src/HOL/Divides.thy
author nipkow
Wed Jan 28 16:29:16 2009 +0100 (2009-01-28)
changeset 29667 53103fc8ffa3
parent 29509 1ff0f3f08a7b
child 29925 17d1e32ef867
permissions -rw-r--r--
Replaced group_ and ring_simps by algebra_simps;
removed compare_rls - use algebra_simps now
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(*  Title:      HOL/Divides.thy
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    ID:         $Id$
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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 Power Product_Type
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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 + 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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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 unfold]: "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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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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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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end
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class ring_div = semiring_div + comm_ring_1
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begin
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text {* Negation respects modular equivalence. *}
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lemma mod_minus_eq: "(- a) mod b = (- (a mod b)) mod b"
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proof -
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  have "(- a) mod b = (- (a div b * b + a mod b)) mod b"
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    by (simp only: mod_div_equality)
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  also have "\<dots> = (- (a mod b) + - (a div b) * b) mod b"
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    by (simp only: minus_add_distrib minus_mult_left add_ac)
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  also have "\<dots> = (- (a mod b)) mod b"
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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_minus_cong:
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  assumes "a mod b = a' mod b"
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  shows "(- a) mod b = (- a') mod b"
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proof -
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  have "(- (a mod b)) mod b = (- (a' mod b)) mod b"
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    unfolding assms ..
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  thus ?thesis
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    by (simp only: mod_minus_eq [symmetric])
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qed
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text {* Subtraction respects modular equivalence. *}
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lemma mod_diff_left_eq: "(a - b) mod c = (a mod c - b) mod c"
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  unfolding diff_minus
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  by (intro mod_add_cong mod_minus_cong) simp_all
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lemma mod_diff_right_eq: "(a - b) mod c = (a - b mod c) mod c"
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  unfolding diff_minus
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  by (intro mod_add_cong mod_minus_cong) simp_all
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lemma mod_diff_eq: "(a - b) mod c = (a mod c - b mod c) mod c"
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  unfolding diff_minus
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  by (intro mod_add_cong mod_minus_cong) simp_all
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lemma mod_diff_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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  unfolding diff_minus using assms
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  by (intro mod_add_cong mod_minus_cong)
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huffman@29405
   317
end
huffman@29405
   318
haftmann@25942
   319
haftmann@26100
   320
subsection {* Division on @{typ nat} *}
haftmann@26100
   321
haftmann@26100
   322
text {*
haftmann@26100
   323
  We define @{const div} and @{const mod} on @{typ nat} by means
haftmann@26100
   324
  of a characteristic relation with two input arguments
haftmann@26100
   325
  @{term "m\<Colon>nat"}, @{term "n\<Colon>nat"} and two output arguments
haftmann@26100
   326
  @{term "q\<Colon>nat"}(uotient) and @{term "r\<Colon>nat"}(emainder).
haftmann@26100
   327
*}
haftmann@26100
   328
haftmann@26100
   329
definition divmod_rel :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool" where
haftmann@26100
   330
  "divmod_rel m n q r \<longleftrightarrow> m = q * n + r \<and> (if n > 0 then 0 \<le> r \<and> r < n else q = 0)"
haftmann@26100
   331
haftmann@26100
   332
text {* @{const divmod_rel} is total: *}
haftmann@26100
   333
haftmann@26100
   334
lemma divmod_rel_ex:
haftmann@26100
   335
  obtains q r where "divmod_rel m n q r"
haftmann@26100
   336
proof (cases "n = 0")
haftmann@26100
   337
  case True with that show thesis
haftmann@26100
   338
    by (auto simp add: divmod_rel_def)
haftmann@26100
   339
next
haftmann@26100
   340
  case False
haftmann@26100
   341
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   342
  proof (induct m)
haftmann@26100
   343
    case 0 with `n \<noteq> 0`
haftmann@26100
   344
    have "(0\<Colon>nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   345
    then show ?case by blast
haftmann@26100
   346
  next
haftmann@26100
   347
    case (Suc m) then obtain q' r'
haftmann@26100
   348
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   349
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   350
      case True
haftmann@26100
   351
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   352
      with True show ?thesis by blast
haftmann@26100
   353
    next
haftmann@26100
   354
      case False then have "n \<le> Suc r'" by auto
haftmann@26100
   355
      moreover from n have "Suc r' \<le> n" by auto
haftmann@26100
   356
      ultimately have "n = Suc r'" by auto
haftmann@26100
   357
      with m have "Suc m = Suc q' * n + 0" by simp
haftmann@26100
   358
      with `n \<noteq> 0` show ?thesis by blast
haftmann@26100
   359
    qed
haftmann@26100
   360
  qed
haftmann@26100
   361
  with that show thesis
haftmann@26100
   362
    using `n \<noteq> 0` by (auto simp add: divmod_rel_def)
haftmann@26100
   363
qed
haftmann@26100
   364
haftmann@26100
   365
text {* @{const divmod_rel} is injective: *}
haftmann@26100
   366
haftmann@26100
   367
lemma divmod_rel_unique_div:
haftmann@26100
   368
  assumes "divmod_rel m n q r"
haftmann@26100
   369
    and "divmod_rel m n q' r'"
haftmann@26100
   370
  shows "q = q'"
haftmann@26100
   371
proof (cases "n = 0")
haftmann@26100
   372
  case True with assms show ?thesis
haftmann@26100
   373
    by (simp add: divmod_rel_def)
haftmann@26100
   374
next
haftmann@26100
   375
  case False
haftmann@26100
   376
  have aux: "\<And>q r q' r'. q' * n + r' = q * n + r \<Longrightarrow> r < n \<Longrightarrow> q' \<le> (q\<Colon>nat)"
haftmann@26100
   377
  apply (rule leI)
haftmann@26100
   378
  apply (subst less_iff_Suc_add)
haftmann@26100
   379
  apply (auto simp add: add_mult_distrib)
haftmann@26100
   380
  done
haftmann@26100
   381
  from `n \<noteq> 0` assms show ?thesis
haftmann@26100
   382
    by (auto simp add: divmod_rel_def
haftmann@26100
   383
      intro: order_antisym dest: aux sym)
haftmann@26100
   384
qed
haftmann@26100
   385
haftmann@26100
   386
lemma divmod_rel_unique_mod:
haftmann@26100
   387
  assumes "divmod_rel m n q r"
haftmann@26100
   388
    and "divmod_rel m n q' r'"
haftmann@26100
   389
  shows "r = r'"
haftmann@26100
   390
proof -
haftmann@26100
   391
  from assms have "q = q'" by (rule divmod_rel_unique_div)
haftmann@26100
   392
  with assms show ?thesis by (simp add: divmod_rel_def)
haftmann@26100
   393
qed
haftmann@26100
   394
haftmann@26100
   395
text {*
haftmann@26100
   396
  We instantiate divisibility on the natural numbers by
haftmann@26100
   397
  means of @{const divmod_rel}:
haftmann@26100
   398
*}
haftmann@25942
   399
haftmann@25942
   400
instantiation nat :: semiring_div
haftmann@25571
   401
begin
haftmann@25571
   402
haftmann@26100
   403
definition divmod :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@28562
   404
  [code del]: "divmod m n = (THE (q, r). divmod_rel m n q r)"
haftmann@26100
   405
haftmann@26100
   406
definition div_nat where
haftmann@26100
   407
  "m div n = fst (divmod m n)"
haftmann@26100
   408
haftmann@26100
   409
definition mod_nat where
haftmann@26100
   410
  "m mod n = snd (divmod m n)"
haftmann@25571
   411
haftmann@26100
   412
lemma divmod_div_mod:
haftmann@26100
   413
  "divmod m n = (m div n, m mod n)"
haftmann@26100
   414
  unfolding div_nat_def mod_nat_def by simp
haftmann@26100
   415
haftmann@26100
   416
lemma divmod_eq:
haftmann@26100
   417
  assumes "divmod_rel m n q r" 
haftmann@26100
   418
  shows "divmod m n = (q, r)"
haftmann@26100
   419
  using assms by (auto simp add: divmod_def
haftmann@26100
   420
    dest: divmod_rel_unique_div divmod_rel_unique_mod)
haftmann@25942
   421
haftmann@26100
   422
lemma div_eq:
haftmann@26100
   423
  assumes "divmod_rel m n q r" 
haftmann@26100
   424
  shows "m div n = q"
haftmann@26100
   425
  using assms by (auto dest: divmod_eq simp add: div_nat_def)
haftmann@26100
   426
haftmann@26100
   427
lemma mod_eq:
haftmann@26100
   428
  assumes "divmod_rel m n q r" 
haftmann@26100
   429
  shows "m mod n = r"
haftmann@26100
   430
  using assms by (auto dest: divmod_eq simp add: mod_nat_def)
haftmann@25571
   431
haftmann@26100
   432
lemma divmod_rel: "divmod_rel m n (m div n) (m mod n)"
haftmann@26100
   433
proof -
haftmann@26100
   434
  from divmod_rel_ex
haftmann@26100
   435
    obtain q r where rel: "divmod_rel m n q r" .
haftmann@26100
   436
  moreover with div_eq mod_eq have "m div n = q" and "m mod n = r"
haftmann@26100
   437
    by simp_all
haftmann@26100
   438
  ultimately show ?thesis by simp
haftmann@26100
   439
qed
paulson@14267
   440
haftmann@26100
   441
lemma divmod_zero:
haftmann@26100
   442
  "divmod m 0 = (0, m)"
haftmann@26100
   443
proof -
haftmann@26100
   444
  from divmod_rel [of m 0] show ?thesis
haftmann@26100
   445
    unfolding divmod_div_mod divmod_rel_def by simp
haftmann@26100
   446
qed
haftmann@25942
   447
haftmann@26100
   448
lemma divmod_base:
haftmann@26100
   449
  assumes "m < n"
haftmann@26100
   450
  shows "divmod m n = (0, m)"
haftmann@26100
   451
proof -
haftmann@26100
   452
  from divmod_rel [of m n] show ?thesis
haftmann@26100
   453
    unfolding divmod_div_mod divmod_rel_def
haftmann@26100
   454
    using assms by (cases "m div n = 0")
haftmann@26100
   455
      (auto simp add: gr0_conv_Suc [of "m div n"])
haftmann@26100
   456
qed
haftmann@25942
   457
haftmann@26100
   458
lemma divmod_step:
haftmann@26100
   459
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   460
  shows "divmod m n = (Suc ((m - n) div n), (m - n) mod n)"
haftmann@26100
   461
proof -
haftmann@26100
   462
  from divmod_rel have divmod_m_n: "divmod_rel m n (m div n) (m mod n)" .
haftmann@26100
   463
  with assms have m_div_n: "m div n \<ge> 1"
haftmann@26100
   464
    by (cases "m div n") (auto simp add: divmod_rel_def)
haftmann@26100
   465
  from assms divmod_m_n have "divmod_rel (m - n) n (m div n - 1) (m mod n)"
haftmann@26100
   466
    by (cases "m div n") (auto simp add: divmod_rel_def)
haftmann@26100
   467
  with divmod_eq have "divmod (m - n) n = (m div n - 1, m mod n)" by simp
haftmann@26100
   468
  moreover from divmod_div_mod have "divmod (m - n) n = ((m - n) div n, (m - n) mod n)" .
haftmann@26100
   469
  ultimately have "m div n = Suc ((m - n) div n)"
haftmann@26100
   470
    and "m mod n = (m - n) mod n" using m_div_n by simp_all
haftmann@26100
   471
  then show ?thesis using divmod_div_mod by simp
haftmann@26100
   472
qed
haftmann@25942
   473
wenzelm@26300
   474
text {* The ''recursion'' equations for @{const div} and @{const mod} *}
haftmann@26100
   475
haftmann@26100
   476
lemma div_less [simp]:
haftmann@26100
   477
  fixes m n :: nat
haftmann@26100
   478
  assumes "m < n"
haftmann@26100
   479
  shows "m div n = 0"
haftmann@26100
   480
  using assms divmod_base divmod_div_mod by simp
haftmann@25942
   481
haftmann@26100
   482
lemma le_div_geq:
haftmann@26100
   483
  fixes m n :: nat
haftmann@26100
   484
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   485
  shows "m div n = Suc ((m - n) div n)"
haftmann@26100
   486
  using assms divmod_step divmod_div_mod by simp
paulson@14267
   487
haftmann@26100
   488
lemma mod_less [simp]:
haftmann@26100
   489
  fixes m n :: nat
haftmann@26100
   490
  assumes "m < n"
haftmann@26100
   491
  shows "m mod n = m"
haftmann@26100
   492
  using assms divmod_base divmod_div_mod by simp
haftmann@26100
   493
haftmann@26100
   494
lemma le_mod_geq:
haftmann@26100
   495
  fixes m n :: nat
haftmann@26100
   496
  assumes "n \<le> m"
haftmann@26100
   497
  shows "m mod n = (m - n) mod n"
haftmann@26100
   498
  using assms divmod_step divmod_div_mod by (cases "n = 0") simp_all
paulson@14267
   499
haftmann@25942
   500
instance proof
haftmann@26100
   501
  fix m n :: nat show "m div n * n + m mod n = m"
haftmann@26100
   502
    using divmod_rel [of m n] by (simp add: divmod_rel_def)
haftmann@25942
   503
next
haftmann@26100
   504
  fix n :: nat show "n div 0 = 0"
haftmann@26100
   505
    using divmod_zero divmod_div_mod [of n 0] by simp
haftmann@25942
   506
next
haftmann@27651
   507
  fix n :: nat show "0 div n = 0"
haftmann@27651
   508
    using divmod_rel [of 0 n] by (cases n) (simp_all add: divmod_rel_def)
haftmann@27651
   509
next
haftmann@27651
   510
  fix m n q :: nat assume "n \<noteq> 0" then show "(q + m * n) div n = m + q div n"
haftmann@25942
   511
    by (induct m) (simp_all add: le_div_geq)
haftmann@25942
   512
qed
haftmann@26100
   513
haftmann@25942
   514
end
paulson@14267
   515
haftmann@26100
   516
text {* Simproc for cancelling @{const div} and @{const mod} *}
haftmann@25942
   517
haftmann@27651
   518
(*lemmas mod_div_equality_nat = semiring_div_class.times_div_mod_plus_zero_one.mod_div_equality [of "m\<Colon>nat" n, standard]
haftmann@27651
   519
lemmas mod_div_equality2_nat = mod_div_equality2 [of "n\<Colon>nat" m, standard*)
haftmann@25942
   520
haftmann@25942
   521
ML {*
haftmann@25942
   522
structure CancelDivModData =
haftmann@25942
   523
struct
haftmann@25942
   524
haftmann@26100
   525
val div_name = @{const_name div};
haftmann@26100
   526
val mod_name = @{const_name mod};
haftmann@25942
   527
val mk_binop = HOLogic.mk_binop;
haftmann@26100
   528
val mk_sum = ArithData.mk_sum;
haftmann@26100
   529
val dest_sum = ArithData.dest_sum;
haftmann@25942
   530
haftmann@25942
   531
(*logic*)
paulson@14267
   532
haftmann@25942
   533
val div_mod_eqs = map mk_meta_eq [@{thm div_mod_equality}, @{thm div_mod_equality2}]
haftmann@25942
   534
haftmann@25942
   535
val trans = trans
haftmann@25942
   536
haftmann@25942
   537
val prove_eq_sums =
haftmann@25942
   538
  let val simps = @{thm add_0} :: @{thm add_0_right} :: @{thms add_ac}
haftmann@26100
   539
  in ArithData.prove_conv all_tac (ArithData.simp_all_tac simps) end;
haftmann@25942
   540
haftmann@25942
   541
end;
haftmann@25942
   542
haftmann@25942
   543
structure CancelDivMod = CancelDivModFun(CancelDivModData);
haftmann@25942
   544
wenzelm@28262
   545
val cancel_div_mod_proc = Simplifier.simproc (the_context ())
haftmann@26100
   546
  "cancel_div_mod" ["(m::nat) + n"] (K CancelDivMod.proc);
haftmann@25942
   547
haftmann@25942
   548
Addsimprocs[cancel_div_mod_proc];
haftmann@25942
   549
*}
haftmann@25942
   550
haftmann@26100
   551
text {* code generator setup *}
haftmann@26100
   552
haftmann@26100
   553
lemma divmod_if [code]: "divmod m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@26100
   554
  let (q, r) = divmod (m - n) n in (Suc q, r))"
nipkow@29667
   555
by (simp add: divmod_zero divmod_base divmod_step)
haftmann@26100
   556
    (simp add: divmod_div_mod)
haftmann@26100
   557
haftmann@26100
   558
code_modulename SML
haftmann@26100
   559
  Divides Nat
haftmann@26100
   560
haftmann@26100
   561
code_modulename OCaml
haftmann@26100
   562
  Divides Nat
haftmann@26100
   563
haftmann@26100
   564
code_modulename Haskell
haftmann@26100
   565
  Divides Nat
haftmann@26100
   566
haftmann@26100
   567
haftmann@26100
   568
subsubsection {* Quotient *}
haftmann@26100
   569
haftmann@26100
   570
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   571
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   572
haftmann@26100
   573
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   574
by (simp add: div_geq)
haftmann@26100
   575
haftmann@26100
   576
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   577
by simp
haftmann@26100
   578
haftmann@26100
   579
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   580
by simp
haftmann@26100
   581
haftmann@25942
   582
haftmann@25942
   583
subsubsection {* Remainder *}
haftmann@25942
   584
haftmann@26100
   585
lemma mod_less_divisor [simp]:
haftmann@26100
   586
  fixes m n :: nat
haftmann@26100
   587
  assumes "n > 0"
haftmann@26100
   588
  shows "m mod n < (n::nat)"
haftmann@26100
   589
  using assms divmod_rel unfolding divmod_rel_def by auto
paulson@14267
   590
haftmann@26100
   591
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   592
  fixes m n :: nat
haftmann@26100
   593
  shows "m mod n \<le> m"
haftmann@26100
   594
proof (rule add_leD2)
haftmann@26100
   595
  from mod_div_equality have "m div n * n + m mod n = m" .
haftmann@26100
   596
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   597
qed
haftmann@26100
   598
haftmann@26100
   599
lemma mod_geq: "\<not> m < (n\<Colon>nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
   600
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   601
haftmann@26100
   602
lemma mod_if: "m mod (n\<Colon>nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
   603
by (simp add: le_mod_geq)
haftmann@26100
   604
paulson@14267
   605
lemma mod_1 [simp]: "m mod Suc 0 = 0"
nipkow@29667
   606
by (induct m) (simp_all add: mod_geq)
paulson@14267
   607
haftmann@26100
   608
lemma mod_mult_distrib: "(m mod n) * (k\<Colon>nat) = (m * k) mod (n * k)"
wenzelm@22718
   609
  apply (cases "n = 0", simp)
wenzelm@22718
   610
  apply (cases "k = 0", simp)
wenzelm@22718
   611
  apply (induct m rule: nat_less_induct)
wenzelm@22718
   612
  apply (subst mod_if, simp)
wenzelm@22718
   613
  apply (simp add: mod_geq diff_mult_distrib)
wenzelm@22718
   614
  done
paulson@14267
   615
paulson@14267
   616
lemma mod_mult_distrib2: "(k::nat) * (m mod n) = (k*m) mod (k*n)"
nipkow@29667
   617
by (simp add: mult_commute [of k] mod_mult_distrib)
paulson@14267
   618
paulson@14267
   619
(* a simple rearrangement of mod_div_equality: *)
paulson@14267
   620
lemma mult_div_cancel: "(n::nat) * (m div n) = m - (m mod n)"
nipkow@29667
   621
by (cut_tac a = m and b = n in mod_div_equality2, arith)
paulson@14267
   622
nipkow@15439
   623
lemma mod_le_divisor[simp]: "0 < n \<Longrightarrow> m mod n \<le> (n::nat)"
wenzelm@22718
   624
  apply (drule mod_less_divisor [where m = m])
wenzelm@22718
   625
  apply simp
wenzelm@22718
   626
  done
paulson@14267
   627
haftmann@26100
   628
subsubsection {* Quotient and Remainder *}
paulson@14267
   629
haftmann@26100
   630
lemma divmod_rel_mult1_eq:
haftmann@26100
   631
  "[| divmod_rel b c q r; c > 0 |]
haftmann@26100
   632
   ==> divmod_rel (a*b) c (a*q + a*r div c) (a*r mod c)"
nipkow@29667
   633
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
paulson@14267
   634
paulson@14267
   635
lemma div_mult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::nat)"
nipkow@25134
   636
apply (cases "c = 0", simp)
haftmann@26100
   637
apply (blast intro: divmod_rel [THEN divmod_rel_mult1_eq, THEN div_eq])
nipkow@25134
   638
done
paulson@14267
   639
paulson@14267
   640
lemma mod_mult1_eq: "(a*b) mod c = a*(b mod c) mod (c::nat)"
nipkow@29667
   641
by (rule mod_mult_right_eq)
paulson@14267
   642
paulson@14267
   643
lemma mod_mult1_eq': "(a*b) mod (c::nat) = ((a mod c) * b) mod c"
nipkow@29667
   644
by (rule mod_mult_left_eq)
paulson@14267
   645
nipkow@25162
   646
lemma mod_mult_distrib_mod:
nipkow@25162
   647
  "(a*b) mod (c::nat) = ((a mod c) * (b mod c)) mod c"
nipkow@29667
   648
by (rule mod_mult_eq)
paulson@14267
   649
haftmann@26100
   650
lemma divmod_rel_add1_eq:
haftmann@26100
   651
  "[| divmod_rel a c aq ar; divmod_rel b c bq br;  c > 0 |]
haftmann@26100
   652
   ==> divmod_rel (a + b) c (aq + bq + (ar+br) div c) ((ar + br) mod c)"
nipkow@29667
   653
by (auto simp add: split_ifs divmod_rel_def algebra_simps)
paulson@14267
   654
paulson@14267
   655
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   656
lemma div_add1_eq:
nipkow@25134
   657
  "(a+b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
nipkow@25134
   658
apply (cases "c = 0", simp)
haftmann@26100
   659
apply (blast intro: divmod_rel_add1_eq [THEN div_eq] divmod_rel)
nipkow@25134
   660
done
paulson@14267
   661
paulson@14267
   662
lemma mod_add1_eq: "(a+b) mod (c::nat) = (a mod c + b mod c) mod c"
nipkow@29667
   663
by (rule mod_add_eq)
paulson@14267
   664
paulson@14267
   665
lemma mod_lemma: "[| (0::nat) < c; r < b |] ==> b * (q mod c) + r < b * c"
wenzelm@22718
   666
  apply (cut_tac m = q and n = c in mod_less_divisor)
wenzelm@22718
   667
  apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
wenzelm@22718
   668
  apply (erule_tac P = "%x. ?lhs < ?rhs x" in ssubst)
wenzelm@22718
   669
  apply (simp add: add_mult_distrib2)
wenzelm@22718
   670
  done
paulson@10559
   671
haftmann@26100
   672
lemma divmod_rel_mult2_eq: "[| divmod_rel a b q r;  0 < b;  0 < c |]
haftmann@26100
   673
      ==> divmod_rel a (b*c) (q div c) (b*(q mod c) + r)"
nipkow@29667
   674
by (auto simp add: mult_ac divmod_rel_def add_mult_distrib2 [symmetric] mod_lemma)
paulson@14267
   675
paulson@14267
   676
lemma div_mult2_eq: "a div (b*c) = (a div b) div (c::nat)"
wenzelm@22718
   677
  apply (cases "b = 0", simp)
wenzelm@22718
   678
  apply (cases "c = 0", simp)
haftmann@26100
   679
  apply (force simp add: divmod_rel [THEN divmod_rel_mult2_eq, THEN div_eq])
wenzelm@22718
   680
  done
paulson@14267
   681
paulson@14267
   682
lemma mod_mult2_eq: "a mod (b*c) = b*(a div b mod c) + a mod (b::nat)"
wenzelm@22718
   683
  apply (cases "b = 0", simp)
wenzelm@22718
   684
  apply (cases "c = 0", simp)
haftmann@26100
   685
  apply (auto simp add: mult_commute divmod_rel [THEN divmod_rel_mult2_eq, THEN mod_eq])
wenzelm@22718
   686
  done
paulson@14267
   687
paulson@14267
   688
haftmann@25942
   689
subsubsection{*Cancellation of Common Factors in Division*}
paulson@14267
   690
paulson@14267
   691
lemma div_mult_mult_lemma:
wenzelm@22718
   692
    "[| (0::nat) < b;  0 < c |] ==> (c*a) div (c*b) = a div b"
nipkow@29667
   693
by (auto simp add: div_mult2_eq)
paulson@14267
   694
paulson@14267
   695
lemma div_mult_mult1 [simp]: "(0::nat) < c ==> (c*a) div (c*b) = a div b"
wenzelm@22718
   696
  apply (cases "b = 0")
wenzelm@22718
   697
  apply (auto simp add: linorder_neq_iff [of b] div_mult_mult_lemma)
wenzelm@22718
   698
  done
paulson@14267
   699
paulson@14267
   700
lemma div_mult_mult2 [simp]: "(0::nat) < c ==> (a*c) div (b*c) = a div b"
wenzelm@22718
   701
  apply (drule div_mult_mult1)
wenzelm@22718
   702
  apply (auto simp add: mult_commute)
wenzelm@22718
   703
  done
paulson@14267
   704
paulson@14267
   705
haftmann@25942
   706
subsubsection{*Further Facts about Quotient and Remainder*}
paulson@14267
   707
paulson@14267
   708
lemma div_1 [simp]: "m div Suc 0 = m"
nipkow@29667
   709
by (induct m) (simp_all add: div_geq)
paulson@14267
   710
paulson@14267
   711
paulson@14267
   712
(* Monotonicity of div in first argument *)
paulson@14267
   713
lemma div_le_mono [rule_format (no_asm)]:
wenzelm@22718
   714
    "\<forall>m::nat. m \<le> n --> (m div k) \<le> (n div k)"
paulson@14267
   715
apply (case_tac "k=0", simp)
paulson@15251
   716
apply (induct "n" rule: nat_less_induct, clarify)
paulson@14267
   717
apply (case_tac "n<k")
paulson@14267
   718
(* 1  case n<k *)
paulson@14267
   719
apply simp
paulson@14267
   720
(* 2  case n >= k *)
paulson@14267
   721
apply (case_tac "m<k")
paulson@14267
   722
(* 2.1  case m<k *)
paulson@14267
   723
apply simp
paulson@14267
   724
(* 2.2  case m>=k *)
nipkow@15439
   725
apply (simp add: div_geq diff_le_mono)
paulson@14267
   726
done
paulson@14267
   727
paulson@14267
   728
(* Antimonotonicity of div in second argument *)
paulson@14267
   729
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   730
apply (subgoal_tac "0<n")
wenzelm@22718
   731
 prefer 2 apply simp
paulson@15251
   732
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   733
apply (rename_tac "k")
paulson@14267
   734
apply (case_tac "k<n", simp)
paulson@14267
   735
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   736
 prefer 2 apply simp
paulson@14267
   737
apply (simp add: div_geq)
paulson@15251
   738
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   739
 prefer 2
paulson@14267
   740
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   741
apply (rule le_trans, simp)
nipkow@15439
   742
apply (simp)
paulson@14267
   743
done
paulson@14267
   744
paulson@14267
   745
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   746
apply (case_tac "n=0", simp)
paulson@14267
   747
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   748
apply (rule div_le_mono2)
paulson@14267
   749
apply (simp_all (no_asm_simp))
paulson@14267
   750
done
paulson@14267
   751
wenzelm@22718
   752
(* Similar for "less than" *)
paulson@17085
   753
lemma div_less_dividend [rule_format]:
paulson@14267
   754
     "!!n::nat. 1<n ==> 0 < m --> m div n < m"
paulson@15251
   755
apply (induct_tac m rule: nat_less_induct)
paulson@14267
   756
apply (rename_tac "m")
paulson@14267
   757
apply (case_tac "m<n", simp)
paulson@14267
   758
apply (subgoal_tac "0<n")
wenzelm@22718
   759
 prefer 2 apply simp
paulson@14267
   760
apply (simp add: div_geq)
paulson@14267
   761
apply (case_tac "n<m")
paulson@15251
   762
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   763
  apply (rule impI less_trans_Suc)+
paulson@14267
   764
apply assumption
nipkow@15439
   765
  apply (simp_all)
paulson@14267
   766
done
paulson@14267
   767
paulson@17085
   768
declare div_less_dividend [simp]
paulson@17085
   769
paulson@14267
   770
text{*A fact for the mutilated chess board*}
paulson@14267
   771
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   772
apply (case_tac "n=0", simp)
paulson@15251
   773
apply (induct "m" rule: nat_less_induct)
paulson@14267
   774
apply (case_tac "Suc (na) <n")
paulson@14267
   775
(* case Suc(na) < n *)
paulson@14267
   776
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   777
(* case n \<le> Suc(na) *)
paulson@16796
   778
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   779
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   780
done
paulson@14267
   781
huffman@29403
   782
lemma nat_mod_div_trivial: "m mod n div n = (0 :: nat)"
nipkow@29667
   783
by simp
paulson@14437
   784
huffman@29403
   785
lemma nat_mod_mod_trivial: "m mod n mod n = (m mod n :: nat)"
nipkow@29667
   786
by simp
paulson@14437
   787
paulson@14267
   788
haftmann@27651
   789
subsubsection {* The Divides Relation *}
paulson@24286
   790
paulson@14267
   791
lemma dvd_1_left [iff]: "Suc 0 dvd k"
wenzelm@22718
   792
  unfolding dvd_def by simp
paulson@14267
   793
paulson@14267
   794
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
nipkow@29667
   795
by (simp add: dvd_def)
paulson@14267
   796
paulson@14267
   797
lemma dvd_anti_sym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
wenzelm@22718
   798
  unfolding dvd_def
wenzelm@22718
   799
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc mult_eq_1_iff)
paulson@14267
   800
haftmann@23684
   801
text {* @{term "op dvd"} is a partial order *}
haftmann@23684
   802
haftmann@29509
   803
interpretation dvd!: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
haftmann@28823
   804
  proof qed (auto intro: dvd_refl dvd_trans dvd_anti_sym)
paulson@14267
   805
paulson@14267
   806
lemma dvd_diff: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
wenzelm@22718
   807
  unfolding dvd_def
wenzelm@22718
   808
  by (blast intro: diff_mult_distrib2 [symmetric])
paulson@14267
   809
paulson@14267
   810
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
wenzelm@22718
   811
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   812
  apply (blast intro: dvd_add)
wenzelm@22718
   813
  done
paulson@14267
   814
paulson@14267
   815
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
nipkow@29667
   816
by (drule_tac m = m in dvd_diff, auto)
paulson@14267
   817
paulson@14267
   818
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
wenzelm@22718
   819
  apply (rule iffI)
wenzelm@22718
   820
   apply (erule_tac [2] dvd_add)
wenzelm@22718
   821
   apply (rule_tac [2] dvd_refl)
wenzelm@22718
   822
  apply (subgoal_tac "n = (n+k) -k")
wenzelm@22718
   823
   prefer 2 apply simp
wenzelm@22718
   824
  apply (erule ssubst)
wenzelm@22718
   825
  apply (erule dvd_diff)
wenzelm@22718
   826
  apply (rule dvd_refl)
wenzelm@22718
   827
  done
paulson@14267
   828
paulson@14267
   829
lemma dvd_mod: "!!n::nat. [| f dvd m; f dvd n |] ==> f dvd m mod n"
wenzelm@22718
   830
  unfolding dvd_def
wenzelm@22718
   831
  apply (case_tac "n = 0", auto)
wenzelm@22718
   832
  apply (blast intro: mod_mult_distrib2 [symmetric])
wenzelm@22718
   833
  done
paulson@14267
   834
paulson@14267
   835
lemma dvd_mod_imp_dvd: "[| (k::nat) dvd m mod n;  k dvd n |] ==> k dvd m"
wenzelm@22718
   836
  apply (subgoal_tac "k dvd (m div n) *n + m mod n")
wenzelm@22718
   837
   apply (simp add: mod_div_equality)
wenzelm@22718
   838
  apply (simp only: dvd_add dvd_mult)
wenzelm@22718
   839
  done
paulson@14267
   840
paulson@14267
   841
lemma dvd_mod_iff: "k dvd n ==> ((k::nat) dvd m mod n) = (k dvd m)"
nipkow@29667
   842
by (blast intro: dvd_mod_imp_dvd dvd_mod)
paulson@14267
   843
paulson@14267
   844
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
wenzelm@22718
   845
  unfolding dvd_def
wenzelm@22718
   846
  apply (erule exE)
wenzelm@22718
   847
  apply (simp add: mult_ac)
wenzelm@22718
   848
  done
paulson@14267
   849
paulson@14267
   850
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
wenzelm@22718
   851
  apply auto
wenzelm@22718
   852
   apply (subgoal_tac "m*n dvd m*1")
wenzelm@22718
   853
   apply (drule dvd_mult_cancel, auto)
wenzelm@22718
   854
  done
paulson@14267
   855
paulson@14267
   856
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
wenzelm@22718
   857
  apply (subst mult_commute)
wenzelm@22718
   858
  apply (erule dvd_mult_cancel1)
wenzelm@22718
   859
  done
paulson@14267
   860
paulson@14267
   861
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
wenzelm@22718
   862
  apply (unfold dvd_def, clarify)
wenzelm@22718
   863
  apply (simp_all (no_asm_use) add: zero_less_mult_iff)
wenzelm@22718
   864
  apply (erule conjE)
wenzelm@22718
   865
  apply (rule le_trans)
wenzelm@22718
   866
   apply (rule_tac [2] le_refl [THEN mult_le_mono])
wenzelm@22718
   867
   apply (erule_tac [2] Suc_leI, simp)
wenzelm@22718
   868
  done
paulson@14267
   869
paulson@14267
   870
lemma dvd_mult_div_cancel: "n dvd m ==> n * (m div n) = (m::nat)"
wenzelm@22718
   871
  apply (subgoal_tac "m mod n = 0")
wenzelm@22718
   872
   apply (simp add: mult_div_cancel)
wenzelm@22718
   873
  apply (simp only: dvd_eq_mod_eq_0)
wenzelm@22718
   874
  done
paulson@14267
   875
haftmann@21408
   876
lemma le_imp_power_dvd: "!!i::nat. m \<le> n ==> i^m dvd i^n"
wenzelm@22718
   877
  apply (unfold dvd_def)
wenzelm@22718
   878
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
wenzelm@22718
   879
  apply (simp add: power_add)
wenzelm@22718
   880
  done
haftmann@21408
   881
nipkow@25162
   882
lemma nat_zero_less_power_iff [simp]: "(x^n > 0) = (x > (0::nat) | n=0)"
wenzelm@22718
   883
  by (induct n) auto
haftmann@21408
   884
haftmann@21408
   885
lemma power_le_dvd [rule_format]: "k^j dvd n --> i\<le>j --> k^i dvd (n::nat)"
wenzelm@22718
   886
  apply (induct j)
wenzelm@22718
   887
   apply (simp_all add: le_Suc_eq)
wenzelm@22718
   888
  apply (blast dest!: dvd_mult_right)
wenzelm@22718
   889
  done
haftmann@21408
   890
haftmann@21408
   891
lemma power_dvd_imp_le: "[|i^m dvd i^n;  (1::nat) < i|] ==> m \<le> n"
wenzelm@22718
   892
  apply (rule power_le_imp_le_exp, assumption)
wenzelm@22718
   893
  apply (erule dvd_imp_le, simp)
wenzelm@22718
   894
  done
haftmann@21408
   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@26100
   963
  then have "divmod_rel m n q (m - n * q)"
haftmann@26100
   964
    unfolding divmod_rel_def by (auto simp add: mult_ac)
haftmann@26100
   965
  then show ?rhs using divmod_rel by (rule divmod_rel_unique_div)
haftmann@26100
   966
qed
berghofe@13882
   967
berghofe@13882
   968
theorem split_div':
berghofe@13882
   969
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   970
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
berghofe@13882
   971
  apply (case_tac "0 < n")
berghofe@13882
   972
  apply (simp only: add: split_div_lemma)
haftmann@27651
   973
  apply simp_all
berghofe@13882
   974
  done
berghofe@13882
   975
nipkow@13189
   976
lemma split_mod:
nipkow@13189
   977
 "P(n mod k :: nat) =
nipkow@13189
   978
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   979
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   980
proof
nipkow@13189
   981
  assume P: ?P
nipkow@13189
   982
  show ?Q
nipkow@13189
   983
  proof (cases)
nipkow@13189
   984
    assume "k = 0"
haftmann@27651
   985
    with P show ?Q by simp
nipkow@13189
   986
  next
nipkow@13189
   987
    assume not0: "k \<noteq> 0"
nipkow@13189
   988
    thus ?Q
nipkow@13189
   989
    proof (simp, intro allI impI)
nipkow@13189
   990
      fix i j
nipkow@13189
   991
      assume "n = k*i + j" "j < k"
nipkow@13189
   992
      thus "P j" using not0 P by(simp add:add_ac mult_ac)
nipkow@13189
   993
    qed
nipkow@13189
   994
  qed
nipkow@13189
   995
next
nipkow@13189
   996
  assume Q: ?Q
nipkow@13189
   997
  show ?P
nipkow@13189
   998
  proof (cases)
nipkow@13189
   999
    assume "k = 0"
haftmann@27651
  1000
    with Q show ?P by simp
nipkow@13189
  1001
  next
nipkow@13189
  1002
    assume not0: "k \<noteq> 0"
nipkow@13189
  1003
    with Q have R: ?R by simp
nipkow@13189
  1004
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1005
    show ?P by simp
nipkow@13189
  1006
  qed
nipkow@13189
  1007
qed
nipkow@13189
  1008
berghofe@13882
  1009
theorem mod_div_equality': "(m::nat) mod n = m - (m div n) * n"
berghofe@13882
  1010
  apply (rule_tac P="%x. m mod n = x - (m div n) * n" in
berghofe@13882
  1011
    subst [OF mod_div_equality [of _ n]])
berghofe@13882
  1012
  apply arith
berghofe@13882
  1013
  done
berghofe@13882
  1014
haftmann@22800
  1015
lemma div_mod_equality':
haftmann@22800
  1016
  fixes m n :: nat
haftmann@22800
  1017
  shows "m div n * n = m - m mod n"
haftmann@22800
  1018
proof -
haftmann@22800
  1019
  have "m mod n \<le> m mod n" ..
haftmann@22800
  1020
  from div_mod_equality have 
haftmann@22800
  1021
    "m div n * n + m mod n - m mod n = m - m mod n" by simp
haftmann@22800
  1022
  with diff_add_assoc [OF `m mod n \<le> m mod n`, of "m div n * n"] have
haftmann@22800
  1023
    "m div n * n + (m mod n - m mod n) = m - m mod n"
haftmann@22800
  1024
    by simp
haftmann@22800
  1025
  then show ?thesis by simp
haftmann@22800
  1026
qed
haftmann@22800
  1027
haftmann@22800
  1028
haftmann@25942
  1029
subsubsection {*An ``induction'' law for modulus arithmetic.*}
paulson@14640
  1030
paulson@14640
  1031
lemma mod_induct_0:
paulson@14640
  1032
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1033
  and base: "P i" and i: "i<p"
paulson@14640
  1034
  shows "P 0"
paulson@14640
  1035
proof (rule ccontr)
paulson@14640
  1036
  assume contra: "\<not>(P 0)"
paulson@14640
  1037
  from i have p: "0<p" by simp
paulson@14640
  1038
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1039
  proof
paulson@14640
  1040
    fix k
paulson@14640
  1041
    show "?A k"
paulson@14640
  1042
    proof (induct k)
paulson@14640
  1043
      show "?A 0" by simp  -- "by contradiction"
paulson@14640
  1044
    next
paulson@14640
  1045
      fix n
paulson@14640
  1046
      assume ih: "?A n"
paulson@14640
  1047
      show "?A (Suc n)"
paulson@14640
  1048
      proof (clarsimp)
wenzelm@22718
  1049
        assume y: "P (p - Suc n)"
wenzelm@22718
  1050
        have n: "Suc n < p"
wenzelm@22718
  1051
        proof (rule ccontr)
wenzelm@22718
  1052
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1053
          hence "p - Suc n = 0"
wenzelm@22718
  1054
            by simp
wenzelm@22718
  1055
          with y contra show "False"
wenzelm@22718
  1056
            by simp
wenzelm@22718
  1057
        qed
wenzelm@22718
  1058
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1059
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1060
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1061
          by blast
wenzelm@22718
  1062
        show "False"
wenzelm@22718
  1063
        proof (cases "n=0")
wenzelm@22718
  1064
          case True
wenzelm@22718
  1065
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1066
        next
wenzelm@22718
  1067
          case False
wenzelm@22718
  1068
          with p have "p-n < p" by arith
wenzelm@22718
  1069
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1070
        qed
paulson@14640
  1071
      qed
paulson@14640
  1072
    qed
paulson@14640
  1073
  qed
paulson@14640
  1074
  moreover
paulson@14640
  1075
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1076
    by (blast dest: less_imp_add_positive)
paulson@14640
  1077
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1078
  moreover
paulson@14640
  1079
  note base
paulson@14640
  1080
  ultimately
paulson@14640
  1081
  show "False" by blast
paulson@14640
  1082
qed
paulson@14640
  1083
paulson@14640
  1084
lemma mod_induct:
paulson@14640
  1085
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1086
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1087
  shows "P j"
paulson@14640
  1088
proof -
paulson@14640
  1089
  have "\<forall>j<p. P j"
paulson@14640
  1090
  proof
paulson@14640
  1091
    fix j
paulson@14640
  1092
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1093
    proof (induct j)
paulson@14640
  1094
      from step base i show "?A 0"
wenzelm@22718
  1095
        by (auto elim: mod_induct_0)
paulson@14640
  1096
    next
paulson@14640
  1097
      fix k
paulson@14640
  1098
      assume ih: "?A k"
paulson@14640
  1099
      show "?A (Suc k)"
paulson@14640
  1100
      proof
wenzelm@22718
  1101
        assume suc: "Suc k < p"
wenzelm@22718
  1102
        hence k: "k<p" by simp
wenzelm@22718
  1103
        with ih have "P k" ..
wenzelm@22718
  1104
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1105
          by blast
wenzelm@22718
  1106
        moreover
wenzelm@22718
  1107
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1108
          by simp
wenzelm@22718
  1109
        ultimately
wenzelm@22718
  1110
        show "P (Suc k)" by simp
paulson@14640
  1111
      qed
paulson@14640
  1112
    qed
paulson@14640
  1113
  qed
paulson@14640
  1114
  with j show ?thesis by blast
paulson@14640
  1115
qed
paulson@14640
  1116
paulson@3366
  1117
end