
  assume "a\<noteq>0" and le_a: "0\<le>a"   
haftmann@33361
  2088
  hence a_pos: "1 \<le> a" by arith
haftmann@33361
  2089
  hence one_less_a2: "1 < 2 * a" by arith
haftmann@33361
  2090
  hence le_2a: "2 * (1 + b mod a) \<le> 2 * a"
haftmann@33361
  2091
    unfolding mult_le_cancel_left
haftmann@33361
  2092
    by (simp add: add1_zle_eq add_commute [of 1])
haftmann@33361
  2093
  with a_pos have "0 \<le> b mod a" by simp
haftmann@33361
  2094
  hence le_addm: "0 \<le> 1 mod (2*a) + 2*(b mod a)"
haftmann@33361
  2095
    by (simp add: mod_pos_pos_trivial one_less_a2)
haftmann@33361
  2096
  with  le_2a
haftmann@33361
  2097
  have "(1 mod (2*a) + 2*(b mod a)) div (2*a) = 0"
haftmann@33361
  2098
    by (simp add: div_pos_pos_trivial le_addm mod_pos_pos_trivial one_less_a2
haftmann@33361
  2099
                  right_distrib) 
haftmann@33361
  2100
  thus ?thesis
haftmann@33361
  2101
    by (subst zdiv_zadd1_eq,
haftmann@33361
  2102
        simp add: mod_mult_mult1 one_less_a2
haftmann@33361
  2103
                  div_pos_pos_trivial)
haftmann@33361
  2104
qed
haftmann@33361
  2105
boehmes@35815
  2106
lemma neg_zdiv_mult_2: 
boehmes@35815
  2107
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
boehmes@35815
  2108
proof -
boehmes@35815
  2109
  have R: "1 + - (2 * (b + 1)) = - (1 + 2 * b)" by simp
boehmes@35815
  2110
  have "(1 + 2 * (-b - 1)) div (2 * (-a)) = (-b - 1) div (-a)"
boehmes@35815
  2111
    by (rule pos_zdiv_mult_2, simp add: A)
boehmes@35815
  2112
  thus ?thesis
boehmes@35815
  2113
    by (simp only: R zdiv_zminus_zminus diff_minus
boehmes@35815
  2114
      minus_add_distrib [symmetric] mult_minus_right)
boehmes@35815
  2115
qed
haftmann@33361
  2116
haftmann@33361
  2117
lemma zdiv_number_of_Bit0 [simp]:
haftmann@33361
  2118
     "number_of (Int.Bit0 v) div number_of (Int.Bit0 w) =  
haftmann@33361
  2119
          number_of v div (number_of w :: int)"
haftmann@33361
  2120
by (simp only: number_of_eq numeral_simps) (simp add: mult_2 [symmetric])
haftmann@33361
  2121
haftmann@33361
  2122
lemma zdiv_number_of_Bit1 [simp]:
haftmann@33361
  2123
     "number_of (Int.Bit1 v) div number_of (Int.Bit0 w) =  
haftmann@33361
  2124
          (if (0::int) \<le> number_of w                    
haftmann@33361
  2125
           then number_of v div (number_of w)     
haftmann@33361
  2126
           else (number_of v + (1::int)) div (number_of w))"
haftmann@33361
  2127
apply (simp only: number_of_eq numeral_simps UNIV_I split: split_if) 
haftmann@33361
  2128
apply (simp add: pos_zdiv_mult_2 neg_zdiv_mult_2 add_ac mult_2 [symmetric])
haftmann@33361
  2129
done
haftmann@33361
  2130
haftmann@33361
  2131
haftmann@33361
  2132
subsubsection{*Computing mod by Shifting (proofs resemble those for div)*}
haftmann@33361
  2133
haftmann@33361
  2134
lemma pos_zmod_mult_2:
haftmann@33361
  2135
  fixes a b :: int
haftmann@33361
  2136
  assumes "0 \<le> a"
haftmann@33361
  2137
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
haftmann@33361
  2138
proof (cases "0 < a")
haftmann@33361
  2139
  case False with assms show ?thesis by simp
haftmann@33361
  2140
next
haftmann@33361
  2141
  case True
haftmann@33361
  2142
  then have "b mod a < a" by (rule pos_mod_bound)
haftmann@33361
  2143
  then have "1 + b mod a \<le> a" by simp
haftmann@33361
  2144
  then have A: "2 * (1 + b mod a) \<le> 2 * a" by simp
haftmann@33361
  2145
  from `0 < a` have "0 \<le> b mod a" by (rule pos_mod_sign)
haftmann@33361
  2146
  then have B: "0 \<le> 1 + 2 * (b mod a)" by simp
haftmann@33361
  2147
  have "((1\<Colon>int) mod ((2\<Colon>int) * a) + (2\<Colon>int) * b mod ((2\<Colon>int) * a)) mod ((2\<Colon>int) * a) = (1\<Colon>int) + (2\<Colon>int) * (b mod a)"
haftmann@33361
  2148
    using `0 < a` and A
haftmann@33361
  2149
    by (auto simp add: mod_mult_mult1 mod_pos_pos_trivial ring_distribs intro!: mod_pos_pos_trivial B)
haftmann@33361
  2150
  then show ?thesis by (subst mod_add_eq)
haftmann@33361
  2151
qed
haftmann@33361
  2152
haftmann@33361
  2153
lemma neg_zmod_mult_2:
haftmann@33361
  2154
  fixes a b :: int
haftmann@33361
  2155
  assumes "a \<le> 0"
haftmann@33361
  2156
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
haftmann@33361
  2157
proof -
haftmann@33361
  2158
  from assms have "0 \<le> - a" by auto
haftmann@33361
  2159
  then have "(1 + 2 * (- b - 1)) mod (2 * (- a)) = 1 + 2 * ((- b - 1) mod (- a))"
haftmann@33361
  2160
    by (rule pos_zmod_mult_2)
haftmann@33361
  2161
  then show ?thesis by (simp add: zmod_zminus2 algebra_simps)
haftmann@33361
  2162
     (simp add: diff_minus add_ac)
haftmann@33361
  2163
qed
haftmann@33361
  2164
haftmann@33361
  2165
lemma zmod_number_of_Bit0 [simp]:
haftmann@33361
  2166
     "number_of (Int.Bit0 v) mod number_of (Int.Bit0 w) =  
haftmann@33361
  2167
      (2::int) * (number_of v mod number_of w)"
haftmann@33361
  2168
apply (simp only: number_of_eq numeral_simps) 
haftmann@33361
  2169
apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
haftmann@33361
  2170
                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
haftmann@33361
  2171
done
haftmann@33361
  2172
haftmann@33361
  2173
lemma zmod_number_of_Bit1 [simp]:
haftmann@33361
  2174
     "number_of (Int.Bit1 v) mod number_of (Int.Bit0 w) =  
haftmann@33361
  2175
      (if (0::int) \<le> number_of w  
haftmann@33361
  2176
                then 2 * (number_of v mod number_of w) + 1     
haftmann@33361
  2177
                else 2 * ((number_of v + (1::int)) mod number_of w) - 1)"
haftmann@33361
  2178
apply (simp only: number_of_eq numeral_simps) 
haftmann@33361
  2179
apply (simp add: mod_mult_mult1 pos_zmod_mult_2 
haftmann@33361
  2180
                 neg_zmod_mult_2 add_ac mult_2 [symmetric])
haftmann@33361
  2181
done
haftmann@33361
  2182
haftmann@33361
  2183
haftmann@33361
  2184
subsubsection{*Quotients of Signs*}
haftmann@33361
  2185
haftmann@33361
  2186
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  2187
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  2188
apply (rule order_trans)
haftmann@33361
  2189
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  2190
apply (auto simp add: div_eq_minus1)
haftmann@33361
  2191
done
haftmann@33361
  2192
haftmann@33361
  2193
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  2194
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  2195
haftmann@33361
  2196
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  2197
by (drule zdiv_mono1, auto)
haftmann@33361
  2198
nipkow@33804
  2199
text{* Now for some equivalences of the form @{text"a div b >=< 0 \<longleftrightarrow> \<dots>"}
nipkow@33804
  2200
conditional upon the sign of @{text a} or @{text b}. There are many more.
nipkow@33804
  2201
They should all be simp rules unless that causes too much search. *}
nipkow@33804
  2202
haftmann@33361
  2203
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2204
apply auto
haftmann@33361
  2205
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2206
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2207
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2208
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2209
done
haftmann@33361
  2210
haftmann@33361
  2211
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2212
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
haftmann@33361
  2213
apply (subst zdiv_zminus_zminus [symmetric])
haftmann@33361
  2214
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2215
done
haftmann@33361
  2216
haftmann@33361
  2217
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2218
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2219
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2220
haftmann@33361
  2221
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2222
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2223
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2224
nipkow@33804
  2225
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2226
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2227
apply rule
nipkow@33804
  2228
 apply rule
nipkow@33804
  2229
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2230
 apply(cases "b=0")apply simp
nipkow@33804
  2231
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2232
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2233
done
nipkow@33804
  2234
haftmann@33361
  2235
haftmann@33361
  2236
subsubsection {* The Divides Relation *}
haftmann@33361
  2237
haftmann@33361
  2238
lemmas zdvd_iff_zmod_eq_0_number_of [simp] =
haftmann@33361
  2239
  dvd_eq_mod_eq_0 [of "number_of x::int" "number_of y::int", standard]
haftmann@33361
  2240
haftmann@33361
  2241
lemma zdvd_zmod: "f dvd m ==> f dvd (n::int) ==> f dvd m mod n"
haftmann@33361
  2242
  by (rule dvd_mod) (* TODO: remove *)
haftmann@33361
  2243
haftmann@33361
  2244
lemma zdvd_zmod_imp_zdvd: "k dvd m mod n ==> k dvd n ==> k dvd (m::int)"
haftmann@33361
  2245
  by (rule dvd_mod_imp_dvd) (* TODO: remove *)
haftmann@33361
  2246
haftmann@33361
  2247
lemma zmult_div_cancel: "(n::int) * (m div n) = m - (m mod n)"
haftmann@33361
  2248
  using zmod_zdiv_equality[where a="m" and b="n"]
haftmann@33361
  2249
  by (simp add: algebra_simps)
haftmann@33361
  2250
haftmann@33361
  2251
lemma zpower_zmod: "((x::int) mod m)^y mod m = x^y mod m"
haftmann@33361
  2252
apply (induct "y", auto)
haftmann@33361
  2253
apply (rule zmod_zmult1_eq [THEN trans])
haftmann@33361
  2254
apply (simp (no_asm_simp))
haftmann@33361
  2255
apply (rule mod_mult_eq [symmetric])
haftmann@33361
  2256
done
haftmann@33361
  2257
haftmann@33361
  2258
lemma zdiv_int: "int (a div b) = (int a) div (int b)"
haftmann@33361
  2259
apply (subst split_div, auto)
haftmann@33361
  2260
apply (subst split_zdiv, auto)
haftmann@33361
  2261
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
  2262
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2263
done
haftmann@33361
  2264
haftmann@33361
  2265
lemma zmod_int: "int (a mod b) = (int a) mod (int b)"
haftmann@33361
  2266
apply (subst split_mod, auto)
haftmann@33361
  2267
apply (subst split_zmod, auto)
haftmann@33361
  2268
apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia 
haftmann@33361
  2269
       in unique_remainder)
haftmann@33361
  2270
apply (auto simp add: divmod_int_rel_def of_nat_mult)
haftmann@33361
  2271
done
haftmann@33361
  2272
haftmann@33361
  2273
lemma abs_div: "(y::int) dvd x \<Longrightarrow> abs (x div y) = abs x div abs y"
haftmann@33361
  2274
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2275
haftmann@33361
  2276
lemma zdvd_mult_div_cancel:"(n::int) dvd m \<Longrightarrow> n * (m div n) = m"
haftmann@33361
  2277
apply (subgoal_tac "m mod n = 0")
haftmann@33361
  2278
 apply (simp add: zmult_div_cancel)
haftmann@33361
  2279
apply (simp only: dvd_eq_mod_eq_0)
haftmann@33361
  2280
done
haftmann@33361
  2281
haftmann@33361
  2282
text{*Suggested by Matthias Daum*}
haftmann@33361
  2283
lemma int_power_div_base:
haftmann@33361
  2284
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2285
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2286
 apply (erule ssubst)
haftmann@33361
  2287
 apply (simp only: power_add)
haftmann@33361
  2288
 apply simp_all
haftmann@33361
  2289
done
haftmann@33361
  2290
haftmann@33361
  2291
text {* by Brian Huffman *}
haftmann@33361
  2292
lemma zminus_zmod: "- ((x::int) mod m) mod m = - x mod m"
haftmann@33361
  2293
by (rule mod_minus_eq [symmetric])
haftmann@33361
  2294
haftmann@33361
  2295
lemma zdiff_zmod_left: "(x mod m - y) mod m = (x - y) mod (m::int)"
haftmann@33361
  2296
by (rule mod_diff_left_eq [symmetric])
haftmann@33361
  2297
haftmann@33361
  2298
lemma zdiff_zmod_right: "(x - y mod m) mod m = (x - y) mod (m::int)"
haftmann@33361
  2299
by (rule mod_diff_right_eq [symmetric])
haftmann@33361
  2300
haftmann@33361
  2301
lemmas zmod_simps =
haftmann@33361
  2302
  mod_add_left_eq  [symmetric]
haftmann@33361
  2303
  mod_add_right_eq [symmetric]
haftmann@33361
  2304
  zmod_zmult1_eq   [symmetric]
haftmann@33361
  2305
  mod_mult_left_eq [symmetric]
haftmann@33361
  2306
  zpower_zmod
haftmann@33361
  2307
  zminus_zmod zdiff_zmod_left zdiff_zmod_right
haftmann@33361
  2308
haftmann@33361
  2309
text {* Distributive laws for function @{text nat}. *}
haftmann@33361
  2310
haftmann@33361
  2311
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2312
apply (rule linorder_cases [of y 0])
haftmann@33361
  2313
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2314
apply simp
haftmann@33361
  2315
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2316
done
haftmann@33361
  2317
haftmann@33361
  2318
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2319
lemma nat_mod_distrib:
haftmann@33361
  2320
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2321
apply (case_tac "y = 0", simp)
haftmann@33361
  2322
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2323
done
haftmann@33361
  2324
haftmann@33361
  2325
text  {* transfer setup *}
haftmann@33361
  2326
haftmann@33361
  2327
lemma transfer_nat_int_functions:
haftmann@33361
  2328
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2329
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2330
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2331
haftmann@33361
  2332
lemma transfer_nat_int_function_closures:
haftmann@33361
  2333
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2334
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2335
  apply (cases "y = 0")
haftmann@33361
  2336
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2337
  apply (cases "y = 0")
haftmann@33361
  2338
  apply auto
haftmann@33361
  2339
done
haftmann@33361
  2340
haftmann@35644
  2341
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2342
  transfer_nat_int_functions
haftmann@33361
  2343
  transfer_nat_int_function_closures
haftmann@33361
  2344
]
haftmann@33361
  2345
haftmann@33361
  2346
lemma transfer_int_nat_functions:
haftmann@33361
  2347
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2348
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2349
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2350
haftmann@33361
  2351
lemma transfer_int_nat_function_closures:
haftmann@33361
  2352
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  2353
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  2354
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  2355
haftmann@35644
  2356
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  2357
  transfer_int_nat_functions
haftmann@33361
  2358
  transfer_int_nat_function_closures
haftmann@33361
  2359
]
haftmann@33361
  2360
haftmann@33361
  2361
text{*Suggested by Matthias Daum*}
haftmann@33361
  2362
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  2363
apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225
  2364
 apply (simp add: nat_div_distrib [symmetric])
haftmann@33361
  2365
apply (rule Divides.div_less_dividend, simp_all)
haftmann@33361
  2366
done
haftmann@33361
  2367
haftmann@33361
  2368
lemma zmod_eq_dvd_iff: "(x::int) mod n = y mod n \<longleftrightarrow> n dvd x - y"
haftmann@33361
  2369
proof
haftmann@33361
  2370
  assume H: "x mod n = y mod n"
haftmann@33361
  2371
  hence "x mod n - y mod n = 0" by simp
haftmann@33361
  2372
  hence "(x mod n - y mod n) mod n = 0" by simp 
haftmann@33361
  2373
  hence "(x - y) mod n = 0" by (simp add: mod_diff_eq[symmetric])
haftmann@33361
  2374
  thus "n dvd x - y" by (simp add: dvd_eq_mod_eq_0)
haftmann@33361
  2375
next
haftmann@33361
  2376
  assume H: "n dvd x - y"
haftmann@33361
  2377
  then obtain k where k: "x-y = n*k" unfolding dvd_def by blast
haftmann@33361
  2378
  hence "x = n*k + y" by simp
haftmann@33361
  2379
  hence "x mod n = (n*k + y) mod n" by simp
haftmann@33361
  2380
  thus "x mod n = y mod n" by (simp add: mod_add_left_eq)
haftmann@33361
  2381
qed
haftmann@33361
  2382
haftmann@33361
  2383
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y  mod n" and xy:"y \<le> x"
haftmann@33361
  2384
  shows "\<exists>q. x = y + n * q"
haftmann@33361
  2385
proof-
haftmann@33361
  2386
  from xy have th: "int x - int y = int (x - y)" by simp 
haftmann@33361
  2387
  from xyn have "int x mod int n = int y mod int n" 
haftmann@33361
  2388
    by (simp add: zmod_int[symmetric])
haftmann@33361
  2389
  hence "int n dvd int x - int y" by (simp only: zmod_eq_dvd_iff[symmetric]) 
haftmann@33361
  2390
  hence "n dvd x - y" by (simp add: th zdvd_int)
haftmann@33361
  2391
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
haftmann@33361
  2392
qed
haftmann@33361
  2393
haftmann@33361
  2394
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)" 
haftmann@33361
  2395
  (is "?lhs = ?rhs")
haftmann@33361
  2396
proof
haftmann@33361
  2397
  assume H: "x mod n = y mod n"
haftmann@33361
  2398
  {assume xy: "x \<le> y"
haftmann@33361
  2399
    from H have th: "y mod n = x mod n" by simp
haftmann@33361
  2400
    from nat_mod_eq_lemma[OF th xy] have ?rhs 
haftmann@33361
  2401
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361
  2402
  moreover
haftmann@33361
  2403
  {assume xy: "y \<le> x"
haftmann@33361
  2404
    from nat_mod_eq_lemma[OF H xy] have ?rhs 
haftmann@33361
  2405
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
haftmann@33361
  2406
  ultimately  show ?rhs using linear[of x y] by blast  
haftmann@33361
  2407
next
haftmann@33361
  2408
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
haftmann@33361
  2409
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
haftmann@33361
  2410
  thus  ?lhs by simp
haftmann@33361
  2411
qed
haftmann@33361
  2412
haftmann@33361
  2413
lemma div_nat_number_of [simp]:
haftmann@33361
  2414
     "(number_of v :: nat)  div  number_of v' =  
haftmann@33361
  2415
          (if neg (number_of v :: int) then 0  
haftmann@33361
  2416
           else nat (number_of v div number_of v'))"
haftmann@33361
  2417
  unfolding nat_number_of_def number_of_is_id neg_def
haftmann@33361
  2418
  by (simp add: nat_div_distrib)
haftmann@33361
  2419
haftmann@33361
  2420
lemma one_div_nat_number_of [simp]:
haftmann@33361
  2421
     "Suc 0 div number_of v' = nat (1 div number_of v')" 
haftmann@33361
  2422
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
haftmann@33361
  2423
haftmann@33361
  2424
lemma mod_nat_number_of [simp]:
haftmann@33361
  2425
     "(number_of v :: nat)  mod  number_of v' =  
haftmann@33361
  2426
        (if neg (number_of v :: int) then 0  
haftmann@33361
  2427
         else if neg (number_of v' :: int) then number_of v  
haftmann@33361
  2428
         else nat (number_of v mod number_of v'))"
haftmann@33361
  2429
  unfolding nat_number_of_def number_of_is_id neg_def
haftmann@33361
  2430
  by (simp add: nat_mod_distrib)
haftmann@33361
  2431
haftmann@33361
  2432
lemma one_mod_nat_number_of [simp]:
haftmann@33361
  2433
     "Suc 0 mod number_of v' =  
haftmann@33361
  2434
        (if neg (number_of v' :: int) then Suc 0
haftmann@33361
  2435
         else nat (1 mod number_of v'))"
haftmann@33361
  2436
by (simp del: nat_numeral_1_eq_1 add: numeral_1_eq_Suc_0 [symmetric]) 
haftmann@33361
  2437
haftmann@33361
  2438
lemmas dvd_eq_mod_eq_0_number_of =
haftmann@33361
  2439
  dvd_eq_mod_eq_0 [of "number_of x" "number_of y", standard]
haftmann@33361
  2440
haftmann@33361
  2441
declare dvd_eq_mod_eq_0_number_of [simp]
haftmann@33361
  2442
haftmann@33361
  2443
blanchet@34126
  2444
subsubsection {* Nitpick *}
blanchet@34126
  2445
blanchet@34126
  2446
lemma zmod_zdiv_equality':
blanchet@34126
  2447
"(m\<Colon>int) mod n = m - (m div n) * n"
blanchet@34126
  2448
by (rule_tac P="%x. m mod n = x - (m div n) * n"
blanchet@34126
  2449
    in subst [OF mod_div_equality [of _ n]])
blanchet@34126
  2450
   arith
blanchet@34126
  2451
blanchet@34982
  2452
lemmas [nitpick_def] = dvd_eq_mod_eq_0 [THEN eq_reflection]
blanchet@34982
  2453
                       mod_div_equality' [THEN eq_reflection]
blanchet@34126
  2454
                       zmod_zdiv_equality' [THEN eq_reflection]
blanchet@34126
  2455
haftmann@35673
  2456
haftmann@33361
  2457
subsubsection {* Code generation *}
haftmann@33361
  2458
haftmann@33361
  2459
definition pdivmod :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
haftmann@33361
  2460
  "pdivmod k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
haftmann@33361
  2461
haftmann@33361
  2462
lemma pdivmod_posDivAlg [code]:
haftmann@33361
  2463
  "pdivmod k l = (if l = 0 then (0, \<bar>k\<bar>) else posDivAlg \<bar>k\<bar> \<bar>l\<bar>)"
haftmann@33361
  2464
by (subst posDivAlg_div_mod) (simp_all add: pdivmod_def)
haftmann@33361
  2465
haftmann@33361
  2466
lemma divmod_int_pdivmod: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@33361
  2467
  apsnd ((op *) (sgn l)) (if 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0
haftmann@33361
  2468
    then pdivmod k l
haftmann@33361
  2469
    else (let (r, s) = pdivmod k l in
haftmann@33361
  2470
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@33361
  2471
proof -
haftmann@33361
  2472
  have aux: "\<And>q::int. - k = l * q \<longleftrightarrow> k = l * - q" by auto
haftmann@33361
  2473
  show ?thesis
haftmann@33361
  2474
    by (simp add: divmod_int_mod_div pdivmod_def)
haftmann@33361
  2475
      (auto simp add: aux not_less not_le zdiv_zminus1_eq_if
haftmann@33361
  2476
      zmod_zminus1_eq_if zdiv_zminus2_eq_if zmod_zminus2_eq_if)
haftmann@33361
  2477
qed
haftmann@33361
  2478
haftmann@33361
  2479
lemma divmod_int_code [code]: "divmod_int k l = (if k = 0 then (0, 0) else if l = 0 then (0, k) else
haftmann@33361
  2480
  apsnd ((op *) (sgn l)) (if sgn k = sgn l
haftmann@33361
  2481
    then pdivmod k l
haftmann@33361
  2482
    else (let (r, s) = pdivmod k l in
haftmann@33361
  2483
      if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
haftmann@33361
  2484
proof -
haftmann@33361
  2485
  have "k \<noteq> 0 \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> 0 < l \<and> 0 \<le> k \<or> l < 0 \<and> k < 0 \<longleftrightarrow> sgn k = sgn l"
haftmann@33361
  2486
    by (auto simp add: not_less sgn_if)
haftmann@33361
  2487
  then show ?thesis by (simp add: divmod_int_pdivmod)
haftmann@33361
  2488
qed
haftmann@33361
  2489
haftmann@35673
  2490
context ring_1
haftmann@35673
  2491
begin
haftmann@35673
  2492
haftmann@35673
  2493
lemma of_int_num [code]:
haftmann@35673
  2494
  "of_int k = (if k = 0 then 0 else if k < 0 then
haftmann@35673
  2495
     - of_int (- k) else let
haftmann@35673
  2496
       (l, m) = divmod_int k 2;
haftmann@35673
  2497
       l' = of_int l
haftmann@35673
  2498
     in if m = 0 then l' + l' else l' + l' + 1)"
haftmann@35673
  2499
proof -
haftmann@35673
  2500
  have aux1: "k mod (2\<Colon>int) \<noteq> (0\<Colon>int) \<Longrightarrow> 
haftmann@35673
  2501
    of_int k = of_int (k div 2 * 2 + 1)"
haftmann@35673
  2502
  proof -
haftmann@35673
  2503
    have "k mod 2 < 2" by (auto intro: pos_mod_bound)
haftmann@35673
  2504
    moreover have "0 \<le> k mod 2" by (auto intro: pos_mod_sign)