src/HOL/Divides.thy
author haftmann
Sun Oct 08 22:28:21 2017 +0200 (20 months ago)
changeset 66806 a4e82b58d833
parent 66801 f3fda9777f9a
child 66808 1907167b6038
permissions -rw-r--r--
abolished (semi)ring_div in favour of euclidean_(semi)ring_cancel
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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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section \<open>More on quotient and remainder\<close>
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theory Divides
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imports Parity
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begin
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subsection \<open>Parity\<close>
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class unique_euclidean_semiring_parity = unique_euclidean_semiring +
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  assumes parity: "a mod 2 = 0 \<or> a mod 2 = 1"
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  assumes one_mod_two_eq_one [simp]: "1 mod 2 = 1"
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  assumes zero_not_eq_two: "0 \<noteq> 2"
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begin
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lemma parity_cases [case_names even odd]:
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  assumes "a mod 2 = 0 \<Longrightarrow> P"
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  assumes "a mod 2 = 1 \<Longrightarrow> P"
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  shows P
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  using assms parity by blast
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lemma one_div_two_eq_zero [simp]:
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  "1 div 2 = 0"
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proof (cases "2 = 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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  from div_mult_mod_eq have "1 div 2 * 2 + 1 mod 2 = 1" .
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  with one_mod_two_eq_one have "1 div 2 * 2 + 1 = 1" by simp
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  then have "1 div 2 * 2 = 0" by (simp add: ac_simps add_left_imp_eq del: mult_eq_0_iff)
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  then have "1 div 2 = 0 \<or> 2 = 0" by simp
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  with False show ?thesis by auto
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qed
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lemma not_mod_2_eq_0_eq_1 [simp]:
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  "a mod 2 \<noteq> 0 \<longleftrightarrow> a mod 2 = 1"
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  by (cases a rule: parity_cases) simp_all
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lemma not_mod_2_eq_1_eq_0 [simp]:
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  "a mod 2 \<noteq> 1 \<longleftrightarrow> a mod 2 = 0"
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  by (cases a rule: parity_cases) simp_all
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subclass semiring_parity
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proof (unfold_locales, unfold dvd_eq_mod_eq_0 not_mod_2_eq_0_eq_1)
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  show "1 mod 2 = 1"
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    by (fact one_mod_two_eq_one)
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next
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  fix a b
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  assume "a mod 2 = 1"
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  moreover assume "b mod 2 = 1"
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  ultimately show "(a + b) mod 2 = 0"
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    using mod_add_eq [of a 2 b] by simp
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next
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  fix a b
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  assume "(a * b) mod 2 = 0"
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  then have "(a mod 2) * (b mod 2) mod 2 = 0"
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    by (simp add: mod_mult_eq)
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  then have "(a mod 2) * (b mod 2) = 0"
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    by (cases "a mod 2 = 0") simp_all
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  then show "a mod 2 = 0 \<or> b mod 2 = 0"
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    by (rule divisors_zero)
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next
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  fix a
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  assume "a mod 2 = 1"
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  then have "a = a div 2 * 2 + 1"
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    using div_mult_mod_eq [of a 2] by simp
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  then show "\<exists>b. a = b + 1" ..
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qed
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lemma even_iff_mod_2_eq_zero:
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  "even a \<longleftrightarrow> a mod 2 = 0"
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  by (fact dvd_eq_mod_eq_0)
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lemma odd_iff_mod_2_eq_one:
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  "odd a \<longleftrightarrow> a mod 2 = 1"
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  by (simp add: even_iff_mod_2_eq_zero)
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lemma even_succ_div_two [simp]:
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  "even a \<Longrightarrow> (a + 1) div 2 = a div 2"
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  by (cases "a = 0") (auto elim!: evenE dest: mult_not_zero)
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lemma odd_succ_div_two [simp]:
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  "odd a \<Longrightarrow> (a + 1) div 2 = a div 2 + 1"
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  by (auto elim!: oddE simp add: zero_not_eq_two [symmetric] add.assoc)
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lemma even_two_times_div_two:
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  "even a \<Longrightarrow> 2 * (a div 2) = a"
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  by (fact dvd_mult_div_cancel)
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lemma odd_two_times_div_two_succ [simp]:
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  "odd a \<Longrightarrow> 2 * (a div 2) + 1 = a"
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  using mult_div_mod_eq [of 2 a] by (simp add: even_iff_mod_2_eq_zero)
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end
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subsection \<open>Numeral division with a pragmatic type class\<close>
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text \<open>
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  The following type class contains everything necessary to formulate
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  a division algorithm in ring structures with numerals, restricted
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  to its positive segments.  This is its primary motivation, and it
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  could surely be formulated using a more fine-grained, more algebraic
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  and less technical class hierarchy.
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\<close>
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class unique_euclidean_semiring_numeral = unique_euclidean_semiring + linordered_semidom +
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  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
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    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
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    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
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    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
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    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
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    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
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    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
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    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
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  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
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  fixes divmod :: "num \<Rightarrow> num \<Rightarrow> 'a \<times> 'a"
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    and divmod_step :: "num \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'a \<times> 'a"
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  assumes divmod_def: "divmod m n = (numeral m div numeral n, numeral m mod numeral n)"
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    and divmod_step_def: "divmod_step l qr = (let (q, r) = qr
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    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
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    else (2 * q, r))"
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    \<comment> \<open>These are conceptually definitions but force generated code
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    to be monomorphic wrt. particular instances of this class which
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    yields a significant speedup.\<close>
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begin
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subclass unique_euclidean_semiring_parity
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proof
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  fix a
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  show "a mod 2 = 0 \<or> a mod 2 = 1"
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  proof (rule ccontr)
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    assume "\<not> (a mod 2 = 0 \<or> a mod 2 = 1)"
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    then have "a mod 2 \<noteq> 0" and "a mod 2 \<noteq> 1" by simp_all
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    have "0 < 2" by simp
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    with pos_mod_bound pos_mod_sign have "0 \<le> a mod 2" "a mod 2 < 2" by simp_all
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    with \<open>a mod 2 \<noteq> 0\<close> have "0 < a mod 2" by simp
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    with discrete have "1 \<le> a mod 2" by simp
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    with \<open>a mod 2 \<noteq> 1\<close> have "1 < a mod 2" by simp
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    with discrete have "2 \<le> a mod 2" by simp
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    with \<open>a mod 2 < 2\<close> show False by simp
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  qed
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next
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  show "1 mod 2 = 1"
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    by (rule mod_less) simp_all
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next
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  show "0 \<noteq> 2"
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    by simp
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qed
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lemma divmod_digit_1:
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  assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
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  shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
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    and "a mod (2 * b) - b = a mod b" (is "?Q")
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proof -
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  from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
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    by (auto intro: trans)
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  with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
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  then have [simp]: "1 \<le> a div b" by (simp add: discrete)
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  with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
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  define w where "w = a div b mod 2"
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  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
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  have mod_w: "a mod (2 * b) = a mod b + b * w"
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    by (simp add: w_def mod_mult2_eq ac_simps)
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  from assms w_exhaust have "w = 1"
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    by (auto simp add: mod_w) (insert mod_less, auto)
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  with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
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  have "2 * (a div (2 * b)) = a div b - w"
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    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
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  with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
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  then show ?P and ?Q
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    by (simp_all add: div mod add_implies_diff [symmetric])
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qed
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lemma divmod_digit_0:
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  assumes "0 < b" and "a mod (2 * b) < b"
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  shows "2 * (a div (2 * b)) = a div b" (is "?P")
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    and "a mod (2 * b) = a mod b" (is "?Q")
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proof -
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  define w where "w = a div b mod 2"
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  with parity have w_exhaust: "w = 0 \<or> w = 1" by auto
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  have mod_w: "a mod (2 * b) = a mod b + b * w"
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    by (simp add: w_def mod_mult2_eq ac_simps)
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  moreover have "b \<le> a mod b + b"
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  proof -
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    from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
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    then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
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    then show ?thesis by simp
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  qed
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  moreover note assms w_exhaust
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  ultimately have "w = 0" by auto
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  with mod_w have mod: "a mod (2 * b) = a mod b" by simp
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  have "2 * (a div (2 * b)) = a div b - w"
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    by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
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  with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
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  then show ?P and ?Q
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    by (simp_all add: div mod)
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qed
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lemma fst_divmod:
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  "fst (divmod m n) = numeral m div numeral n"
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  by (simp add: divmod_def)
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lemma snd_divmod:
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  "snd (divmod m n) = numeral m mod numeral n"
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  by (simp add: divmod_def)
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text \<open>
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  This is a formulation of one step (referring to one digit position)
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  in school-method division: compare the dividend at the current
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  digit position with the remainder from previous division steps
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  and evaluate accordingly.
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\<close>
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lemma divmod_step_eq [simp]:
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  "divmod_step l (q, r) = (if numeral l \<le> r
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    then (2 * q + 1, r - numeral l) else (2 * q, r))"
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  by (simp add: divmod_step_def)
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text \<open>
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  This is a formulation of school-method division.
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  If the divisor is smaller than the dividend, terminate.
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  If not, shift the dividend to the right until termination
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  occurs and then reiterate single division steps in the
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  opposite direction.
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\<close>
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lemma divmod_divmod_step:
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  "divmod m n = (if m < n then (0, numeral m)
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    else divmod_step n (divmod m (Num.Bit0 n)))"
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proof (cases "m < n")
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  case True then have "numeral m < numeral n" by simp
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  then show ?thesis
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    by (simp add: prod_eq_iff div_less mod_less fst_divmod snd_divmod)
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next
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  case False
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  have "divmod m n =
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    divmod_step n (numeral m div (2 * numeral n),
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      numeral m mod (2 * numeral n))"
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  proof (cases "numeral n \<le> numeral m mod (2 * numeral n)")
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    case True
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    with divmod_step_eq
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      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
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        (2 * (numeral m div (2 * numeral n)) + 1, numeral m mod (2 * numeral n) - numeral n)"
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        by simp
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    moreover from True divmod_digit_1 [of "numeral m" "numeral n"]
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      have "2 * (numeral m div (2 * numeral n)) + 1 = numeral m div numeral n"
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      and "numeral m mod (2 * numeral n) - numeral n = numeral m mod numeral n"
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      by simp_all
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    ultimately show ?thesis by (simp only: divmod_def)
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  next
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    case False then have *: "numeral m mod (2 * numeral n) < numeral n"
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      by (simp add: not_le)
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    with divmod_step_eq
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      have "divmod_step n (numeral m div (2 * numeral n), numeral m mod (2 * numeral n)) =
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        (2 * (numeral m div (2 * numeral n)), numeral m mod (2 * numeral n))"
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        by auto
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    moreover from * divmod_digit_0 [of "numeral n" "numeral m"]
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      have "2 * (numeral m div (2 * numeral n)) = numeral m div numeral n"
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      and "numeral m mod (2 * numeral n) = numeral m mod numeral n"
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      by (simp_all only: zero_less_numeral)
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    ultimately show ?thesis by (simp only: divmod_def)
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  qed
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  then have "divmod m n =
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    divmod_step n (numeral m div numeral (Num.Bit0 n),
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      numeral m mod numeral (Num.Bit0 n))"
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    by (simp only: numeral.simps distrib mult_1)
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  then have "divmod m n = divmod_step n (divmod m (Num.Bit0 n))"
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    by (simp add: divmod_def)
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  with False show ?thesis by simp
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qed
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text \<open>The division rewrite proper -- first, trivial results involving \<open>1\<close>\<close>
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lemma divmod_trivial [simp]:
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  "divmod Num.One Num.One = (numeral Num.One, 0)"
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  "divmod (Num.Bit0 m) Num.One = (numeral (Num.Bit0 m), 0)"
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  "divmod (Num.Bit1 m) Num.One = (numeral (Num.Bit1 m), 0)"
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  "divmod num.One (num.Bit0 n) = (0, Numeral1)"
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  "divmod num.One (num.Bit1 n) = (0, Numeral1)"
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  using divmod_divmod_step [of "Num.One"] by (simp_all add: divmod_def)
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text \<open>Division by an even number is a right-shift\<close>
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lemma divmod_cancel [simp]:
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  "divmod (Num.Bit0 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r))" (is ?P)
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  "divmod (Num.Bit1 m) (Num.Bit0 n) = (case divmod m n of (q, r) \<Rightarrow> (q, 2 * r + 1))" (is ?Q)
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proof -
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  have *: "\<And>q. numeral (Num.Bit0 q) = 2 * numeral q"
haftmann@53069
   294
    "\<And>q. numeral (Num.Bit1 q) = 2 * numeral q + 1"
haftmann@53069
   295
    by (simp_all only: numeral_mult numeral.simps distrib) simp_all
haftmann@53069
   296
  have "1 div 2 = 0" "1 mod 2 = 1" by (auto intro: div_less mod_less)
haftmann@53069
   297
  then show ?P and ?Q
haftmann@60867
   298
    by (simp_all add: fst_divmod snd_divmod prod_eq_iff split_def * [of m] * [of n] mod_mult_mult1
haftmann@60867
   299
      div_mult2_eq [of _ _ 2] mod_mult2_eq [of _ _ 2]
haftmann@60867
   300
      add.commute del: numeral_times_numeral)
haftmann@58953
   301
qed
haftmann@58953
   302
haftmann@60867
   303
text \<open>The really hard work\<close>
haftmann@60867
   304
haftmann@61275
   305
lemma divmod_steps [simp]:
haftmann@60867
   306
  "divmod (num.Bit0 m) (num.Bit1 n) =
haftmann@60867
   307
      (if m \<le> n then (0, numeral (num.Bit0 m))
haftmann@60867
   308
       else divmod_step (num.Bit1 n)
haftmann@60867
   309
             (divmod (num.Bit0 m)
haftmann@60867
   310
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   311
  "divmod (num.Bit1 m) (num.Bit1 n) =
haftmann@60867
   312
      (if m < n then (0, numeral (num.Bit1 m))
haftmann@60867
   313
       else divmod_step (num.Bit1 n)
haftmann@60867
   314
             (divmod (num.Bit1 m)
haftmann@60867
   315
               (num.Bit0 (num.Bit1 n))))"
haftmann@60867
   316
  by (simp_all add: divmod_divmod_step)
haftmann@60867
   317
haftmann@61275
   318
lemmas divmod_algorithm_code = divmod_step_eq divmod_trivial divmod_cancel divmod_steps  
haftmann@61275
   319
wenzelm@60758
   320
text \<open>Special case: divisibility\<close>
haftmann@58953
   321
haftmann@58953
   322
definition divides_aux :: "'a \<times> 'a \<Rightarrow> bool"
haftmann@58953
   323
where
haftmann@58953
   324
  "divides_aux qr \<longleftrightarrow> snd qr = 0"
haftmann@58953
   325
haftmann@58953
   326
lemma divides_aux_eq [simp]:
haftmann@58953
   327
  "divides_aux (q, r) \<longleftrightarrow> r = 0"
haftmann@58953
   328
  by (simp add: divides_aux_def)
haftmann@58953
   329
haftmann@58953
   330
lemma dvd_numeral_simp [simp]:
haftmann@58953
   331
  "numeral m dvd numeral n \<longleftrightarrow> divides_aux (divmod n m)"
haftmann@58953
   332
  by (simp add: divmod_def mod_eq_0_iff_dvd)
haftmann@53069
   333
haftmann@60867
   334
text \<open>Generic computation of quotient and remainder\<close>  
haftmann@60867
   335
haftmann@60867
   336
lemma numeral_div_numeral [simp]: 
haftmann@60867
   337
  "numeral k div numeral l = fst (divmod k l)"
haftmann@60867
   338
  by (simp add: fst_divmod)
haftmann@60867
   339
haftmann@60867
   340
lemma numeral_mod_numeral [simp]: 
haftmann@60867
   341
  "numeral k mod numeral l = snd (divmod k l)"
haftmann@60867
   342
  by (simp add: snd_divmod)
haftmann@60867
   343
haftmann@60867
   344
lemma one_div_numeral [simp]:
haftmann@60867
   345
  "1 div numeral n = fst (divmod num.One n)"
haftmann@60867
   346
  by (simp add: fst_divmod)
haftmann@60867
   347
haftmann@60867
   348
lemma one_mod_numeral [simp]:
haftmann@60867
   349
  "1 mod numeral n = snd (divmod num.One n)"
haftmann@60867
   350
  by (simp add: snd_divmod)
haftmann@64630
   351
haftmann@64630
   352
text \<open>Computing congruences modulo \<open>2 ^ q\<close>\<close>
haftmann@64630
   353
haftmann@64630
   354
lemma cong_exp_iff_simps:
haftmann@64630
   355
  "numeral n mod numeral Num.One = 0
haftmann@64630
   356
    \<longleftrightarrow> True"
haftmann@64630
   357
  "numeral (Num.Bit0 n) mod numeral (Num.Bit0 q) = 0
haftmann@64630
   358
    \<longleftrightarrow> numeral n mod numeral q = 0"
haftmann@64630
   359
  "numeral (Num.Bit1 n) mod numeral (Num.Bit0 q) = 0
haftmann@64630
   360
    \<longleftrightarrow> False"
haftmann@64630
   361
  "numeral m mod numeral Num.One = (numeral n mod numeral Num.One)
haftmann@64630
   362
    \<longleftrightarrow> True"
haftmann@64630
   363
  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
haftmann@64630
   364
    \<longleftrightarrow> True"
haftmann@64630
   365
  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
haftmann@64630
   366
    \<longleftrightarrow> False"
haftmann@64630
   367
  "numeral Num.One mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
haftmann@64630
   368
    \<longleftrightarrow> (numeral n mod numeral q) = 0"
haftmann@64630
   369
  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
haftmann@64630
   370
    \<longleftrightarrow> False"
haftmann@64630
   371
  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
haftmann@64630
   372
    \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
haftmann@64630
   373
  "numeral (Num.Bit0 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
haftmann@64630
   374
    \<longleftrightarrow> False"
haftmann@64630
   375
  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral Num.One mod numeral (Num.Bit0 q))
haftmann@64630
   376
    \<longleftrightarrow> (numeral m mod numeral q) = 0"
haftmann@64630
   377
  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit0 n) mod numeral (Num.Bit0 q))
haftmann@64630
   378
    \<longleftrightarrow> False"
haftmann@64630
   379
  "numeral (Num.Bit1 m) mod numeral (Num.Bit0 q) = (numeral (Num.Bit1 n) mod numeral (Num.Bit0 q))
haftmann@64630
   380
    \<longleftrightarrow> numeral m mod numeral q = (numeral n mod numeral q)"
haftmann@64630
   381
  by (auto simp add: case_prod_beta dest: arg_cong [of _ _ even])
haftmann@64630
   382
haftmann@53067
   383
end
haftmann@53067
   384
lp15@60562
   385
wenzelm@60758
   386
subsection \<open>Division on @{typ nat}\<close>
wenzelm@60758
   387
haftmann@61433
   388
context
haftmann@61433
   389
begin
haftmann@61433
   390
wenzelm@60758
   391
text \<open>
haftmann@63950
   392
  We define @{const divide} and @{const modulo} on @{typ nat} by means
haftmann@26100
   393
  of a characteristic relation with two input arguments
wenzelm@61076
   394
  @{term "m::nat"}, @{term "n::nat"} and two output arguments
wenzelm@61076
   395
  @{term "q::nat"}(uotient) and @{term "r::nat"}(emainder).
wenzelm@60758
   396
\<close>
haftmann@26100
   397
haftmann@64635
   398
inductive eucl_rel_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat \<Rightarrow> bool"
haftmann@64635
   399
  where eucl_rel_nat_by0: "eucl_rel_nat m 0 (0, m)"
haftmann@64635
   400
  | eucl_rel_natI: "r < n \<Longrightarrow> m = q * n + r \<Longrightarrow> eucl_rel_nat m n (q, r)"
haftmann@64635
   401
haftmann@64635
   402
text \<open>@{const eucl_rel_nat} is total:\<close>
haftmann@64635
   403
haftmann@64635
   404
qualified lemma eucl_rel_nat_ex:
haftmann@64635
   405
  obtains q r where "eucl_rel_nat m n (q, r)"
haftmann@26100
   406
proof (cases "n = 0")
haftmann@64635
   407
  case True
haftmann@64635
   408
  with that eucl_rel_nat_by0 show thesis
haftmann@64635
   409
    by blast
haftmann@26100
   410
next
haftmann@26100
   411
  case False
haftmann@26100
   412
  have "\<exists>q r. m = q * n + r \<and> r < n"
haftmann@26100
   413
  proof (induct m)
wenzelm@60758
   414
    case 0 with \<open>n \<noteq> 0\<close>
wenzelm@61076
   415
    have "(0::nat) = 0 * n + 0 \<and> 0 < n" by simp
haftmann@26100
   416
    then show ?case by blast
haftmann@26100
   417
  next
haftmann@26100
   418
    case (Suc m) then obtain q' r'
haftmann@26100
   419
      where m: "m = q' * n + r'" and n: "r' < n" by auto
haftmann@26100
   420
    then show ?case proof (cases "Suc r' < n")
haftmann@26100
   421
      case True
haftmann@26100
   422
      from m n have "Suc m = q' * n + Suc r'" by simp
haftmann@26100
   423
      with True show ?thesis by blast
haftmann@26100
   424
    next
haftmann@64592
   425
      case False then have "n \<le> Suc r'"
haftmann@64592
   426
        by (simp add: not_less)
haftmann@64592
   427
      moreover from n have "Suc r' \<le> n"
haftmann@64592
   428
        by (simp add: Suc_le_eq)
haftmann@26100
   429
      ultimately have "n = Suc r'" by auto
haftmann@26100
   430
      with m have "Suc m = Suc q' * n + 0" by simp
wenzelm@60758
   431
      with \<open>n \<noteq> 0\<close> show ?thesis by blast
haftmann@26100
   432
    qed
haftmann@26100
   433
  qed
haftmann@64635
   434
  with that \<open>n \<noteq> 0\<close> eucl_rel_natI show thesis
haftmann@64635
   435
    by blast
haftmann@26100
   436
qed
haftmann@26100
   437
haftmann@64635
   438
text \<open>@{const eucl_rel_nat} is injective:\<close>
haftmann@64635
   439
haftmann@64635
   440
qualified lemma eucl_rel_nat_unique_div:
haftmann@64635
   441
  assumes "eucl_rel_nat m n (q, r)"
haftmann@64635
   442
    and "eucl_rel_nat m n (q', r')"
haftmann@64635
   443
  shows "q = q'"
haftmann@26100
   444
proof (cases "n = 0")
haftmann@26100
   445
  case True with assms show ?thesis
haftmann@64635
   446
    by (auto elim: eucl_rel_nat.cases)
haftmann@26100
   447
next
haftmann@26100
   448
  case False
haftmann@64635
   449
  have *: "q' \<le> q" if "q' * n + r' = q * n + r" "r < n" for q r q' r' :: nat
haftmann@64635
   450
  proof (rule ccontr)
haftmann@64635
   451
    assume "\<not> q' \<le> q"
haftmann@64635
   452
    then have "q < q'"
haftmann@64635
   453
      by (simp add: not_le)
haftmann@64635
   454
    with that show False
haftmann@64635
   455
      by (auto simp add: less_iff_Suc_add algebra_simps)
haftmann@64635
   456
  qed
haftmann@64635
   457
  from \<open>n \<noteq> 0\<close> assms show ?thesis
haftmann@64635
   458
    by (auto intro: order_antisym elim: eucl_rel_nat.cases dest: * sym split: if_splits)
haftmann@64635
   459
qed
haftmann@64635
   460
haftmann@64635
   461
qualified lemma eucl_rel_nat_unique_mod:
haftmann@64635
   462
  assumes "eucl_rel_nat m n (q, r)"
haftmann@64635
   463
    and "eucl_rel_nat m n (q', r')"
haftmann@64635
   464
  shows "r = r'"
haftmann@64635
   465
proof -
haftmann@64635
   466
  from assms have "q' = q"
haftmann@64635
   467
    by (auto intro: eucl_rel_nat_unique_div)
haftmann@64635
   468
  with assms show ?thesis
haftmann@64635
   469
    by (auto elim!: eucl_rel_nat.cases)
haftmann@26100
   470
qed
haftmann@26100
   471
wenzelm@60758
   472
text \<open>
haftmann@26100
   473
  We instantiate divisibility on the natural numbers by
haftmann@64635
   474
  means of @{const eucl_rel_nat}:
wenzelm@60758
   475
\<close>
haftmann@25942
   476
haftmann@61433
   477
qualified definition divmod_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@64635
   478
  "divmod_nat m n = (THE qr. eucl_rel_nat m n qr)"
haftmann@64635
   479
haftmann@64635
   480
qualified lemma eucl_rel_nat_divmod_nat:
haftmann@64635
   481
  "eucl_rel_nat m n (divmod_nat m n)"
haftmann@30923
   482
proof -
haftmann@64635
   483
  from eucl_rel_nat_ex
haftmann@64635
   484
    obtain q r where rel: "eucl_rel_nat m n (q, r)" .
haftmann@30923
   485
  then show ?thesis
haftmann@64635
   486
    by (auto simp add: divmod_nat_def intro: theI
haftmann@64635
   487
      elim: eucl_rel_nat_unique_div eucl_rel_nat_unique_mod)
haftmann@30923
   488
qed
haftmann@30923
   489
haftmann@61433
   490
qualified lemma divmod_nat_unique:
haftmann@64635
   491
  "divmod_nat m n = (q, r)" if "eucl_rel_nat m n (q, r)"
haftmann@64635
   492
  using that
haftmann@64635
   493
  by (auto simp add: divmod_nat_def intro: eucl_rel_nat_divmod_nat elim: eucl_rel_nat_unique_div eucl_rel_nat_unique_mod)
haftmann@64635
   494
haftmann@64635
   495
qualified lemma divmod_nat_zero:
haftmann@64635
   496
  "divmod_nat m 0 = (0, m)"
haftmann@64635
   497
  by (rule divmod_nat_unique) (fact eucl_rel_nat_by0)
haftmann@64635
   498
haftmann@64635
   499
qualified lemma divmod_nat_zero_left:
haftmann@64635
   500
  "divmod_nat 0 n = (0, 0)"
haftmann@64635
   501
  by (rule divmod_nat_unique) 
haftmann@64635
   502
    (cases n, auto intro: eucl_rel_nat_by0 eucl_rel_natI)
haftmann@64635
   503
haftmann@64635
   504
qualified lemma divmod_nat_base:
haftmann@64635
   505
  "m < n \<Longrightarrow> divmod_nat m n = (0, m)"
haftmann@64635
   506
  by (rule divmod_nat_unique) 
haftmann@64635
   507
    (cases n, auto intro: eucl_rel_nat_by0 eucl_rel_natI)
haftmann@61433
   508
haftmann@61433
   509
qualified lemma divmod_nat_step:
haftmann@61433
   510
  assumes "0 < n" and "n \<le> m"
haftmann@64635
   511
  shows "divmod_nat m n =
haftmann@64635
   512
    (Suc (fst (divmod_nat (m - n) n)), snd (divmod_nat (m - n) n))"
haftmann@61433
   513
proof (rule divmod_nat_unique)
haftmann@64635
   514
  have "eucl_rel_nat (m - n) n (divmod_nat (m - n) n)"
haftmann@64635
   515
    by (fact eucl_rel_nat_divmod_nat)
haftmann@64635
   516
  then show "eucl_rel_nat m n (Suc
haftmann@64635
   517
    (fst (divmod_nat (m - n) n)), snd (divmod_nat (m - n) n))"
haftmann@64635
   518
    using assms
haftmann@64635
   519
      by (auto split: if_splits intro: eucl_rel_natI elim!: eucl_rel_nat.cases simp add: algebra_simps)
haftmann@61433
   520
qed
haftmann@61433
   521
haftmann@61433
   522
end
haftmann@64592
   523
haftmann@64592
   524
instantiation nat :: "{semidom_modulo, normalization_semidom}"
haftmann@60352
   525
begin
haftmann@60352
   526
haftmann@64592
   527
definition normalize_nat :: "nat \<Rightarrow> nat"
haftmann@64592
   528
  where [simp]: "normalize = (id :: nat \<Rightarrow> nat)"
haftmann@64592
   529
haftmann@64592
   530
definition unit_factor_nat :: "nat \<Rightarrow> nat"
haftmann@64592
   531
  where "unit_factor n = (if n = 0 then 0 else 1 :: nat)"
haftmann@64592
   532
haftmann@64592
   533
lemma unit_factor_simps [simp]:
haftmann@64592
   534
  "unit_factor 0 = (0::nat)"
haftmann@64592
   535
  "unit_factor (Suc n) = 1"
haftmann@64592
   536
  by (simp_all add: unit_factor_nat_def)
haftmann@64592
   537
haftmann@64592
   538
definition divide_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@64592
   539
  where div_nat_def: "m div n = fst (Divides.divmod_nat m n)"
haftmann@64592
   540
haftmann@64592
   541
definition modulo_nat :: "nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@64592
   542
  where mod_nat_def: "m mod n = snd (Divides.divmod_nat m n)"
huffman@46551
   543
huffman@46551
   544
lemma fst_divmod_nat [simp]:
haftmann@61433
   545
  "fst (Divides.divmod_nat m n) = m div n"
huffman@46551
   546
  by (simp add: div_nat_def)
huffman@46551
   547
huffman@46551
   548
lemma snd_divmod_nat [simp]:
haftmann@61433
   549
  "snd (Divides.divmod_nat m n) = m mod n"
huffman@46551
   550
  by (simp add: mod_nat_def)
huffman@46551
   551
haftmann@33340
   552
lemma divmod_nat_div_mod:
haftmann@61433
   553
  "Divides.divmod_nat m n = (m div n, m mod n)"
huffman@46551
   554
  by (simp add: prod_eq_iff)
haftmann@26100
   555
huffman@47135
   556
lemma div_nat_unique:
haftmann@64635
   557
  assumes "eucl_rel_nat m n (q, r)"
haftmann@26100
   558
  shows "m div n = q"
haftmann@64592
   559
  using assms
haftmann@64592
   560
  by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
huffman@47135
   561
huffman@47135
   562
lemma mod_nat_unique:
haftmann@64635
   563
  assumes "eucl_rel_nat m n (q, r)"
haftmann@26100
   564
  shows "m mod n = r"
haftmann@64592
   565
  using assms
haftmann@64592
   566
  by (auto dest!: Divides.divmod_nat_unique simp add: prod_eq_iff)
haftmann@25571
   567
haftmann@64635
   568
lemma eucl_rel_nat: "eucl_rel_nat m n (m div n, m mod n)"
haftmann@64635
   569
  using Divides.eucl_rel_nat_divmod_nat
haftmann@64592
   570
  by (simp add: divmod_nat_div_mod)
haftmann@25942
   571
haftmann@63950
   572
text \<open>The ''recursion'' equations for @{const divide} and @{const modulo}\<close>
haftmann@26100
   573
haftmann@26100
   574
lemma div_less [simp]:
haftmann@26100
   575
  fixes m n :: nat
haftmann@26100
   576
  assumes "m < n"
haftmann@26100
   577
  shows "m div n = 0"
haftmann@61433
   578
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
haftmann@25942
   579
haftmann@26100
   580
lemma le_div_geq:
haftmann@26100
   581
  fixes m n :: nat
haftmann@26100
   582
  assumes "0 < n" and "n \<le> m"
haftmann@26100
   583
  shows "m div n = Suc ((m - n) div n)"
haftmann@61433
   584
  using assms Divides.divmod_nat_step by (simp add: prod_eq_iff)
paulson@14267
   585
haftmann@26100
   586
lemma mod_less [simp]:
haftmann@26100
   587
  fixes m n :: nat
haftmann@26100
   588
  assumes "m < n"
haftmann@26100
   589
  shows "m mod n = m"
haftmann@61433
   590
  using assms Divides.divmod_nat_base by (simp add: prod_eq_iff)
haftmann@26100
   591
haftmann@26100
   592
lemma le_mod_geq:
haftmann@26100
   593
  fixes m n :: nat
haftmann@26100
   594
  assumes "n \<le> m"
haftmann@26100
   595
  shows "m mod n = (m - n) mod n"
haftmann@61433
   596
  using assms Divides.divmod_nat_step by (cases "n = 0") (simp_all add: prod_eq_iff)
paulson@14267
   597
haftmann@64592
   598
lemma mod_less_divisor [simp]:
haftmann@64592
   599
  fixes m n :: nat
haftmann@64592
   600
  assumes "n > 0"
haftmann@64592
   601
  shows "m mod n < n"
haftmann@64635
   602
  using assms eucl_rel_nat [of m n]
haftmann@64635
   603
    by (auto elim: eucl_rel_nat.cases)
haftmann@64592
   604
haftmann@64592
   605
lemma mod_le_divisor [simp]:
haftmann@64592
   606
  fixes m n :: nat
haftmann@64592
   607
  assumes "n > 0"
haftmann@64592
   608
  shows "m mod n \<le> n"
haftmann@64635
   609
  using assms eucl_rel_nat [of m n]
haftmann@64635
   610
    by (auto elim: eucl_rel_nat.cases)
haftmann@64592
   611
huffman@47136
   612
instance proof
huffman@47136
   613
  fix m n :: nat
huffman@47136
   614
  show "m div n * n + m mod n = m"
haftmann@64635
   615
    using eucl_rel_nat [of m n]
haftmann@64635
   616
    by (auto elim: eucl_rel_nat.cases)
huffman@47136
   617
next
haftmann@64592
   618
  fix n :: nat show "n div 0 = 0"
haftmann@64592
   619
    by (simp add: div_nat_def Divides.divmod_nat_zero)
haftmann@64592
   620
next
haftmann@64592
   621
  fix m n :: nat
haftmann@64592
   622
  assume "n \<noteq> 0"
haftmann@64592
   623
  then show "m * n div n = m"
haftmann@64635
   624
    by (auto intro!: eucl_rel_natI div_nat_unique [of _ _ _ 0])
haftmann@64592
   625
qed (simp_all add: unit_factor_nat_def)
haftmann@64592
   626
haftmann@64592
   627
end
haftmann@64592
   628
haftmann@63950
   629
text \<open>Simproc for cancelling @{const divide} and @{const modulo}\<close>
haftmann@25942
   630
haftmann@64592
   631
lemma (in semiring_modulo) cancel_div_mod_rules:
haftmann@64592
   632
  "((a div b) * b + a mod b) + c = a + c"
haftmann@64592
   633
  "(b * (a div b) + a mod b) + c = a + c"
haftmann@64592
   634
  by (simp_all add: div_mult_mod_eq mult_div_mod_eq)
haftmann@64592
   635
wenzelm@51299
   636
ML_file "~~/src/Provers/Arith/cancel_div_mod.ML"
wenzelm@51299
   637
wenzelm@60758
   638
ML \<open>
wenzelm@43594
   639
structure Cancel_Div_Mod_Nat = Cancel_Div_Mod
wenzelm@41550
   640
(
haftmann@60352
   641
  val div_name = @{const_name divide};
haftmann@63950
   642
  val mod_name = @{const_name modulo};
haftmann@30934
   643
  val mk_binop = HOLogic.mk_binop;
huffman@48561
   644
  val mk_plus = HOLogic.mk_binop @{const_name Groups.plus};
huffman@48561
   645
  val dest_plus = HOLogic.dest_bin @{const_name Groups.plus} HOLogic.natT;
huffman@48561
   646
  fun mk_sum [] = HOLogic.zero
huffman@48561
   647
    | mk_sum [t] = t
huffman@48561
   648
    | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
huffman@48561
   649
  fun dest_sum tm =
huffman@48561
   650
    if HOLogic.is_zero tm then []
huffman@48561
   651
    else
huffman@48561
   652
      (case try HOLogic.dest_Suc tm of
huffman@48561
   653
        SOME t => HOLogic.Suc_zero :: dest_sum t
huffman@48561
   654
      | NONE =>
huffman@48561
   655
          (case try dest_plus tm of
huffman@48561
   656
            SOME (t, u) => dest_sum t @ dest_sum u
huffman@48561
   657
          | NONE => [tm]));
haftmann@25942
   658
haftmann@64250
   659
  val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
haftmann@64250
   660
haftmann@64250
   661
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac
haftmann@64250
   662
    (Arith_Data.simp_all_tac @{thms add_0_left add_0_right ac_simps})
wenzelm@41550
   663
)
wenzelm@60758
   664
\<close>
wenzelm@60758
   665
haftmann@64592
   666
simproc_setup cancel_div_mod_nat ("(m::nat) + n") =
haftmann@64592
   667
  \<open>K Cancel_Div_Mod_Nat.proc\<close>
haftmann@64592
   668
haftmann@66806
   669
lemma div_by_Suc_0 [simp]:
haftmann@66806
   670
  "m div Suc 0 = m"
haftmann@66806
   671
  using div_by_1 [of m] by simp
haftmann@66806
   672
haftmann@66806
   673
lemma mod_by_Suc_0 [simp]:
haftmann@66806
   674
  "m mod Suc 0 = 0"
haftmann@66806
   675
  using mod_by_1 [of m] by simp
haftmann@66806
   676
haftmann@66806
   677
lemma mod_greater_zero_iff_not_dvd:
haftmann@66806
   678
  fixes m n :: nat
haftmann@66806
   679
  shows "m mod n > 0 \<longleftrightarrow> \<not> n dvd m"
haftmann@66806
   680
  by (simp add: dvd_eq_mod_eq_0)
haftmann@66806
   681
haftmann@66806
   682
instantiation nat :: unique_euclidean_semiring
haftmann@66806
   683
begin
haftmann@66806
   684
haftmann@66806
   685
definition [simp]:
haftmann@66806
   686
  "euclidean_size_nat = (id :: nat \<Rightarrow> nat)"
haftmann@66806
   687
haftmann@66806
   688
definition [simp]:
haftmann@66806
   689
  "uniqueness_constraint_nat = (top :: nat \<Rightarrow> nat \<Rightarrow> bool)"
haftmann@66806
   690
haftmann@66806
   691
instance proof
haftmann@66806
   692
  fix n q r :: nat
haftmann@66806
   693
  assume "euclidean_size r < euclidean_size n"
haftmann@66806
   694
  then have "n > r"
haftmann@66806
   695
    by simp_all
haftmann@66806
   696
  then have "eucl_rel_nat (q * n + r) n (q, r)"
haftmann@66806
   697
    by (rule eucl_rel_natI) rule
haftmann@66806
   698
  then show "(q * n + r) div n = q"
haftmann@66806
   699
    by (rule div_nat_unique)
haftmann@66806
   700
qed (use mult_le_mono2 [of 1] in \<open>simp_all\<close>)
haftmann@66806
   701
haftmann@66806
   702
end
haftmann@66806
   703
  
haftmann@64592
   704
lemma divmod_nat_if [code]:
haftmann@64592
   705
  "Divides.divmod_nat m n = (if n = 0 \<or> m < n then (0, m) else
haftmann@64592
   706
    let (q, r) = Divides.divmod_nat (m - n) n in (Suc q, r))"
haftmann@64592
   707
  by (simp add: prod_eq_iff case_prod_beta not_less le_div_geq le_mod_geq)
wenzelm@60758
   708
haftmann@64593
   709
lemma mod_Suc_eq [mod_simps]:
haftmann@64593
   710
  "Suc (m mod n) mod n = Suc m mod n"
haftmann@64593
   711
proof -
haftmann@64593
   712
  have "(m mod n + 1) mod n = (m + 1) mod n"
haftmann@64593
   713
    by (simp only: mod_simps)
haftmann@64593
   714
  then show ?thesis
haftmann@64593
   715
    by simp
haftmann@64593
   716
qed
haftmann@64593
   717
haftmann@64593
   718
lemma mod_Suc_Suc_eq [mod_simps]:
haftmann@64593
   719
  "Suc (Suc (m mod n)) mod n = Suc (Suc m) mod n"
haftmann@64593
   720
proof -
haftmann@64593
   721
  have "(m mod n + 2) mod n = (m + 2) mod n"
haftmann@64593
   722
    by (simp only: mod_simps)
haftmann@64593
   723
  then show ?thesis
haftmann@64593
   724
    by simp
haftmann@64593
   725
qed
haftmann@64593
   726
wenzelm@60758
   727
wenzelm@60758
   728
subsubsection \<open>Quotient\<close>
haftmann@26100
   729
haftmann@26100
   730
lemma div_geq: "0 < n \<Longrightarrow>  \<not> m < n \<Longrightarrow> m div n = Suc ((m - n) div n)"
nipkow@29667
   731
by (simp add: le_div_geq linorder_not_less)
haftmann@26100
   732
haftmann@26100
   733
lemma div_if: "0 < n \<Longrightarrow> m div n = (if m < n then 0 else Suc ((m - n) div n))"
nipkow@29667
   734
by (simp add: div_geq)
haftmann@26100
   735
haftmann@26100
   736
lemma div_mult_self_is_m [simp]: "0<n ==> (m*n) div n = (m::nat)"
nipkow@29667
   737
by simp
haftmann@26100
   738
haftmann@26100
   739
lemma div_mult_self1_is_m [simp]: "0<n ==> (n*m) div n = (m::nat)"
nipkow@29667
   740
by simp
haftmann@26100
   741
haftmann@53066
   742
lemma div_positive:
haftmann@53066
   743
  fixes m n :: nat
haftmann@53066
   744
  assumes "n > 0"
haftmann@53066
   745
  assumes "m \<ge> n"
haftmann@53066
   746
  shows "m div n > 0"
haftmann@53066
   747
proof -
wenzelm@60758
   748
  from \<open>m \<ge> n\<close> obtain q where "m = n + q"
haftmann@53066
   749
    by (auto simp add: le_iff_add)
eberlm@63499
   750
  with \<open>n > 0\<close> show ?thesis by (simp add: div_add_self1)
haftmann@53066
   751
qed
haftmann@53066
   752
hoelzl@59000
   753
lemma div_eq_0_iff: "(a div b::nat) = 0 \<longleftrightarrow> a < b \<or> b = 0"
haftmann@64592
   754
  by auto (metis div_positive less_numeral_extra(3) not_less)
haftmann@64592
   755
haftmann@25942
   756
wenzelm@60758
   757
subsubsection \<open>Remainder\<close>
haftmann@25942
   758
haftmann@51173
   759
lemma mod_Suc_le_divisor [simp]:
haftmann@51173
   760
  "m mod Suc n \<le> n"
haftmann@51173
   761
  using mod_less_divisor [of "Suc n" m] by arith
haftmann@51173
   762
haftmann@26100
   763
lemma mod_less_eq_dividend [simp]:
haftmann@26100
   764
  fixes m n :: nat
haftmann@26100
   765
  shows "m mod n \<le> m"
haftmann@26100
   766
proof (rule add_leD2)
haftmann@64242
   767
  from div_mult_mod_eq have "m div n * n + m mod n = m" .
haftmann@26100
   768
  then show "m div n * n + m mod n \<le> m" by auto
haftmann@26100
   769
qed
haftmann@26100
   770
wenzelm@61076
   771
lemma mod_geq: "\<not> m < (n::nat) \<Longrightarrow> m mod n = (m - n) mod n"
nipkow@29667
   772
by (simp add: le_mod_geq linorder_not_less)
paulson@14267
   773
wenzelm@61076
   774
lemma mod_if: "m mod (n::nat) = (if m < n then m else (m - n) mod n)"
nipkow@29667
   775
by (simp add: le_mod_geq)
haftmann@26100
   776
paulson@14267
   777
wenzelm@60758
   778
subsubsection \<open>Quotient and Remainder\<close>
paulson@14267
   779
haftmann@30923
   780
lemma div_mult1_eq:
haftmann@30923
   781
  "(a * b) div c = a * (b div c) + a * (b mod c) div (c::nat)"
haftmann@64635
   782
  by (cases "c = 0")
haftmann@64635
   783
     (auto simp add: algebra_simps distrib_left [symmetric]
haftmann@64635
   784
     intro!: div_nat_unique [of _ _ _ "(a * (b mod c)) mod c"] eucl_rel_natI)
haftmann@64635
   785
haftmann@64635
   786
lemma eucl_rel_nat_add1_eq:
haftmann@64635
   787
  "eucl_rel_nat a c (aq, ar) \<Longrightarrow> eucl_rel_nat b c (bq, br)
haftmann@64635
   788
   \<Longrightarrow> eucl_rel_nat (a + b) c (aq + bq + (ar + br) div c, (ar + br) mod c)"
haftmann@64635
   789
  by (auto simp add: split_ifs algebra_simps elim!: eucl_rel_nat.cases intro: eucl_rel_nat_by0 eucl_rel_natI)
paulson@14267
   790
paulson@14267
   791
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
paulson@14267
   792
lemma div_add1_eq:
haftmann@64635
   793
  "(a + b) div (c::nat) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@64635
   794
by (blast intro: eucl_rel_nat_add1_eq [THEN div_nat_unique] eucl_rel_nat)
haftmann@64635
   795
haftmann@64635
   796
lemma eucl_rel_nat_mult2_eq:
haftmann@64635
   797
  assumes "eucl_rel_nat a b (q, r)"
haftmann@64635
   798
  shows "eucl_rel_nat a (b * c) (q div c, b *(q mod c) + r)"
haftmann@64635
   799
proof (cases "c = 0")
haftmann@64635
   800
  case True
haftmann@64635
   801
  with assms show ?thesis
haftmann@64635
   802
    by (auto intro: eucl_rel_nat_by0 elim!: eucl_rel_nat.cases simp add: ac_simps)
haftmann@64635
   803
next
haftmann@64635
   804
  case False
haftmann@64635
   805
  { assume "r < b"
haftmann@64635
   806
    with False have "b * (q mod c) + r < b * c"
haftmann@60352
   807
      apply (cut_tac m = q and n = c in mod_less_divisor)
haftmann@60352
   808
      apply (drule_tac [2] m = "q mod c" in less_imp_Suc_add, auto)
haftmann@60352
   809
      apply (erule_tac P = "%x. lhs < rhs x" for lhs rhs in ssubst)
haftmann@60352
   810
      apply (simp add: add_mult_distrib2)
haftmann@60352
   811
      done
haftmann@60352
   812
    then have "r + b * (q mod c) < b * c"
haftmann@60352
   813
      by (simp add: ac_simps)
haftmann@64635
   814
  } with assms False show ?thesis
haftmann@64635
   815
    by (auto simp add: algebra_simps add_mult_distrib2 [symmetric] elim!: eucl_rel_nat.cases intro: eucl_rel_nat.intros)
haftmann@60352
   816
qed
lp15@60562
   817
blanchet@55085
   818
lemma div_mult2_eq: "a div (b * c) = (a div b) div (c::nat)"
haftmann@64635
   819
by (force simp add: eucl_rel_nat [THEN eucl_rel_nat_mult2_eq, THEN div_nat_unique])
paulson@14267
   820
blanchet@55085
   821
lemma mod_mult2_eq: "a mod (b * c) = b * (a div b mod c) + a mod (b::nat)"
haftmann@64635
   822
by (auto simp add: mult.commute eucl_rel_nat [THEN eucl_rel_nat_mult2_eq, THEN mod_nat_unique])
paulson@14267
   823
haftmann@66806
   824
instantiation nat :: unique_euclidean_semiring_numeral
haftmann@61275
   825
begin
haftmann@61275
   826
haftmann@61275
   827
definition divmod_nat :: "num \<Rightarrow> num \<Rightarrow> nat \<times> nat"
haftmann@61275
   828
where
haftmann@61275
   829
  divmod'_nat_def: "divmod_nat m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
   830
haftmann@61275
   831
definition divmod_step_nat :: "num \<Rightarrow> nat \<times> nat \<Rightarrow> nat \<times> nat"
haftmann@61275
   832
where
haftmann@61275
   833
  "divmod_step_nat l qr = (let (q, r) = qr
haftmann@61275
   834
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
   835
    else (2 * q, r))"
haftmann@61275
   836
haftmann@61275
   837
instance
haftmann@61275
   838
  by standard (auto intro: div_positive simp add: divmod'_nat_def divmod_step_nat_def mod_mult2_eq div_mult2_eq)
haftmann@61275
   839
haftmann@61275
   840
end
haftmann@61275
   841
haftmann@61275
   842
declare divmod_algorithm_code [where ?'a = nat, code]
haftmann@61275
   843
  
paulson@14267
   844
wenzelm@60758
   845
subsubsection \<open>Further Facts about Quotient and Remainder\<close>
paulson@14267
   846
haftmann@64592
   847
lemma div_le_mono:
haftmann@64592
   848
  fixes m n k :: nat
haftmann@64592
   849
  assumes "m \<le> n"
haftmann@64592
   850
  shows "m div k \<le> n div k"
haftmann@64592
   851
proof -
haftmann@64592
   852
  from assms obtain q where "n = m + q"
haftmann@64592
   853
    by (auto simp add: le_iff_add)
haftmann@64592
   854
  then show ?thesis
haftmann@64592
   855
    by (simp add: div_add1_eq [of m q k])
haftmann@64592
   856
qed
paulson@14267
   857
paulson@14267
   858
(* Antimonotonicity of div in second argument *)
paulson@14267
   859
lemma div_le_mono2: "!!m::nat. [| 0<m; m\<le>n |] ==> (k div n) \<le> (k div m)"
paulson@14267
   860
apply (subgoal_tac "0<n")
wenzelm@22718
   861
 prefer 2 apply simp
paulson@15251
   862
apply (induct_tac k rule: nat_less_induct)
paulson@14267
   863
apply (rename_tac "k")
paulson@14267
   864
apply (case_tac "k<n", simp)
paulson@14267
   865
apply (subgoal_tac "~ (k<m) ")
wenzelm@22718
   866
 prefer 2 apply simp
paulson@14267
   867
apply (simp add: div_geq)
paulson@15251
   868
apply (subgoal_tac "(k-n) div n \<le> (k-m) div n")
paulson@14267
   869
 prefer 2
paulson@14267
   870
 apply (blast intro: div_le_mono diff_le_mono2)
paulson@14267
   871
apply (rule le_trans, simp)
nipkow@15439
   872
apply (simp)
paulson@14267
   873
done
paulson@14267
   874
paulson@14267
   875
lemma div_le_dividend [simp]: "m div n \<le> (m::nat)"
paulson@14267
   876
apply (case_tac "n=0", simp)
paulson@14267
   877
apply (subgoal_tac "m div n \<le> m div 1", simp)
paulson@14267
   878
apply (rule div_le_mono2)
paulson@14267
   879
apply (simp_all (no_asm_simp))
paulson@14267
   880
done
paulson@14267
   881
wenzelm@22718
   882
(* Similar for "less than" *)
huffman@47138
   883
lemma div_less_dividend [simp]:
huffman@47138
   884
  "\<lbrakk>(1::nat) < n; 0 < m\<rbrakk> \<Longrightarrow> m div n < m"
huffman@47138
   885
apply (induct m rule: nat_less_induct)
paulson@14267
   886
apply (rename_tac "m")
paulson@14267
   887
apply (case_tac "m<n", simp)
paulson@14267
   888
apply (subgoal_tac "0<n")
wenzelm@22718
   889
 prefer 2 apply simp
paulson@14267
   890
apply (simp add: div_geq)
paulson@14267
   891
apply (case_tac "n<m")
paulson@15251
   892
 apply (subgoal_tac "(m-n) div n < (m-n) ")
paulson@14267
   893
  apply (rule impI less_trans_Suc)+
paulson@14267
   894
apply assumption
nipkow@15439
   895
  apply (simp_all)
paulson@14267
   896
done
paulson@14267
   897
wenzelm@60758
   898
text\<open>A fact for the mutilated chess board\<close>
paulson@14267
   899
lemma mod_Suc: "Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"
paulson@14267
   900
apply (case_tac "n=0", simp)
paulson@15251
   901
apply (induct "m" rule: nat_less_induct)
paulson@14267
   902
apply (case_tac "Suc (na) <n")
paulson@14267
   903
(* case Suc(na) < n *)
paulson@14267
   904
apply (frule lessI [THEN less_trans], simp add: less_not_refl3)
paulson@14267
   905
(* case n \<le> Suc(na) *)
paulson@16796
   906
apply (simp add: linorder_not_less le_Suc_eq mod_geq)
nipkow@15439
   907
apply (auto simp add: Suc_diff_le le_mod_geq)
paulson@14267
   908
done
paulson@14267
   909
paulson@14267
   910
lemma mod_eq_0_iff: "(m mod d = 0) = (\<exists>q::nat. m = d*q)"
nipkow@29667
   911
by (auto simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
paulson@17084
   912
wenzelm@22718
   913
lemmas mod_eq_0D [dest!] = mod_eq_0_iff [THEN iffD1]
paulson@14267
   914
paulson@14267
   915
(*Loses information, namely we also have r<d provided d is nonzero*)
haftmann@57514
   916
lemma mod_eqD:
haftmann@57514
   917
  fixes m d r q :: nat
haftmann@57514
   918
  assumes "m mod d = r"
haftmann@57514
   919
  shows "\<exists>q. m = r + q * d"
haftmann@57514
   920
proof -
haftmann@64242
   921
  from div_mult_mod_eq obtain q where "q * d + m mod d = m" by blast
haftmann@57514
   922
  with assms have "m = r + q * d" by simp
haftmann@57514
   923
  then show ?thesis ..
haftmann@57514
   924
qed
paulson@14267
   925
nipkow@13152
   926
lemma split_div:
nipkow@13189
   927
 "P(n div k :: nat) =
nipkow@13189
   928
 ((k = 0 \<longrightarrow> P 0) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P i)))"
nipkow@13189
   929
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
   930
proof
nipkow@13189
   931
  assume P: ?P
nipkow@13189
   932
  show ?Q
nipkow@13189
   933
  proof (cases)
nipkow@13189
   934
    assume "k = 0"
haftmann@27651
   935
    with P show ?Q by simp
nipkow@13189
   936
  next
nipkow@13189
   937
    assume not0: "k \<noteq> 0"
nipkow@13189
   938
    thus ?Q
nipkow@13189
   939
    proof (simp, intro allI impI)
nipkow@13189
   940
      fix i j
nipkow@13189
   941
      assume n: "n = k*i + j" and j: "j < k"
nipkow@13189
   942
      show "P i"
nipkow@13189
   943
      proof (cases)
wenzelm@22718
   944
        assume "i = 0"
wenzelm@22718
   945
        with n j P show "P i" by simp
nipkow@13189
   946
      next
wenzelm@22718
   947
        assume "i \<noteq> 0"
haftmann@57514
   948
        with not0 n j P show "P i" by(simp add:ac_simps)
nipkow@13189
   949
      qed
nipkow@13189
   950
    qed
nipkow@13189
   951
  qed
nipkow@13189
   952
next
nipkow@13189
   953
  assume Q: ?Q
nipkow@13189
   954
  show ?P
nipkow@13189
   955
  proof (cases)
nipkow@13189
   956
    assume "k = 0"
haftmann@27651
   957
    with Q show ?P by simp
nipkow@13189
   958
  next
nipkow@13189
   959
    assume not0: "k \<noteq> 0"
nipkow@13189
   960
    with Q have R: ?R by simp
nipkow@13189
   961
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
   962
    show ?P by simp
nipkow@13189
   963
  qed
nipkow@13189
   964
qed
nipkow@13189
   965
berghofe@13882
   966
lemma split_div_lemma:
haftmann@26100
   967
  assumes "0 < n"
wenzelm@61076
   968
  shows "n * q \<le> m \<and> m < n * Suc q \<longleftrightarrow> q = ((m::nat) div n)" (is "?lhs \<longleftrightarrow> ?rhs")
haftmann@26100
   969
proof
haftmann@26100
   970
  assume ?rhs
haftmann@64246
   971
  with minus_mod_eq_mult_div [symmetric] have nq: "n * q = m - (m mod n)" by simp
haftmann@26100
   972
  then have A: "n * q \<le> m" by simp
haftmann@26100
   973
  have "n - (m mod n) > 0" using mod_less_divisor assms by auto
haftmann@26100
   974
  then have "m < m + (n - (m mod n))" by simp
haftmann@26100
   975
  then have "m < n + (m - (m mod n))" by simp
haftmann@26100
   976
  with nq have "m < n + n * q" by simp
haftmann@26100
   977
  then have B: "m < n * Suc q" by simp
haftmann@26100
   978
  from A B show ?lhs ..
haftmann@26100
   979
next
haftmann@26100
   980
  assume P: ?lhs
haftmann@64635
   981
  then have "eucl_rel_nat m n (q, m - n * q)"
haftmann@64635
   982
    by (auto intro: eucl_rel_natI simp add: ac_simps)
haftmann@61433
   983
  then have "m div n = q"
haftmann@61433
   984
    by (rule div_nat_unique)
haftmann@30923
   985
  then show ?rhs by simp
haftmann@26100
   986
qed
berghofe@13882
   987
berghofe@13882
   988
theorem split_div':
berghofe@13882
   989
  "P ((m::nat) div n) = ((n = 0 \<and> P 0) \<or>
paulson@14267
   990
   (\<exists>q. (n * q \<le> m \<and> m < n * (Suc q)) \<and> P q))"
haftmann@61433
   991
  apply (cases "0 < n")
berghofe@13882
   992
  apply (simp only: add: split_div_lemma)
haftmann@27651
   993
  apply simp_all
berghofe@13882
   994
  done
berghofe@13882
   995
nipkow@13189
   996
lemma split_mod:
nipkow@13189
   997
 "P(n mod k :: nat) =
nipkow@13189
   998
 ((k = 0 \<longrightarrow> P n) \<and> (k \<noteq> 0 \<longrightarrow> (!i. !j<k. n = k*i + j \<longrightarrow> P j)))"
nipkow@13189
   999
 (is "?P = ?Q" is "_ = (_ \<and> (_ \<longrightarrow> ?R))")
nipkow@13189
  1000
proof
nipkow@13189
  1001
  assume P: ?P
nipkow@13189
  1002
  show ?Q
nipkow@13189
  1003
  proof (cases)
nipkow@13189
  1004
    assume "k = 0"
haftmann@27651
  1005
    with P show ?Q by simp
nipkow@13189
  1006
  next
nipkow@13189
  1007
    assume not0: "k \<noteq> 0"
nipkow@13189
  1008
    thus ?Q
nipkow@13189
  1009
    proof (simp, intro allI impI)
nipkow@13189
  1010
      fix i j
nipkow@13189
  1011
      assume "n = k*i + j" "j < k"
haftmann@58786
  1012
      thus "P j" using not0 P by (simp add: ac_simps)
nipkow@13189
  1013
    qed
nipkow@13189
  1014
  qed
nipkow@13189
  1015
next
nipkow@13189
  1016
  assume Q: ?Q
nipkow@13189
  1017
  show ?P
nipkow@13189
  1018
  proof (cases)
nipkow@13189
  1019
    assume "k = 0"
haftmann@27651
  1020
    with Q show ?P by simp
nipkow@13189
  1021
  next
nipkow@13189
  1022
    assume not0: "k \<noteq> 0"
nipkow@13189
  1023
    with Q have R: ?R by simp
nipkow@13189
  1024
    from not0 R[THEN spec,of "n div k",THEN spec, of "n mod k"]
nipkow@13517
  1025
    show ?P by simp
nipkow@13189
  1026
  qed
nipkow@13189
  1027
qed
nipkow@13189
  1028
noschinl@52398
  1029
lemma div_eq_dividend_iff: "a \<noteq> 0 \<Longrightarrow> (a :: nat) div b = a \<longleftrightarrow> b = 1"
noschinl@52398
  1030
  apply rule
noschinl@52398
  1031
  apply (cases "b = 0")
noschinl@52398
  1032
  apply simp_all
noschinl@52398
  1033
  apply (metis (full_types) One_nat_def Suc_lessI div_less_dividend less_not_refl3)
noschinl@52398
  1034
  done
noschinl@52398
  1035
haftmann@63417
  1036
lemma (in field_char_0) of_nat_div:
haftmann@63417
  1037
  "of_nat (m div n) = ((of_nat m - of_nat (m mod n)) / of_nat n)"
haftmann@63417
  1038
proof -
haftmann@63417
  1039
  have "of_nat (m div n) = ((of_nat (m div n * n + m mod n) - of_nat (m mod n)) / of_nat n :: 'a)"
haftmann@63417
  1040
    unfolding of_nat_add by (cases "n = 0") simp_all
haftmann@63417
  1041
  then show ?thesis
haftmann@63417
  1042
    by simp
haftmann@63417
  1043
qed
haftmann@63417
  1044
haftmann@22800
  1045
wenzelm@60758
  1046
subsubsection \<open>An ``induction'' law for modulus arithmetic.\<close>
paulson@14640
  1047
paulson@14640
  1048
lemma mod_induct_0:
paulson@14640
  1049
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1050
  and base: "P i" and i: "i<p"
paulson@14640
  1051
  shows "P 0"
paulson@14640
  1052
proof (rule ccontr)
paulson@14640
  1053
  assume contra: "\<not>(P 0)"
paulson@14640
  1054
  from i have p: "0<p" by simp
paulson@14640
  1055
  have "\<forall>k. 0<k \<longrightarrow> \<not> P (p-k)" (is "\<forall>k. ?A k")
paulson@14640
  1056
  proof
paulson@14640
  1057
    fix k
paulson@14640
  1058
    show "?A k"
paulson@14640
  1059
    proof (induct k)
wenzelm@61799
  1060
      show "?A 0" by simp  \<comment> "by contradiction"
paulson@14640
  1061
    next
paulson@14640
  1062
      fix n
paulson@14640
  1063
      assume ih: "?A n"
paulson@14640
  1064
      show "?A (Suc n)"
paulson@14640
  1065
      proof (clarsimp)
wenzelm@22718
  1066
        assume y: "P (p - Suc n)"
wenzelm@22718
  1067
        have n: "Suc n < p"
wenzelm@22718
  1068
        proof (rule ccontr)
wenzelm@22718
  1069
          assume "\<not>(Suc n < p)"
wenzelm@22718
  1070
          hence "p - Suc n = 0"
wenzelm@22718
  1071
            by simp
wenzelm@22718
  1072
          with y contra show "False"
wenzelm@22718
  1073
            by simp
wenzelm@22718
  1074
        qed
wenzelm@22718
  1075
        hence n2: "Suc (p - Suc n) = p-n" by arith
wenzelm@22718
  1076
        from p have "p - Suc n < p" by arith
wenzelm@22718
  1077
        with y step have z: "P ((Suc (p - Suc n)) mod p)"
wenzelm@22718
  1078
          by blast
wenzelm@22718
  1079
        show "False"
wenzelm@22718
  1080
        proof (cases "n=0")
wenzelm@22718
  1081
          case True
wenzelm@22718
  1082
          with z n2 contra show ?thesis by simp
wenzelm@22718
  1083
        next
wenzelm@22718
  1084
          case False
wenzelm@22718
  1085
          with p have "p-n < p" by arith
wenzelm@22718
  1086
          with z n2 False ih show ?thesis by simp
wenzelm@22718
  1087
        qed
paulson@14640
  1088
      qed
paulson@14640
  1089
    qed
paulson@14640
  1090
  qed
paulson@14640
  1091
  moreover
paulson@14640
  1092
  from i obtain k where "0<k \<and> i+k=p"
paulson@14640
  1093
    by (blast dest: less_imp_add_positive)
paulson@14640
  1094
  hence "0<k \<and> i=p-k" by auto
paulson@14640
  1095
  moreover
paulson@14640
  1096
  note base
paulson@14640
  1097
  ultimately
paulson@14640
  1098
  show "False" by blast
paulson@14640
  1099
qed
paulson@14640
  1100
paulson@14640
  1101
lemma mod_induct:
paulson@14640
  1102
  assumes step: "\<forall>i<p. P i \<longrightarrow> P ((Suc i) mod p)"
paulson@14640
  1103
  and base: "P i" and i: "i<p" and j: "j<p"
paulson@14640
  1104
  shows "P j"
paulson@14640
  1105
proof -
paulson@14640
  1106
  have "\<forall>j<p. P j"
paulson@14640
  1107
  proof
paulson@14640
  1108
    fix j
paulson@14640
  1109
    show "j<p \<longrightarrow> P j" (is "?A j")
paulson@14640
  1110
    proof (induct j)
paulson@14640
  1111
      from step base i show "?A 0"
wenzelm@22718
  1112
        by (auto elim: mod_induct_0)
paulson@14640
  1113
    next
paulson@14640
  1114
      fix k
paulson@14640
  1115
      assume ih: "?A k"
paulson@14640
  1116
      show "?A (Suc k)"
paulson@14640
  1117
      proof
wenzelm@22718
  1118
        assume suc: "Suc k < p"
wenzelm@22718
  1119
        hence k: "k<p" by simp
wenzelm@22718
  1120
        with ih have "P k" ..
wenzelm@22718
  1121
        with step k have "P (Suc k mod p)"
wenzelm@22718
  1122
          by blast
wenzelm@22718
  1123
        moreover
wenzelm@22718
  1124
        from suc have "Suc k mod p = Suc k"
wenzelm@22718
  1125
          by simp
wenzelm@22718
  1126
        ultimately
wenzelm@22718
  1127
        show "P (Suc k)" by simp
paulson@14640
  1128
      qed
paulson@14640
  1129
    qed
paulson@14640
  1130
  qed
paulson@14640
  1131
  with j show ?thesis by blast
paulson@14640
  1132
qed
paulson@14640
  1133
haftmann@33296
  1134
lemma div2_Suc_Suc [simp]: "Suc (Suc m) div 2 = Suc (m div 2)"
huffman@47138
  1135
  by (simp add: numeral_2_eq_2 le_div_geq)
huffman@47138
  1136
huffman@47138
  1137
lemma mod2_Suc_Suc [simp]: "Suc (Suc m) mod 2 = m mod 2"
huffman@47138
  1138
  by (simp add: numeral_2_eq_2 le_mod_geq)
haftmann@33296
  1139
haftmann@33296
  1140
lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)"
huffman@47217
  1141
by (simp add: mult_2 [symmetric])
haftmann@33296
  1142
wenzelm@61076
  1143
lemma mod2_gr_0 [simp]: "0 < (m::nat) mod 2 \<longleftrightarrow> m mod 2 = 1"
haftmann@33296
  1144
proof -
boehmes@35815
  1145
  { fix n :: nat have  "(n::nat) < 2 \<Longrightarrow> n = 0 \<or> n = 1" by (cases n) simp_all }
haftmann@33296
  1146
  moreover have "m mod 2 < 2" by simp
haftmann@33296
  1147
  ultimately have "m mod 2 = 0 \<or> m mod 2 = 1" .
haftmann@33296
  1148
  then show ?thesis by auto
haftmann@33296
  1149
qed
haftmann@33296
  1150
wenzelm@60758
  1151
text\<open>These lemmas collapse some needless occurrences of Suc:
haftmann@33296
  1152
    at least three Sucs, since two and fewer are rewritten back to Suc again!
wenzelm@60758
  1153
    We already have some rules to simplify operands smaller than 3.\<close>
haftmann@33296
  1154
haftmann@33296
  1155
lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)"
haftmann@33296
  1156
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1157
haftmann@33296
  1158
lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)"
haftmann@33296
  1159
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1160
haftmann@33296
  1161
lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n"
haftmann@33296
  1162
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1163
haftmann@33296
  1164
lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n"
haftmann@33296
  1165
by (simp add: Suc3_eq_add_3)
haftmann@33296
  1166
huffman@47108
  1167
lemmas Suc_div_eq_add3_div_numeral [simp] = Suc_div_eq_add3_div [of _ "numeral v"] for v
huffman@47108
  1168
lemmas Suc_mod_eq_add3_mod_numeral [simp] = Suc_mod_eq_add3_mod [of _ "numeral v"] for v
haftmann@33296
  1169
lp15@60562
  1170
lemma Suc_times_mod_eq: "1<k ==> Suc (k * m) mod k = 1"
haftmann@33361
  1171
apply (induct "m")
haftmann@33361
  1172
apply (simp_all add: mod_Suc)
haftmann@33361
  1173
done
haftmann@33361
  1174
huffman@47108
  1175
declare Suc_times_mod_eq [of "numeral w", simp] for w
haftmann@33361
  1176
huffman@47138
  1177
lemma Suc_div_le_mono [simp]: "n div k \<le> (Suc n) div k"
huffman@47138
  1178
by (simp add: div_le_mono)
haftmann@33361
  1179
haftmann@33361
  1180
lemma Suc_n_div_2_gt_zero [simp]: "(0::nat) < n ==> 0 < (n + 1) div 2"
haftmann@33361
  1181
by (cases n) simp_all
haftmann@33361
  1182
boehmes@35815
  1183
lemma div_2_gt_zero [simp]: assumes A: "(1::nat) < n" shows "0 < n div 2"
boehmes@35815
  1184
proof -
boehmes@35815
  1185
  from A have B: "0 < n - 1" and C: "n - 1 + 1 = n" by simp_all
lp15@60562
  1186
  from Suc_n_div_2_gt_zero [OF B] C show ?thesis by simp
boehmes@35815
  1187
qed
haftmann@33361
  1188
haftmann@66801
  1189
lemma mod_mult_self3' [simp]: "Suc (k * n + m) mod n = Suc m mod n"
haftmann@66801
  1190
  using mod_mult_self3 [of k n "Suc m"] by simp
haftmann@33361
  1191
haftmann@33361
  1192
lemma mod_Suc_eq_Suc_mod: "Suc m mod n = Suc (m mod n) mod n"
lp15@60562
  1193
apply (subst mod_Suc [of m])
lp15@60562
  1194
apply (subst mod_Suc [of "m mod n"], simp)
haftmann@33361
  1195
done
haftmann@33361
  1196
huffman@47108
  1197
lemma mod_2_not_eq_zero_eq_one_nat:
huffman@47108
  1198
  fixes n :: nat
huffman@47108
  1199
  shows "n mod 2 \<noteq> 0 \<longleftrightarrow> n mod 2 = 1"
haftmann@58786
  1200
  by (fact not_mod_2_eq_0_eq_1)
lp15@60562
  1201
haftmann@58778
  1202
lemma even_Suc_div_two [simp]:
haftmann@58778
  1203
  "even n \<Longrightarrow> Suc n div 2 = n div 2"
haftmann@58778
  1204
  using even_succ_div_two [of n] by simp
lp15@60562
  1205
haftmann@58778
  1206
lemma odd_Suc_div_two [simp]:
haftmann@58778
  1207
  "odd n \<Longrightarrow> Suc n div 2 = Suc (n div 2)"
haftmann@58778
  1208
  using odd_succ_div_two [of n] by simp
haftmann@58778
  1209
haftmann@58834
  1210
lemma odd_two_times_div_two_nat [simp]:
haftmann@60352
  1211
  assumes "odd n"
haftmann@60352
  1212
  shows "2 * (n div 2) = n - (1 :: nat)"
haftmann@60352
  1213
proof -
haftmann@60352
  1214
  from assms have "2 * (n div 2) + 1 = n"
haftmann@60352
  1215
    by (rule odd_two_times_div_two_succ)
haftmann@60352
  1216
  then have "Suc (2 * (n div 2)) - 1 = n - 1"
haftmann@60352
  1217
    by simp
haftmann@60352
  1218
  then show ?thesis
haftmann@60352
  1219
    by simp
haftmann@60352
  1220
qed
haftmann@58778
  1221
haftmann@58778
  1222
lemma parity_induct [case_names zero even odd]:
haftmann@58778
  1223
  assumes zero: "P 0"
haftmann@58778
  1224
  assumes even: "\<And>n. P n \<Longrightarrow> P (2 * n)"
haftmann@58778
  1225
  assumes odd: "\<And>n. P n \<Longrightarrow> P (Suc (2 * n))"
haftmann@58778
  1226
  shows "P n"
haftmann@58778
  1227
proof (induct n rule: less_induct)
haftmann@58778
  1228
  case (less n)
haftmann@58778
  1229
  show "P n"
haftmann@58778
  1230
  proof (cases "n = 0")
haftmann@58778
  1231
    case True with zero show ?thesis by simp
haftmann@58778
  1232
  next
haftmann@58778
  1233
    case False
haftmann@58778
  1234
    with less have hyp: "P (n div 2)" by simp
haftmann@58778
  1235
    show ?thesis
haftmann@58778
  1236
    proof (cases "even n")
haftmann@58778
  1237
      case True
haftmann@58778
  1238
      with hyp even [of "n div 2"] show ?thesis
haftmann@58834
  1239
        by simp
haftmann@58778
  1240
    next
haftmann@58778
  1241
      case False
lp15@60562
  1242
      with hyp odd [of "n div 2"] show ?thesis
haftmann@58834
  1243
        by simp
haftmann@58778
  1244
    qed
haftmann@58778
  1245
  qed
haftmann@58778
  1246
qed
haftmann@58778
  1247
haftmann@60868
  1248
lemma Suc_0_div_numeral [simp]:
haftmann@60868
  1249
  fixes k l :: num
haftmann@60868
  1250
  shows "Suc 0 div numeral k = fst (divmod Num.One k)"
haftmann@60868
  1251
  by (simp_all add: fst_divmod)
haftmann@60868
  1252
haftmann@60868
  1253
lemma Suc_0_mod_numeral [simp]:
haftmann@60868
  1254
  fixes k l :: num
haftmann@60868
  1255
  shows "Suc 0 mod numeral k = snd (divmod Num.One k)"
haftmann@60868
  1256
  by (simp_all add: snd_divmod)
haftmann@60868
  1257
haftmann@33361
  1258
wenzelm@60758
  1259
subsection \<open>Division on @{typ int}\<close>
haftmann@33361
  1260
haftmann@64592
  1261
context
haftmann@64592
  1262
begin
haftmann@64592
  1263
haftmann@64635
  1264
inductive eucl_rel_int :: "int \<Rightarrow> int \<Rightarrow> int \<times> int \<Rightarrow> bool"
haftmann@64635
  1265
  where eucl_rel_int_by0: "eucl_rel_int k 0 (0, k)"
haftmann@64635
  1266
  | eucl_rel_int_dividesI: "l \<noteq> 0 \<Longrightarrow> k = q * l \<Longrightarrow> eucl_rel_int k l (q, 0)"
haftmann@64635
  1267
  | eucl_rel_int_remainderI: "sgn r = sgn l \<Longrightarrow> \<bar>r\<bar> < \<bar>l\<bar>
haftmann@64635
  1268
      \<Longrightarrow> k = q * l + r \<Longrightarrow> eucl_rel_int k l (q, r)"
haftmann@64635
  1269
haftmann@64635
  1270
lemma eucl_rel_int_iff:    
haftmann@64635
  1271
  "eucl_rel_int k l (q, r) \<longleftrightarrow> 
haftmann@64635
  1272
    k = l * q + r \<and>
haftmann@64635
  1273
     (if 0 < l then 0 \<le> r \<and> r < l else if l < 0 then l < r \<and> r \<le> 0 else q = 0)"
haftmann@64635
  1274
  by (cases "r = 0")
haftmann@64635
  1275
    (auto elim!: eucl_rel_int.cases intro: eucl_rel_int_by0 eucl_rel_int_dividesI eucl_rel_int_remainderI
haftmann@64635
  1276
    simp add: ac_simps sgn_1_pos sgn_1_neg)
haftmann@33361
  1277
haftmann@33361
  1278
lemma unique_quotient_lemma:
haftmann@60868
  1279
  "b * q' + r' \<le> b * q + r \<Longrightarrow> 0 \<le> r' \<Longrightarrow> r' < b \<Longrightarrow> r < b \<Longrightarrow> q' \<le> (q::int)"
haftmann@33361
  1280
apply (subgoal_tac "r' + b * (q'-q) \<le> r")
haftmann@33361
  1281
 prefer 2 apply (simp add: right_diff_distrib)
haftmann@33361
  1282
apply (subgoal_tac "0 < b * (1 + q - q') ")
haftmann@33361
  1283
apply (erule_tac [2] order_le_less_trans)
webertj@49962
  1284
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1285
apply (subgoal_tac "b * q' < b * (1 + q) ")
webertj@49962
  1286
 prefer 2 apply (simp add: right_diff_distrib distrib_left)
haftmann@33361
  1287
apply (simp add: mult_less_cancel_left)
haftmann@33361
  1288
done
haftmann@33361
  1289
haftmann@33361
  1290
lemma unique_quotient_lemma_neg:
haftmann@60868
  1291
  "b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
haftmann@60868
  1292
  by (rule_tac b = "-b" and r = "-r'" and r' = "-r" in unique_quotient_lemma) auto
haftmann@33361
  1293
haftmann@33361
  1294
lemma unique_quotient:
haftmann@64635
  1295
  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> q = q'"
haftmann@64635
  1296
  apply (simp add: eucl_rel_int_iff linorder_neq_iff split: if_split_asm)
haftmann@64635
  1297
  apply (blast intro: order_antisym
haftmann@64635
  1298
    dest: order_eq_refl [THEN unique_quotient_lemma]
haftmann@64635
  1299
    order_eq_refl [THEN unique_quotient_lemma_neg] sym)+
haftmann@64635
  1300
  done
haftmann@33361
  1301
haftmann@33361
  1302
lemma unique_remainder:
haftmann@64635
  1303
  "eucl_rel_int a b (q, r) \<Longrightarrow> eucl_rel_int a b (q', r') \<Longrightarrow> r = r'"
haftmann@33361
  1304
apply (subgoal_tac "q = q'")
haftmann@64635
  1305
 apply (simp add: eucl_rel_int_iff)
haftmann@33361
  1306
apply (blast intro: unique_quotient)
haftmann@33361
  1307
done
haftmann@33361
  1308
haftmann@64592
  1309
end
haftmann@64592
  1310
haftmann@64592
  1311
instantiation int :: "{idom_modulo, normalization_semidom}"
haftmann@60868
  1312
begin
haftmann@60868
  1313
haftmann@64592
  1314
definition normalize_int :: "int \<Rightarrow> int"
haftmann@64592
  1315
  where [simp]: "normalize = (abs :: int \<Rightarrow> int)"
haftmann@64592
  1316
haftmann@64592
  1317
definition unit_factor_int :: "int \<Rightarrow> int"
haftmann@64592
  1318
  where [simp]: "unit_factor = (sgn :: int \<Rightarrow> int)"
haftmann@64592
  1319
haftmann@64592
  1320
definition divide_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  1321
  where "k div l = (if l = 0 \<or> k = 0 then 0
haftmann@60868
  1322
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
  1323
      then int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
  1324
      else
haftmann@60868
  1325
        if l dvd k then - int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)
haftmann@60868
  1326
        else - int (Suc (nat \<bar>k\<bar> div nat \<bar>l\<bar>)))"
haftmann@60868
  1327
haftmann@64592
  1328
definition modulo_int :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  1329
  where "k mod l = (if l = 0 then k else if l dvd k then 0
haftmann@60868
  1330
    else if k > 0 \<and> l > 0 \<or> k < 0 \<and> l < 0
haftmann@60868
  1331
      then sgn l * int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)
haftmann@60868
  1332
      else sgn l * (\<bar>l\<bar> - int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)))"
haftmann@60868
  1333
haftmann@64635
  1334
lemma eucl_rel_int:
haftmann@64635
  1335
  "eucl_rel_int k l (k div l, k mod l)"
haftmann@64592
  1336
proof (cases k rule: int_cases3)
haftmann@64592
  1337
  case zero
haftmann@64592
  1338
  then show ?thesis
haftmann@64635
  1339
    by (simp add: eucl_rel_int_iff divide_int_def modulo_int_def)
haftmann@64592
  1340
next
haftmann@64592
  1341
  case (pos n)
haftmann@64592
  1342
  then show ?thesis
haftmann@64592
  1343
    using div_mult_mod_eq [of n]
haftmann@64592
  1344
    by (cases l rule: int_cases3)
haftmann@64592
  1345
      (auto simp del: of_nat_mult of_nat_add
haftmann@64592
  1346
        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
haftmann@64635
  1347
        eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
haftmann@64592
  1348
next
haftmann@64592
  1349
  case (neg n)
haftmann@64592
  1350
  then show ?thesis
haftmann@64592
  1351
    using div_mult_mod_eq [of n]
haftmann@64592
  1352
    by (cases l rule: int_cases3)
haftmann@64592
  1353
      (auto simp del: of_nat_mult of_nat_add
haftmann@64592
  1354
        simp add: mod_greater_zero_iff_not_dvd of_nat_mult [symmetric] of_nat_add [symmetric] algebra_simps
haftmann@64635
  1355
        eucl_rel_int_iff divide_int_def modulo_int_def int_dvd_iff)
haftmann@64592
  1356
qed
haftmann@33361
  1357
huffman@47141
  1358
lemma divmod_int_unique:
haftmann@64635
  1359
  assumes "eucl_rel_int k l (q, r)"
haftmann@60868
  1360
  shows div_int_unique: "k div l = q" and mod_int_unique: "k mod l = r"
haftmann@64635
  1361
  using assms eucl_rel_int [of k l]
haftmann@60868
  1362
  using unique_quotient [of k l] unique_remainder [of k l]
haftmann@60868
  1363
  by auto
haftmann@64592
  1364
haftmann@64592
  1365
instance proof
haftmann@64592
  1366
  fix k l :: int
haftmann@64592
  1367
  show "k div l * l + k mod l = k"
haftmann@64635
  1368
    using eucl_rel_int [of k l]
haftmann@64635
  1369
    unfolding eucl_rel_int_iff by (simp add: ac_simps)
huffman@47141
  1370
next
haftmann@64592
  1371
  fix k :: int show "k div 0 = 0"
haftmann@64635
  1372
    by (rule div_int_unique, simp add: eucl_rel_int_iff)
huffman@47141
  1373
next
haftmann@64592
  1374
  fix k l :: int
haftmann@64592
  1375
  assume "l \<noteq> 0"
haftmann@64592
  1376
  then show "k * l div l = k"
haftmann@64635
  1377
    by (auto simp add: eucl_rel_int_iff ac_simps intro: div_int_unique [of _ _ _ 0])
haftmann@64848
  1378
qed (auto simp add: sgn_mult mult_sgn_abs abs_eq_iff')
huffman@47141
  1379
haftmann@60429
  1380
end
haftmann@60429
  1381
haftmann@66806
  1382
ML \<open>
haftmann@66806
  1383
structure Cancel_Div_Mod_Int = Cancel_Div_Mod
haftmann@66806
  1384
(
haftmann@66806
  1385
  val div_name = @{const_name divide};
haftmann@66806
  1386
  val mod_name = @{const_name modulo};
haftmann@66806
  1387
  val mk_binop = HOLogic.mk_binop;
haftmann@66806
  1388
  val mk_sum = Arith_Data.mk_sum HOLogic.intT;
haftmann@66806
  1389
  val dest_sum = Arith_Data.dest_sum;
haftmann@66806
  1390
haftmann@66806
  1391
  val div_mod_eqs = map mk_meta_eq @{thms cancel_div_mod_rules};
haftmann@66806
  1392
haftmann@66806
  1393
  val prove_eq_sums = Arith_Data.prove_conv2 all_tac (Arith_Data.simp_all_tac
haftmann@66806
  1394
    @{thms diff_conv_add_uminus add_0_left add_0_right ac_simps})
haftmann@66806
  1395
)
haftmann@66806
  1396
\<close>
haftmann@66806
  1397
haftmann@66806
  1398
simproc_setup cancel_div_mod_int ("(k::int) + l") =
haftmann@66806
  1399
  \<open>K Cancel_Div_Mod_Int.proc\<close>
haftmann@66806
  1400
haftmann@60517
  1401
lemma is_unit_int:
haftmann@60517
  1402
  "is_unit (k::int) \<longleftrightarrow> k = 1 \<or> k = - 1"
haftmann@60517
  1403
  by auto
haftmann@60517
  1404
haftmann@64715
  1405
lemma zdiv_int:
haftmann@64715
  1406
  "int (a div b) = int a div int b"
haftmann@64715
  1407
  by (simp add: divide_int_def)
haftmann@64715
  1408
haftmann@64715
  1409
lemma zmod_int:
haftmann@64715
  1410
  "int (a mod b) = int a mod int b"
haftmann@64715
  1411
  by (simp add: modulo_int_def int_dvd_iff)
haftmann@64715
  1412
haftmann@64715
  1413
lemma div_abs_eq_div_nat:
haftmann@64715
  1414
  "\<bar>k\<bar> div \<bar>l\<bar> = int (nat \<bar>k\<bar> div nat \<bar>l\<bar>)"
haftmann@64715
  1415
  by (simp add: divide_int_def)
haftmann@64715
  1416
haftmann@64715
  1417
lemma mod_abs_eq_div_nat:
haftmann@64715
  1418
  "\<bar>k\<bar> mod \<bar>l\<bar> = int (nat \<bar>k\<bar> mod nat \<bar>l\<bar>)"
haftmann@64715
  1419
  by (simp add: modulo_int_def dvd_int_unfold_dvd_nat)
haftmann@64715
  1420
haftmann@64715
  1421
lemma div_sgn_abs_cancel:
haftmann@64715
  1422
  fixes k l v :: int
haftmann@64715
  1423
  assumes "v \<noteq> 0"
haftmann@64715
  1424
  shows "(sgn v * \<bar>k\<bar>) div (sgn v * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
  1425
proof -
haftmann@64715
  1426
  from assms have "sgn v = - 1 \<or> sgn v = 1"
haftmann@64715
  1427
    by (cases "v \<ge> 0") auto
haftmann@64715
  1428
  then show ?thesis
blanchet@66630
  1429
    using assms unfolding divide_int_def [of "sgn v * \<bar>k\<bar>" "sgn v * \<bar>l\<bar>"]
blanchet@66630
  1430
    by (fastforce simp add: not_less div_abs_eq_div_nat)
haftmann@64715
  1431
qed
haftmann@64715
  1432
haftmann@64715
  1433
lemma div_eq_sgn_abs:
haftmann@64715
  1434
  fixes k l v :: int
haftmann@64715
  1435
  assumes "sgn k = sgn l"
haftmann@64715
  1436
  shows "k div l = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
  1437
proof (cases "l = 0")
haftmann@64715
  1438
  case True
haftmann@64715
  1439
  then show ?thesis
haftmann@64715
  1440
    by simp
haftmann@64715
  1441
next
haftmann@64715
  1442
  case False
haftmann@64715
  1443
  with assms have "(sgn k * \<bar>k\<bar>) div (sgn l * \<bar>l\<bar>) = \<bar>k\<bar> div \<bar>l\<bar>"
haftmann@64715
  1444
    by (simp add: div_sgn_abs_cancel)
haftmann@64715
  1445
  then show ?thesis
haftmann@64715
  1446
    by (simp add: sgn_mult_abs)
haftmann@64715
  1447
qed
haftmann@64715
  1448
haftmann@64715
  1449
lemma div_dvd_sgn_abs:
haftmann@64715
  1450
  fixes k l :: int
haftmann@64715
  1451
  assumes "l dvd k"
haftmann@64715
  1452
  shows "k div l = (sgn k * sgn l) * (\<bar>k\<bar> div \<bar>l\<bar>)"
haftmann@64715
  1453
proof (cases "k = 0")
haftmann@64715
  1454
  case True
haftmann@64715
  1455
  then show ?thesis
haftmann@64715
  1456
    by simp
haftmann@64715
  1457
next
haftmann@64715
  1458
  case False
haftmann@64715
  1459
  show ?thesis
haftmann@64715
  1460
  proof (cases "sgn l = sgn k")
haftmann@64715
  1461
    case True
haftmann@64715
  1462
    then show ?thesis
haftmann@64715
  1463
      by (simp add: div_eq_sgn_abs)
haftmann@64715
  1464
  next
haftmann@64715
  1465
    case False
haftmann@64715
  1466
    with \<open>k \<noteq> 0\<close> assms show ?thesis
haftmann@64715
  1467
      unfolding divide_int_def [of k l]
haftmann@64715
  1468
        by (auto simp add: zdiv_int)
haftmann@64715
  1469
  qed
haftmann@64715
  1470
qed
haftmann@64715
  1471
haftmann@64715
  1472
lemma div_noneq_sgn_abs:
haftmann@64715
  1473
  fixes k l :: int
haftmann@64715
  1474
  assumes "l \<noteq> 0"
haftmann@64715
  1475
  assumes "sgn k \<noteq> sgn l"
haftmann@64715
  1476
  shows "k div l = - (\<bar>k\<bar> div \<bar>l\<bar>) - of_bool (\<not> l dvd k)"
haftmann@64715
  1477
  using assms
haftmann@64715
  1478
  by (simp only: divide_int_def [of k l], auto simp add: not_less zdiv_int)
haftmann@64715
  1479
  
haftmann@64715
  1480
lemma sgn_mod:
haftmann@64715
  1481
  fixes k l :: int
haftmann@64715
  1482
  assumes "l \<noteq> 0" "\<not> l dvd k"
haftmann@64715
  1483
  shows "sgn (k mod l) = sgn l"
haftmann@64715
  1484
proof -
haftmann@64715
  1485
  from \<open>\<not> l dvd k\<close>
haftmann@64715
  1486
  have "k mod l \<noteq> 0"
haftmann@64715
  1487
    by (simp add: dvd_eq_mod_eq_0)
haftmann@64715
  1488
  show ?thesis
haftmann@64715
  1489
    using \<open>l \<noteq> 0\<close> \<open>\<not> l dvd k\<close>
haftmann@64715
  1490
    unfolding modulo_int_def [of k l]
haftmann@64715
  1491
    by (auto simp add: sgn_1_pos sgn_1_neg mod_greater_zero_iff_not_dvd nat_dvd_iff not_less
haftmann@64715
  1492
      zless_nat_eq_int_zless [symmetric] elim: nonpos_int_cases)
haftmann@64715
  1493
qed
haftmann@64715
  1494
haftmann@64715
  1495
lemma abs_mod_less:
haftmann@64715
  1496
  fixes k l :: int
haftmann@64715
  1497
  assumes "l \<noteq> 0"
haftmann@64715
  1498
  shows "\<bar>k mod l\<bar> < \<bar>l\<bar>"
haftmann@64715
  1499
  using assms unfolding modulo_int_def [of k l]
haftmann@64715
  1500
  by (auto simp add: not_less int_dvd_iff mod_greater_zero_iff_not_dvd elim: pos_int_cases neg_int_cases nonneg_int_cases nonpos_int_cases)
haftmann@64715
  1501
haftmann@66806
  1502
instantiation int :: unique_euclidean_ring
haftmann@66806
  1503
begin
haftmann@66806
  1504
haftmann@66806
  1505
definition [simp]:
haftmann@66806
  1506
  "euclidean_size_int = (nat \<circ> abs :: int \<Rightarrow> nat)"
haftmann@66806
  1507
haftmann@66806
  1508
definition [simp]:
haftmann@66806
  1509
  "uniqueness_constraint_int (k :: int) l \<longleftrightarrow> unit_factor k = unit_factor l"
haftmann@66806
  1510
  
haftmann@66806
  1511
instance proof
haftmann@66806
  1512
  fix l q r:: int
haftmann@66806
  1513
  assume "uniqueness_constraint r l"
haftmann@66806
  1514
    and "euclidean_size r < euclidean_size l"
haftmann@66806
  1515
  then have "sgn r = sgn l" and "\<bar>r\<bar> < \<bar>l\<bar>"
haftmann@66806
  1516
    by simp_all
haftmann@66806
  1517
  then have "eucl_rel_int (q * l + r) l (q, r)"
haftmann@66806
  1518
    by (rule eucl_rel_int_remainderI) simp
haftmann@66806
  1519
  then show "(q * l + r) div l = q"
haftmann@64592
  1520
    by (rule div_int_unique)
haftmann@66806
  1521
qed (use mult_le_mono2 [of 1] in \<open>auto simp add: abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib\<close>)
wenzelm@60758
  1522
haftmann@66806
  1523
end
haftmann@64592
  1524
haftmann@64592
  1525
text\<open>Basic laws about division and remainder\<close>
haftmann@64592
  1526
huffman@47141
  1527
lemma pos_mod_conj: "(0::int) < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b"
haftmann@64635
  1528
  using eucl_rel_int [of a b]
haftmann@64635
  1529
  by (auto simp add: eucl_rel_int_iff prod_eq_iff)
haftmann@33361
  1530
wenzelm@45607
  1531
lemmas pos_mod_sign [simp] = pos_mod_conj [THEN conjunct1]
wenzelm@45607
  1532
   and pos_mod_bound [simp] = pos_mod_conj [THEN conjunct2]
haftmann@33361
  1533
huffman@47141
  1534
lemma neg_mod_conj: "b < (0::int) \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b"
haftmann@64635
  1535
  using eucl_rel_int [of a b]
haftmann@64635
  1536
  by (auto simp add: eucl_rel_int_iff prod_eq_iff)
haftmann@33361
  1537
wenzelm@45607
  1538
lemmas neg_mod_sign [simp] = neg_mod_conj [THEN conjunct1]
wenzelm@45607
  1539
   and neg_mod_bound [simp] = neg_mod_conj [THEN conjunct2]
haftmann@33361
  1540
haftmann@33361
  1541
wenzelm@60758
  1542
subsubsection \<open>General Properties of div and mod\<close>
haftmann@33361
  1543
haftmann@33361
  1544
lemma div_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a div b = 0"
huffman@47140
  1545
apply (rule div_int_unique)
haftmann@64635
  1546
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
  1547
done
haftmann@33361
  1548
haftmann@33361
  1549
lemma div_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a div b = 0"
huffman@47140
  1550
apply (rule div_int_unique)
haftmann@64635
  1551
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
  1552
done
haftmann@33361
  1553
haftmann@33361
  1554
lemma div_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a div b = -1"
huffman@47140
  1555
apply (rule div_int_unique)
haftmann@64635
  1556
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
  1557
done
haftmann@33361
  1558
haftmann@66801
  1559
lemma div_positive_int:
haftmann@66801
  1560
  "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
haftmann@66801
  1561
  using that by (simp add: divide_int_def div_positive)
haftmann@66801
  1562
haftmann@33361
  1563
(*There is no div_neg_pos_trivial because  0 div b = 0 would supersede it*)
haftmann@33361
  1564
haftmann@33361
  1565
lemma mod_pos_pos_trivial: "[| (0::int) \<le> a;  a < b |] ==> a mod b = a"
huffman@47140
  1566
apply (rule_tac q = 0 in mod_int_unique)
haftmann@64635
  1567
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
  1568
done
haftmann@33361
  1569
haftmann@33361
  1570
lemma mod_neg_neg_trivial: "[| a \<le> (0::int);  b < a |] ==> a mod b = a"
huffman@47140
  1571
apply (rule_tac q = 0 in mod_int_unique)
haftmann@64635
  1572
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
  1573
done
haftmann@33361
  1574
haftmann@33361
  1575
lemma mod_pos_neg_trivial: "[| (0::int) < a;  a+b \<le> 0 |] ==> a mod b = a+b"
huffman@47140
  1576
apply (rule_tac q = "-1" in mod_int_unique)
haftmann@64635
  1577
apply (auto simp add: eucl_rel_int_iff)
haftmann@33361
  1578
done
haftmann@33361
  1579
wenzelm@61799
  1580
text\<open>There is no \<open>mod_neg_pos_trivial\<close>.\<close>
wenzelm@60758
  1581
wenzelm@60758
  1582
wenzelm@60758
  1583
subsubsection \<open>Laws for div and mod with Unary Minus\<close>
haftmann@33361
  1584
haftmann@33361
  1585
lemma zminus1_lemma:
haftmann@64635
  1586
     "eucl_rel_int a b (q, r) ==> b \<noteq> 0
haftmann@64635
  1587
      ==> eucl_rel_int (-a) b (if r=0 then -q else -q - 1,
haftmann@33361
  1588
                          if r=0 then 0 else b-r)"
blanchet@66630
  1589
by (force simp add: eucl_rel_int_iff right_diff_distrib)
haftmann@33361
  1590
haftmann@33361
  1591
haftmann@33361
  1592
lemma zdiv_zminus1_eq_if:
lp15@60562
  1593
     "b \<noteq> (0::int)
lp15@60562
  1594
      ==> (-a) div b =
haftmann@33361
  1595
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
haftmann@64635
  1596
by (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN div_int_unique])
haftmann@33361
  1597
haftmann@33361
  1598
lemma zmod_zminus1_eq_if:
haftmann@33361
  1599
     "(-a::int) mod b = (if a mod b = 0 then 0 else  b - (a mod b))"
haftmann@33361
  1600
apply (case_tac "b = 0", simp)
haftmann@64635
  1601
apply (blast intro: eucl_rel_int [THEN zminus1_lemma, THEN mod_int_unique])
haftmann@33361
  1602
done
haftmann@33361
  1603
haftmann@64593
  1604
lemma zmod_zminus1_not_zero:
haftmann@33361
  1605
  fixes k l :: int
haftmann@33361
  1606
  shows "- k mod l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@64592
  1607
  by (simp add: mod_eq_0_iff_dvd)
haftmann@64592
  1608
haftmann@64593
  1609
lemma zmod_zminus2_not_zero:
haftmann@64592
  1610
  fixes k l :: int
haftmann@64592
  1611
  shows "k mod - l \<noteq> 0 \<Longrightarrow> k mod l \<noteq> 0"
haftmann@64592
  1612
  by (simp add: mod_eq_0_iff_dvd)
haftmann@33361
  1613
haftmann@33361
  1614
lemma zdiv_zminus2_eq_if:
lp15@60562
  1615
     "b \<noteq> (0::int)
lp15@60562
  1616
      ==> a div (-b) =
haftmann@33361
  1617
          (if a mod b = 0 then - (a div b) else  - (a div b) - 1)"
huffman@47159
  1618
by (simp add: zdiv_zminus1_eq_if div_minus_right)
haftmann@33361
  1619
haftmann@33361
  1620
lemma zmod_zminus2_eq_if:
haftmann@33361
  1621
     "a mod (-b::int) = (if a mod b = 0 then 0 else  (a mod b) - b)"
huffman@47159
  1622
by (simp add: zmod_zminus1_eq_if mod_minus_right)
haftmann@33361
  1623
haftmann@33361
  1624
wenzelm@60758
  1625
subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
haftmann@33361
  1626
haftmann@33361
  1627
lemma zdiv_mono1: "[| a \<le> a';  0 < (b::int) |] ==> a div b \<le> a' div b"
haftmann@64246
  1628
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1629
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
  1630
apply -
haftmann@33361
  1631
apply (rule unique_quotient_lemma)
haftmann@33361
  1632
apply (erule subst)
haftmann@33361
  1633
apply (erule subst, simp_all)
haftmann@33361
  1634
done
haftmann@33361
  1635
haftmann@33361
  1636
lemma zdiv_mono1_neg: "[| a \<le> a';  (b::int) < 0 |] ==> a' div b \<le> a div b"
haftmann@64246
  1637
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1638
using mult_div_mod_eq [symmetric, of a' b]
haftmann@64246
  1639
apply -
haftmann@33361
  1640
apply (rule unique_quotient_lemma_neg)
haftmann@33361
  1641
apply (erule subst)
haftmann@33361
  1642
apply (erule subst, simp_all)
haftmann@33361
  1643
done
haftmann@33361
  1644
haftmann@33361
  1645
wenzelm@60758
  1646
subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
haftmann@33361
  1647
haftmann@33361
  1648
lemma q_pos_lemma:
haftmann@33361
  1649
     "[| 0 \<le> b'*q' + r'; r' < b';  0 < b' |] ==> 0 \<le> (q'::int)"
haftmann@33361
  1650
apply (subgoal_tac "0 < b'* (q' + 1) ")
haftmann@33361
  1651
 apply (simp add: zero_less_mult_iff)
webertj@49962
  1652
apply (simp add: distrib_left)
haftmann@33361
  1653
done
haftmann@33361
  1654
haftmann@33361
  1655
lemma zdiv_mono2_lemma:
lp15@60562
  1656
     "[| b*q + r = b'*q' + r';  0 \<le> b'*q' + r';
lp15@60562
  1657
         r' < b';  0 \<le> r;  0 < b';  b' \<le> b |]
haftmann@33361
  1658
      ==> q \<le> (q'::int)"
lp15@60562
  1659
apply (frule q_pos_lemma, assumption+)
haftmann@33361
  1660
apply (subgoal_tac "b*q < b* (q' + 1) ")
haftmann@33361
  1661
 apply (simp add: mult_less_cancel_left)
haftmann@33361
  1662
apply (subgoal_tac "b*q = r' - r + b'*q'")
haftmann@33361
  1663
 prefer 2 apply simp
webertj@49962
  1664
apply (simp (no_asm_simp) add: distrib_left)
haftmann@57512
  1665
apply (subst add.commute, rule add_less_le_mono, arith)
haftmann@33361
  1666
apply (rule mult_right_mono, auto)
haftmann@33361
  1667
done
haftmann@33361
  1668
haftmann@33361
  1669
lemma zdiv_mono2:
haftmann@33361
  1670
     "[| (0::int) \<le> a;  0 < b';  b' \<le> b |] ==> a div b \<le> a div b'"
haftmann@33361
  1671
apply (subgoal_tac "b \<noteq> 0")
haftmann@64246
  1672
  prefer 2 apply arith
haftmann@64246
  1673
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1674
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
  1675
apply -
haftmann@33361
  1676
apply (rule zdiv_mono2_lemma)
haftmann@33361
  1677
apply (erule subst)
haftmann@33361
  1678
apply (erule subst, simp_all)
haftmann@33361
  1679
done
haftmann@33361
  1680
haftmann@33361
  1681
lemma q_neg_lemma:
haftmann@33361
  1682
     "[| b'*q' + r' < 0;  0 \<le> r';  0 < b' |] ==> q' \<le> (0::int)"
haftmann@33361
  1683
apply (subgoal_tac "b'*q' < 0")
haftmann@33361
  1684
 apply (simp add: mult_less_0_iff, arith)
haftmann@33361
  1685
done
haftmann@33361
  1686
haftmann@33361
  1687
lemma zdiv_mono2_neg_lemma:
lp15@60562
  1688
     "[| b*q + r = b'*q' + r';  b'*q' + r' < 0;
lp15@60562
  1689
         r < b;  0 \<le> r';  0 < b';  b' \<le> b |]
haftmann@33361
  1690
      ==> q' \<le> (q::int)"
lp15@60562
  1691
apply (frule q_neg_lemma, assumption+)
haftmann@33361
  1692
apply (subgoal_tac "b*q' < b* (q + 1) ")
haftmann@33361
  1693
 apply (simp add: mult_less_cancel_left)
webertj@49962
  1694
apply (simp add: distrib_left)
haftmann@33361
  1695
apply (subgoal_tac "b*q' \<le> b'*q'")
haftmann@33361
  1696
 prefer 2 apply (simp add: mult_right_mono_neg, arith)
haftmann@33361
  1697
done
haftmann@33361
  1698
haftmann@33361
  1699
lemma zdiv_mono2_neg:
haftmann@33361
  1700
     "[| a < (0::int);  0 < b';  b' \<le> b |] ==> a div b' \<le> a div b"
haftmann@64246
  1701
using mult_div_mod_eq [symmetric, of a b]
haftmann@64246
  1702
using mult_div_mod_eq [symmetric, of a b']
haftmann@64246
  1703
apply -
haftmann@33361
  1704
apply (rule zdiv_mono2_neg_lemma)
haftmann@33361
  1705
apply (erule subst)
haftmann@33361
  1706
apply (erule subst, simp_all)
haftmann@33361
  1707
done
haftmann@33361
  1708
haftmann@33361
  1709
wenzelm@60758
  1710
subsubsection \<open>More Algebraic Laws for div and mod\<close>
wenzelm@60758
  1711
wenzelm@60758
  1712
text\<open>proving (a*b) div c = a * (b div c) + a * (b mod c)\<close>
haftmann@33361
  1713
haftmann@33361
  1714
lemma zmult1_lemma:
haftmann@64635
  1715
     "[| eucl_rel_int b c (q, r) |]
haftmann@64635
  1716
      ==> eucl_rel_int (a * b) c (a*q + a*r div c, a*r mod c)"
haftmann@64635
  1717
by (auto simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left ac_simps)
haftmann@33361
  1718
haftmann@33361
  1719
lemma zdiv_zmult1_eq: "(a*b) div c = a*(b div c) + a*(b mod c) div (c::int)"
haftmann@33361
  1720
apply (case_tac "c = 0", simp)
haftmann@64635
  1721
apply (blast intro: eucl_rel_int [THEN zmult1_lemma, THEN div_int_unique])
haftmann@33361
  1722
done
haftmann@33361
  1723
wenzelm@60758
  1724
text\<open>proving (a+b) div c = a div c + b div c + ((a mod c + b mod c) div c)\<close>
haftmann@33361
  1725
haftmann@33361
  1726
lemma zadd1_lemma:
haftmann@64635
  1727
     "[| eucl_rel_int a c (aq, ar);  eucl_rel_int b c (bq, br) |]
haftmann@64635
  1728
      ==> eucl_rel_int (a+b) c (aq + bq + (ar+br) div c, (ar+br) mod c)"
haftmann@64635
  1729
by (force simp add: split_ifs eucl_rel_int_iff linorder_neq_iff distrib_left)
haftmann@33361
  1730
haftmann@33361
  1731
(*NOT suitable for rewriting: the RHS has an instance of the LHS*)
haftmann@33361
  1732
lemma zdiv_zadd1_eq:
haftmann@33361
  1733
     "(a+b) div (c::int) = a div c + b div c + ((a mod c + b mod c) div c)"
haftmann@33361
  1734
apply (case_tac "c = 0", simp)
haftmann@64635
  1735
apply (blast intro: zadd1_lemma [OF eucl_rel_int eucl_rel_int] div_int_unique)
haftmann@33361
  1736
done
haftmann@33361
  1737
haftmann@33361
  1738
lemma zmod_eq_0_iff: "(m mod d = 0) = (EX q::int. m = d*q)"
haftmann@33361
  1739
by (simp add: dvd_eq_mod_eq_0 [symmetric] dvd_def)
haftmann@33361
  1740
haftmann@33361
  1741
(* REVISIT: should this be generalized to all semiring_div types? *)
haftmann@33361
  1742
lemmas zmod_eq_0D [dest!] = zmod_eq_0_iff [THEN iffD1]
haftmann@33361
  1743
haftmann@33361
  1744
wenzelm@60758
  1745
subsubsection \<open>Proving  @{term "a div (b * c) = (a div b) div c"}\<close>
haftmann@33361
  1746
haftmann@33361
  1747
(*The condition c>0 seems necessary.  Consider that 7 div ~6 = ~2 but
haftmann@33361
  1748
  7 div 2 div ~3 = 3 div ~3 = ~1.  The subcase (a div b) mod c = 0 seems
haftmann@33361
  1749
  to cause particular problems.*)
haftmann@33361
  1750
wenzelm@60758
  1751
text\<open>first, four lemmas to bound the remainder for the cases b<0 and b>0\<close>
haftmann@33361
  1752
blanchet@55085
  1753
lemma zmult2_lemma_aux1: "[| (0::int) < c;  b < r;  r \<le> 0 |] ==> b * c < b * (q mod c) + r"
haftmann@33361
  1754
apply (subgoal_tac "b * (c - q mod c) < r * 1")
haftmann@33361
  1755
 apply (simp add: algebra_simps)
haftmann@33361
  1756
apply (rule order_le_less_trans)
haftmann@33361
  1757
 apply (erule_tac [2] mult_strict_right_mono)
haftmann@33361
  1758
 apply (rule mult_left_mono_neg)
huffman@35216
  1759
  using add1_zle_eq[of "q mod c"]apply(simp add: algebra_simps)
haftmann@33361
  1760
 apply (simp)
haftmann@33361
  1761
apply (simp)
haftmann@33361
  1762
done
haftmann@33361
  1763
haftmann@33361
  1764
lemma zmult2_lemma_aux2:
haftmann@33361
  1765
     "[| (0::int) < c;   b < r;  r \<le> 0 |] ==> b * (q mod c) + r \<le> 0"
haftmann@33361
  1766
apply (subgoal_tac "b * (q mod c) \<le> 0")
haftmann@33361
  1767
 apply arith
haftmann@33361
  1768
apply (simp add: mult_le_0_iff)
haftmann@33361
  1769
done
haftmann@33361
  1770
haftmann@33361
  1771
lemma zmult2_lemma_aux3: "[| (0::int) < c;  0 \<le> r;  r < b |] ==> 0 \<le> b * (q mod c) + r"
haftmann@33361
  1772
apply (subgoal_tac "0 \<le> b * (q mod c) ")
haftmann@33361
  1773
apply arith
haftmann@33361
  1774
apply (simp add: zero_le_mult_iff)
haftmann@33361
  1775
done
haftmann@33361
  1776
haftmann@33361
  1777
lemma zmult2_lemma_aux4: "[| (0::int) < c; 0 \<le> r; r < b |] ==> b * (q mod c) + r < b * c"
haftmann@33361
  1778
apply (subgoal_tac "r * 1 < b * (c - q mod c) ")
haftmann@33361
  1779
 apply (simp add: right_diff_distrib)
haftmann@33361
  1780
apply (rule order_less_le_trans)
haftmann@33361
  1781
 apply (erule mult_strict_right_mono)
haftmann@33361
  1782
 apply (rule_tac [2] mult_left_mono)
haftmann@33361
  1783
  apply simp
huffman@35216
  1784
 using add1_zle_eq[of "q mod c"] apply (simp add: algebra_simps)
haftmann@33361
  1785
apply simp
haftmann@33361
  1786
done
haftmann@33361
  1787
haftmann@64635
  1788
lemma zmult2_lemma: "[| eucl_rel_int a b (q, r); 0 < c |]
haftmann@64635
  1789
      ==> eucl_rel_int a (b * c) (q div c, b*(q mod c) + r)"
haftmann@64635
  1790
by (auto simp add: mult.assoc eucl_rel_int_iff linorder_neq_iff
lp15@60562
  1791
                   zero_less_mult_iff distrib_left [symmetric]
nipkow@62390
  1792
                   zmult2_lemma_aux1 zmult2_lemma_aux2 zmult2_lemma_aux3 zmult2_lemma_aux4 mult_less_0_iff split: if_split_asm)
haftmann@33361
  1793
haftmann@53068
  1794
lemma zdiv_zmult2_eq:
haftmann@53068
  1795
  fixes a b c :: int
haftmann@53068
  1796
  shows "0 \<le> c \<Longrightarrow> a div (b * c) = (a div b) div c"
haftmann@33361
  1797
apply (case_tac "b = 0", simp)
haftmann@64635
  1798
apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN div_int_unique])
haftmann@33361
  1799
done
haftmann@33361
  1800
haftmann@33361
  1801
lemma zmod_zmult2_eq:
haftmann@53068
  1802
  fixes a b c :: int
haftmann@53068
  1803
  shows "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
haftmann@33361
  1804
apply (case_tac "b = 0", simp)
haftmann@64635
  1805
apply (force simp add: le_less eucl_rel_int [THEN zmult2_lemma, THEN mod_int_unique])
haftmann@33361
  1806
done
haftmann@33361
  1807
huffman@47108
  1808
lemma div_pos_geq:
huffman@47108
  1809
  fixes k l :: int
huffman@47108
  1810
  assumes "0 < l" and "l \<le> k"
huffman@47108
  1811
  shows "k div l = (k - l) div l + 1"
huffman@47108
  1812
proof -
huffman@47108
  1813
  have "k = (k - l) + l" by simp
huffman@47108
  1814
  then obtain j where k: "k = j + l" ..
eberlm@63499
  1815
  with assms show ?thesis by (simp add: div_add_self2)
huffman@47108
  1816
qed
huffman@47108
  1817
huffman@47108
  1818
lemma mod_pos_geq:
huffman@47108
  1819
  fixes k l :: int
huffman@47108
  1820
  assumes "0 < l" and "l \<le> k"
huffman@47108
  1821
  shows "k mod l = (k - l) mod l"
huffman@47108
  1822
proof -
huffman@47108
  1823
  have "k = (k - l) + l" by simp
huffman@47108
  1824
  then obtain j where k: "k = j + l" ..
huffman@47108
  1825
  with assms show ?thesis by simp
huffman@47108
  1826
qed
huffman@47108
  1827
haftmann@33361
  1828
wenzelm@60758
  1829
subsubsection \<open>Splitting Rules for div and mod\<close>
wenzelm@60758
  1830
wenzelm@60758
  1831
text\<open>The proofs of the two lemmas below are essentially identical\<close>
haftmann@33361
  1832
haftmann@33361
  1833
lemma split_pos_lemma:
lp15@60562
  1834
 "0<k ==>
haftmann@33361
  1835
    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
  1836
apply (rule iffI, clarify)
lp15@60562
  1837
 apply (erule_tac P="P x y" for x y in rev_mp)
haftmann@64593
  1838
 apply (subst mod_add_eq [symmetric])
lp15@60562
  1839
 apply (subst zdiv_zadd1_eq)
lp15@60562
  1840
 apply (simp add: div_pos_pos_trivial mod_pos_pos_trivial)
wenzelm@60758
  1841
txt\<open>converse direction\<close>
lp15@60562
  1842
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  1843
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  1844
done
haftmann@33361
  1845
haftmann@33361
  1846
lemma split_neg_lemma:
haftmann@33361
  1847
 "k<0 ==>
haftmann@33361
  1848
    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
  1849
apply (rule iffI, clarify)
lp15@60562
  1850
 apply (erule_tac P="P x y" for x y in rev_mp)
haftmann@64593
  1851
 apply (subst mod_add_eq [symmetric])
lp15@60562
  1852
 apply (subst zdiv_zadd1_eq)
lp15@60562
  1853
 apply (simp add: div_neg_neg_trivial mod_neg_neg_trivial)
wenzelm@60758
  1854
txt\<open>converse direction\<close>
lp15@60562
  1855
apply (drule_tac x = "n div k" in spec)
haftmann@33361
  1856
apply (drule_tac x = "n mod k" in spec, simp)
haftmann@33361
  1857
done
haftmann@33361
  1858
haftmann@33361
  1859
lemma split_zdiv:
haftmann@33361
  1860
 "P(n div k :: int) =
lp15@60562
  1861
  ((k = 0 --> P 0) &
lp15@60562
  1862
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P i)) &
haftmann@33361
  1863
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P i)))"
haftmann@33361
  1864
apply (case_tac "k=0", simp)
haftmann@33361
  1865
apply (simp only: linorder_neq_iff)
lp15@60562
  1866
apply (erule disjE)
lp15@60562
  1867
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P x"]
haftmann@33361
  1868
                      split_neg_lemma [of concl: "%x y. P x"])
haftmann@33361
  1869
done
haftmann@33361
  1870
haftmann@33361
  1871
lemma split_zmod:
haftmann@33361
  1872
 "P(n mod k :: int) =
lp15@60562
  1873
  ((k = 0 --> P n) &
lp15@60562
  1874
   (0<k --> (\<forall>i j. 0\<le>j & j<k & n = k*i + j --> P j)) &
haftmann@33361
  1875
   (k<0 --> (\<forall>i j. k<j & j\<le>0 & n = k*i + j --> P j)))"
haftmann@33361
  1876
apply (case_tac "k=0", simp)
haftmann@33361
  1877
apply (simp only: linorder_neq_iff)
lp15@60562
  1878
apply (erule disjE)
lp15@60562
  1879
 apply (simp_all add: split_pos_lemma [of concl: "%x y. P y"]
haftmann@33361
  1880
                      split_neg_lemma [of concl: "%x y. P y"])
haftmann@33361
  1881
done
haftmann@33361
  1882
haftmann@63950
  1883
text \<open>Enable (lin)arith to deal with @{const divide} and @{const modulo}
webertj@33730
  1884
  when these are applied to some constant that is of the form
wenzelm@60758
  1885
  @{term "numeral k"}:\<close>
huffman@47108
  1886
declare split_zdiv [of _ _ "numeral k", arith_split] for k
huffman@47108
  1887
declare split_zmod [of _ _ "numeral k", arith_split] for k
haftmann@33361
  1888
haftmann@33361
  1889
wenzelm@61799
  1890
subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
huffman@47166
  1891
haftmann@64635
  1892
lemma pos_eucl_rel_int_mult_2:
huffman@47166
  1893
  assumes "0 \<le> b"
haftmann@64635
  1894
  assumes "eucl_rel_int a b (q, r)"
haftmann@64635
  1895
  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 1 + 2*r)"
haftmann@64635
  1896
  using assms unfolding eucl_rel_int_iff by auto
haftmann@64635
  1897
haftmann@64635
  1898
lemma neg_eucl_rel_int_mult_2:
huffman@47166
  1899
  assumes "b \<le> 0"
haftmann@64635
  1900
  assumes "eucl_rel_int (a + 1) b (q, r)"
haftmann@64635
  1901
  shows "eucl_rel_int (1 + 2*a) (2*b) (q, 2*r - 1)"
haftmann@64635
  1902
  using assms unfolding eucl_rel_int_iff by auto
haftmann@33361
  1903
wenzelm@60758
  1904
text\<open>computing div by shifting\<close>
haftmann@33361
  1905
haftmann@33361
  1906
lemma pos_zdiv_mult_2: "(0::int) \<le> a ==> (1 + 2*b) div (2*a) = b div a"
haftmann@64635
  1907
  using pos_eucl_rel_int_mult_2 [OF _ eucl_rel_int]
huffman@47166
  1908
  by (rule div_int_unique)
haftmann@33361
  1909
lp15@60562
  1910
lemma neg_zdiv_mult_2:
boehmes@35815
  1911
  assumes A: "a \<le> (0::int)" shows "(1 + 2*b) div (2*a) = (b+1) div a"
haftmann@64635
  1912
  using neg_eucl_rel_int_mult_2 [OF A eucl_rel_int]
huffman@47166
  1913
  by (rule div_int_unique)
haftmann@33361
  1914
huffman@47108
  1915
(* FIXME: add rules for negative numerals *)
huffman@47108
  1916
lemma zdiv_numeral_Bit0 [simp]:
huffman@47108
  1917
  "numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
huffman@47108
  1918
    numeral v div (numeral w :: int)"
huffman@47108
  1919
  unfolding numeral.simps unfolding mult_2 [symmetric]
huffman@47108
  1920
  by (rule div_mult_mult1, simp)
huffman@47108
  1921
huffman@47108
  1922
lemma zdiv_numeral_Bit1 [simp]:
lp15@60562
  1923
  "numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
huffman@47108
  1924
    (numeral v div (numeral w :: int))"
huffman@47108
  1925
  unfolding numeral.simps
haftmann@57512
  1926
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  1927
  by (rule pos_zdiv_mult_2, simp)
haftmann@33361
  1928
haftmann@33361
  1929
lemma pos_zmod_mult_2:
haftmann@33361
  1930
  fixes a b :: int
haftmann@33361
  1931
  assumes "0 \<le> a"
haftmann@33361
  1932
  shows "(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)"
haftmann@64635
  1933
  using pos_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
huffman@47166
  1934
  by (rule mod_int_unique)
haftmann@33361
  1935
haftmann@33361
  1936
lemma neg_zmod_mult_2:
haftmann@33361
  1937
  fixes a b :: int
haftmann@33361
  1938
  assumes "a \<le> 0"
haftmann@33361
  1939
  shows "(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1"
haftmann@64635
  1940
  using neg_eucl_rel_int_mult_2 [OF assms eucl_rel_int]
huffman@47166
  1941
  by (rule mod_int_unique)
haftmann@33361
  1942
huffman@47108
  1943
(* FIXME: add rules for negative numerals *)
huffman@47108
  1944
lemma zmod_numeral_Bit0 [simp]:
lp15@60562
  1945
  "numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
huffman@47108
  1946
    (2::int) * (numeral v mod numeral w)"
huffman@47108
  1947
  unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
huffman@47108
  1948
  unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
huffman@47108
  1949
huffman@47108
  1950
lemma zmod_numeral_Bit1 [simp]:
huffman@47108
  1951
  "numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
huffman@47108
  1952
    2 * (numeral v mod numeral w) + (1::int)"
huffman@47108
  1953
  unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
haftmann@57512
  1954
  unfolding mult_2 [symmetric] add.commute [of _ 1]
huffman@47108
  1955
  by (rule pos_zmod_mult_2, simp)
haftmann@33361
  1956
nipkow@39489
  1957
lemma zdiv_eq_0_iff:
nipkow@39489
  1958
 "(i::int) div k = 0 \<longleftrightarrow> k=0 \<or> 0\<le>i \<and> i<k \<or> i\<le>0 \<and> k<i" (is "?L = ?R")
nipkow@39489
  1959
proof
nipkow@39489
  1960
  assume ?L
nipkow@39489
  1961
  have "?L \<longrightarrow> ?R" by (rule split_zdiv[THEN iffD2]) simp
wenzelm@60758
  1962
  with \<open>?L\<close> show ?R by blast
nipkow@39489
  1963
next
nipkow@39489
  1964
  assume ?R thus ?L
nipkow@39489
  1965
    by(auto simp: div_pos_pos_trivial div_neg_neg_trivial)
nipkow@39489
  1966
qed
nipkow@39489
  1967
haftmann@63947
  1968
lemma zmod_trival_iff:
haftmann@63947
  1969
  fixes i k :: int
haftmann@63947
  1970
  shows "i mod k = i \<longleftrightarrow> k = 0 \<or> 0 \<le> i \<and> i < k \<or> i \<le> 0 \<and> k < i"
haftmann@63947
  1971
proof -
haftmann@63947
  1972
  have "i mod k = i \<longleftrightarrow> i div k = 0"
haftmann@64242
  1973
    by safe (insert div_mult_mod_eq [of i k], auto)
haftmann@63947
  1974
  with zdiv_eq_0_iff
haftmann@63947
  1975
  show ?thesis
haftmann@63947
  1976
    by simp
haftmann@63947
  1977
qed
nipkow@39489
  1978
haftmann@64785
  1979
  
wenzelm@60758
  1980
subsubsection \<open>Quotients of Signs\<close>
haftmann@33361
  1981
haftmann@60868
  1982
lemma div_eq_minus1: "(0::int) < b ==> -1 div b = -1"
haftmann@60868
  1983
by (simp add: divide_int_def)
haftmann@60868
  1984
haftmann@60868
  1985
lemma zmod_minus1: "(0::int) < b ==> -1 mod b = b - 1"
haftmann@63950
  1986
by (simp add: modulo_int_def)
haftmann@60868
  1987
haftmann@33361
  1988
lemma div_neg_pos_less0: "[| a < (0::int);  0 < b |] ==> a div b < 0"
haftmann@33361
  1989
apply (subgoal_tac "a div b \<le> -1", force)
haftmann@33361
  1990
apply (rule order_trans)
haftmann@33361
  1991
apply (rule_tac a' = "-1" in zdiv_mono1)
haftmann@33361
  1992
apply (auto simp add: div_eq_minus1)
haftmann@33361
  1993
done
haftmann@33361
  1994
haftmann@33361
  1995
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
haftmann@33361
  1996
by (drule zdiv_mono1_neg, auto)
haftmann@33361
  1997
haftmann@33361
  1998
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
haftmann@33361
  1999
by (drule zdiv_mono1, auto)
haftmann@33361
  2000
wenzelm@61799
  2001
text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
wenzelm@61799
  2002
conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
wenzelm@60758
  2003
They should all be simp rules unless that causes too much search.\<close>
nipkow@33804
  2004
haftmann@33361
  2005
lemma pos_imp_zdiv_nonneg_iff: "(0::int) < b ==> (0 \<le> a div b) = (0 \<le> a)"
haftmann@33361
  2006
apply auto
haftmann@33361
  2007
apply (drule_tac [2] zdiv_mono1)
haftmann@33361
  2008
apply (auto simp add: linorder_neq_iff)
haftmann@33361
  2009
apply (simp (no_asm_use) add: linorder_not_less [symmetric])
haftmann@33361
  2010
apply (blast intro: div_neg_pos_less0)
haftmann@33361
  2011
done
haftmann@33361
  2012
haftmann@60868
  2013
lemma pos_imp_zdiv_pos_iff:
haftmann@60868
  2014
  "0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
haftmann@60868
  2015
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k]
haftmann@60868
  2016
by arith
haftmann@60868
  2017
haftmann@33361
  2018
lemma neg_imp_zdiv_nonneg_iff:
nipkow@33804
  2019
  "b < (0::int) ==> (0 \<le> a div b) = (a \<le> (0::int))"
huffman@47159
  2020
apply (subst div_minus_minus [symmetric])
haftmann@33361
  2021
apply (subst pos_imp_zdiv_nonneg_iff, auto)
haftmann@33361
  2022
done
haftmann@33361
  2023
haftmann@33361
  2024
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
haftmann@33361
  2025
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
haftmann@33361
  2026
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2027
haftmann@33361
  2028
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
haftmann@33361
  2029
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
haftmann@33361
  2030
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
haftmann@33361
  2031
nipkow@33804
  2032
lemma nonneg1_imp_zdiv_pos_iff:
nipkow@33804
  2033
  "(0::int) <= a \<Longrightarrow> (a div b > 0) = (a >= b & b>0)"
nipkow@33804
  2034
apply rule
nipkow@33804
  2035
 apply rule
nipkow@33804
  2036
  using div_pos_pos_trivial[of a b]apply arith
nipkow@33804
  2037
 apply(cases "b=0")apply simp
nipkow@33804
  2038
 using div_nonneg_neg_le0[of a b]apply arith
nipkow@33804
  2039
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b]apply simp
nipkow@33804
  2040
done
nipkow@33804
  2041
nipkow@39489
  2042
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 ==> m mod k \<le> m"
nipkow@39489
  2043
apply (rule split_zmod[THEN iffD2])
nipkow@44890
  2044
apply(fastforce dest: q_pos_lemma intro: split_mult_pos_le)
nipkow@39489
  2045
done
nipkow@39489
  2046
haftmann@60868
  2047
haftmann@60868
  2048
subsubsection \<open>Computation of Division and Remainder\<close>
haftmann@60868
  2049
haftmann@66806
  2050
instantiation int :: unique_euclidean_semiring_numeral
haftmann@61275
  2051
begin
haftmann@61275
  2052
haftmann@61275
  2053
definition divmod_int :: "num \<Rightarrow> num \<Rightarrow> int \<times> int"
haftmann@61275
  2054
where
haftmann@61275
  2055
  "divmod_int m n = (numeral m div numeral n, numeral m mod numeral n)"
haftmann@61275
  2056
haftmann@61275
  2057
definition divmod_step_int :: "num \<Rightarrow> int \<times> int \<Rightarrow> int \<times> int"
haftmann@61275
  2058
where
haftmann@61275
  2059
  "divmod_step_int l qr = (let (q, r) = qr
haftmann@61275
  2060
    in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
haftmann@61275
  2061
    else (2 * q, r))"
haftmann@61275
  2062
haftmann@61275
  2063
instance
haftmann@61275
  2064
  by standard (auto intro: zmod_le_nonneg_dividend simp add: divmod_int_def divmod_step_int_def
haftmann@61275
  2065
    pos_imp_zdiv_pos_iff div_pos_pos_trivial mod_pos_pos_trivial zmod_zmult2_eq zdiv_zmult2_eq)
haftmann@61275
  2066
haftmann@61275
  2067
end
haftmann@61275
  2068
haftmann@61275
  2069
declare divmod_algorithm_code [where ?'a = int, code]
lp15@60562
  2070
haftmann@60930
  2071
context
haftmann@60930
  2072
begin
haftmann@60930
  2073
  
haftmann@60930
  2074
qualified definition adjust_div :: "int \<times> int \<Rightarrow> int"
haftmann@60868
  2075
where
haftmann@60868
  2076
  "adjust_div qr = (let (q, r) = qr in q + of_bool (r \<noteq> 0))"
haftmann@60868
  2077
haftmann@60930
  2078
qualified lemma adjust_div_eq [simp, code]:
haftmann@60868
  2079
  "adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
haftmann@60868
  2080
  by (simp add: adjust_div_def)
haftmann@60868
  2081
haftmann@60930
  2082
qualified definition adjust_mod :: "int \<Rightarrow> int \<Rightarrow> int"
haftmann@60868
  2083
where
haftmann@60868
  2084
  [simp]: "adjust_mod l r = (if r = 0 then 0 else l - r)"
haftmann@60868
  2085
haftmann@60868
  2086
lemma minus_numeral_div_numeral [simp]:
haftmann@60868
  2087
  "- numeral m div numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  2088
proof -
haftmann@60868
  2089
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  2090
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  2091
  then show ?thesis
haftmann@60868
  2092
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  2093
qed
haftmann@60868
  2094
haftmann@60868
  2095
lemma minus_numeral_mod_numeral [simp]:
haftmann@60868
  2096
  "- numeral m mod numeral n = adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  2097
proof -
haftmann@60868
  2098
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@63950
  2099
    using that by (simp only: snd_divmod modulo_int_def) auto
haftmann@60868
  2100
  then show ?thesis
haftmann@63950
  2101
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
haftmann@60868
  2102
qed
haftmann@60868
  2103
haftmann@60868
  2104
lemma numeral_div_minus_numeral [simp]:
haftmann@60868
  2105
  "numeral m div - numeral n = - (adjust_div (divmod m n) :: int)"
haftmann@60868
  2106
proof -
haftmann@60868
  2107
  have "int (fst (divmod m n)) = fst (divmod m n)"
haftmann@60868
  2108
    by (simp only: fst_divmod divide_int_def) auto
haftmann@60868
  2109
  then show ?thesis
haftmann@60868
  2110
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def divide_int_def)
haftmann@60868
  2111
qed
haftmann@60868
  2112
  
haftmann@60868
  2113
lemma numeral_mod_minus_numeral [simp]:
haftmann@60868
  2114
  "numeral m mod - numeral n = - adjust_mod (numeral n) (snd (divmod m n) :: int)"
haftmann@60868
  2115
proof -
haftmann@60868
  2116
  have "int (snd (divmod m n)) = snd (divmod m n)" if "snd (divmod m n) \<noteq> (0::int)"
haftmann@63950
  2117
    using that by (simp only: snd_divmod modulo_int_def) auto
haftmann@60868
  2118
  then show ?thesis
haftmann@63950
  2119
    by (auto simp add: split_def Let_def adjust_div_def divides_aux_def modulo_int_def)
haftmann@60868
  2120
qed
haftmann@60868
  2121
haftmann@60868
  2122
lemma minus_one_div_numeral [simp]:
haftmann@60868
  2123
  "- 1 div numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  2124
  using minus_numeral_div_numeral [of Num.One n] by simp  
haftmann@60868
  2125
haftmann@60868
  2126
lemma minus_one_mod_numeral [simp]:
haftmann@60868
  2127
  "- 1 mod numeral n = adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  2128
  using minus_numeral_mod_numeral [of Num.One n] by simp
haftmann@60868
  2129
haftmann@60868
  2130
lemma one_div_minus_numeral [simp]:
haftmann@60868
  2131
  "1 div - numeral n = - (adjust_div (divmod Num.One n) :: int)"
haftmann@60868
  2132
  using numeral_div_minus_numeral [of Num.One n] by simp
haftmann@60868
  2133
  
haftmann@60868
  2134
lemma one_mod_minus_numeral [simp]:
haftmann@60868
  2135
  "1 mod - numeral n = - adjust_mod (numeral n) (snd (divmod Num.One n) :: int)"
haftmann@60868
  2136
  using numeral_mod_minus_numeral [of Num.One n] by simp
haftmann@60868
  2137
haftmann@60930
  2138
end
haftmann@60930
  2139
haftmann@60868
  2140
haftmann@60868
  2141
subsubsection \<open>Further properties\<close>
haftmann@60868
  2142
haftmann@60868
  2143
text \<open>Simplify expresions in which div and mod combine numerical constants\<close>
haftmann@60868
  2144
haftmann@60868
  2145
lemma int_div_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@64635
  2146
  by (rule div_int_unique [of a b q r]) (simp add: eucl_rel_int_iff)
haftmann@60868
  2147
haftmann@60868
  2148
lemma int_div_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a div b = q"
haftmann@60868
  2149
  by (rule div_int_unique [of a b q r],
haftmann@64635
  2150
    simp add: eucl_rel_int_iff)
haftmann@60868
  2151
haftmann@60868
  2152
lemma int_mod_pos_eq: "\<lbrakk>(a::int) = b * q + r; 0 \<le> r; r < b\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  2153
  by (rule mod_int_unique [of a b q r],
haftmann@64635
  2154
    simp add: eucl_rel_int_iff)
haftmann@60868
  2155
haftmann@60868
  2156
lemma int_mod_neg_eq: "\<lbrakk>(a::int) = b * q + r; r \<le> 0; b < r\<rbrakk> \<Longrightarrow> a mod b = r"
haftmann@60868
  2157
  by (rule mod_int_unique [of a b q r],
haftmann@64635
  2158
    simp add: eucl_rel_int_iff)
haftmann@33361
  2159
wenzelm@61944
  2160
lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
haftmann@33361
  2161
by (unfold dvd_def, cases "y=0", auto simp add: abs_mult)
haftmann@33361
  2162
wenzelm@60758
  2163
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  2164
lemma int_power_div_base:
haftmann@33361
  2165
     "\<lbrakk>0 < m; 0 < k\<rbrakk> \<Longrightarrow> k ^ m div k = (k::int) ^ (m - Suc 0)"
haftmann@33361
  2166
apply (subgoal_tac "k ^ m = k ^ ((m - Suc 0) + Suc 0)")
haftmann@33361
  2167
 apply (erule ssubst)
haftmann@33361
  2168
 apply (simp only: power_add)
haftmann@33361
  2169
 apply simp_all
haftmann@33361
  2170
done
haftmann@33361
  2171
wenzelm@61799
  2172
text \<open>Distributive laws for function \<open>nat\<close>.\<close>
haftmann@33361
  2173
haftmann@33361
  2174
lemma nat_div_distrib: "0 \<le> x \<Longrightarrow> nat (x div y) = nat x div nat y"
haftmann@33361
  2175
apply (rule linorder_cases [of y 0])
haftmann@33361
  2176
apply (simp add: div_nonneg_neg_le0)
haftmann@33361
  2177
apply simp
haftmann@33361
  2178
apply (simp add: nat_eq_iff pos_imp_zdiv_nonneg_iff zdiv_int)
haftmann@33361
  2179
done
haftmann@33361
  2180
haftmann@33361
  2181
(*Fails if y<0: the LHS collapses to (nat z) but the RHS doesn't*)
haftmann@33361
  2182
lemma nat_mod_distrib:
haftmann@33361
  2183
  "\<lbrakk>0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> nat (x mod y) = nat x mod nat y"
haftmann@33361
  2184
apply (case_tac "y = 0", simp)
haftmann@33361
  2185
apply (simp add: nat_eq_iff zmod_int)
haftmann@33361
  2186
done
haftmann@33361
  2187
wenzelm@60758
  2188
text  \<open>transfer setup\<close>
haftmann@33361
  2189
haftmann@33361
  2190
lemma transfer_nat_int_functions:
haftmann@33361
  2191
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) div (nat y) = nat (x div y)"
haftmann@33361
  2192
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> (nat x) mod (nat y) = nat (x mod y)"
haftmann@33361
  2193
  by (auto simp add: nat_div_distrib nat_mod_distrib)
haftmann@33361
  2194
haftmann@33361
  2195
lemma transfer_nat_int_function_closures:
haftmann@33361
  2196
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x div y >= 0"
haftmann@33361
  2197
    "(x::int) >= 0 \<Longrightarrow> y >= 0 \<Longrightarrow> x mod y >= 0"
haftmann@33361
  2198
  apply (cases "y = 0")
haftmann@33361
  2199
  apply (auto simp add: pos_imp_zdiv_nonneg_iff)
haftmann@33361
  2200
  apply (cases "y = 0")
haftmann@33361
  2201
  apply auto
haftmann@33361
  2202
done
haftmann@33361
  2203
haftmann@35644
  2204
declare transfer_morphism_nat_int [transfer add return:
haftmann@33361
  2205
  transfer_nat_int_functions
haftmann@33361
  2206
  transfer_nat_int_function_closures
haftmann@33361
  2207
]
haftmann@33361
  2208
haftmann@33361
  2209
lemma transfer_int_nat_functions:
haftmann@33361
  2210
    "(int x) div (int y) = int (x div y)"
haftmann@33361
  2211
    "(int x) mod (int y) = int (x mod y)"
haftmann@33361
  2212
  by (auto simp add: zdiv_int zmod_int)
haftmann@33361
  2213
haftmann@33361
  2214
lemma transfer_int_nat_function_closures:
haftmann@33361
  2215
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x div y)"
haftmann@33361
  2216
    "is_nat x \<Longrightarrow> is_nat y \<Longrightarrow> is_nat (x mod y)"
haftmann@33361
  2217
  by (simp_all only: is_nat_def transfer_nat_int_function_closures)
haftmann@33361
  2218
haftmann@35644
  2219
declare transfer_morphism_int_nat [transfer add return:
haftmann@33361
  2220
  transfer_int_nat_functions
haftmann@33361
  2221
  transfer_int_nat_function_closures
haftmann@33361
  2222
]
haftmann@33361
  2223
wenzelm@60758
  2224
text\<open>Suggested by Matthias Daum\<close>
haftmann@33361
  2225
lemma int_div_less_self: "\<lbrakk>0 < x; 1 < k\<rbrakk> \<Longrightarrow> x div k < (x::int)"
haftmann@33361
  2226
apply (subgoal_tac "nat x div nat k < nat x")
nipkow@34225
  2227
 apply (simp add: nat_div_distrib [symmetric])
haftmann@33361
  2228
apply (rule Divides.div_less_dividend, simp_all)
haftmann@33361
  2229
done
haftmann@33361
  2230
haftmann@64593
  2231
lemma nat_mod_eq_lemma: assumes xyn: "(x::nat) mod n = y mod n" and xy:"y \<le> x"
haftmann@33361
  2232
  shows "\<exists>q. x = y + n * q"
haftmann@33361
  2233
proof-
lp15@60562
  2234
  from xy have th: "int x - int y = int (x - y)" by simp
lp15@60562
  2235
  from xyn have "int x mod int n = int y mod int n"
huffman@46551
  2236
    by (simp add: zmod_int [symmetric])
haftmann@64593
  2237
  hence "int n dvd int x - int y" by (simp only: mod_eq_dvd_iff [symmetric])
haftmann@33361
  2238
  hence "n dvd x - y" by (simp add: th zdvd_int)
haftmann@33361
  2239
  then show ?thesis using xy unfolding dvd_def apply clarsimp apply (rule_tac x="k" in exI) by arith
haftmann@33361
  2240
qed
haftmann@33361
  2241
lp15@60562
  2242
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
haftmann@33361
  2243
  (is "?lhs = ?rhs")
haftmann@33361
  2244
proof
haftmann@33361
  2245
  assume H: "x mod n = y mod n"
haftmann@33361
  2246
  {assume xy: "x \<le> y"
haftmann@33361
  2247
    from H have th: "y mod n = x mod n" by simp
lp15@60562
  2248
    from nat_mod_eq_lemma[OF th xy] have ?rhs
haftmann@33361
  2249
      apply clarify  apply (rule_tac x="q" in exI) by (rule exI[where x="0"], simp)}
haftmann@33361
  2250
  moreover
haftmann@33361
  2251
  {assume xy: "y \<le> x"
lp15@60562
  2252
    from nat_mod_eq_lemma[OF H xy] have ?rhs
haftmann@33361
  2253
      apply clarify  apply (rule_tac x="0" in exI) by (rule_tac x="q" in exI, simp)}
lp15@60562
  2254
  ultimately  show ?rhs using linear[of x y] by blast
haftmann@33361
  2255
next
haftmann@33361
  2256
  assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
haftmann@33361
  2257
  hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
haftmann@33361
  2258
  thus  ?lhs by simp
haftmann@33361
  2259
qed
haftmann@33361
  2260
haftmann@60868
  2261
subsubsection \<open>Dedicated simproc for calculation\<close>
haftmann@60868
  2262
wenzelm@60758
  2263
text \<open>
haftmann@60868
  2264
  There is space for improvement here: the calculation itself
haftmann@60868
  2265
  could be carried outside the logic, and a generic simproc
haftmann@60868
  2266
  (simplifier setup) for generic calculation would be helpful. 
wenzelm@60758
  2267
\<close>
haftmann@53067
  2268
haftmann@60868
  2269
simproc_setup numeral_divmod
haftmann@66806
  2270
  ("0 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  2271
   "0 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "0 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  2272
   "0 div - 1 :: int" | "0 mod - 1 :: int" |
haftmann@66806
  2273
   "0 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "0 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  2274
   "0 div - numeral b :: int" | "0 mod - numeral b :: int" |
haftmann@66806
  2275
   "1 div 0 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  2276
   "1 div 1 :: 'a :: unique_euclidean_semiring_numeral" | "1 mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  2277
   "1 div - 1 :: int" | "1 mod - 1 :: int" |
haftmann@66806
  2278
   "1 div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "1 mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  2279
   "1 div - numeral b :: int" |"1 mod - numeral b :: int" |
haftmann@60868
  2280
   "- 1 div 0 :: int" | "- 1 mod 0 :: int" | "- 1 div 1 :: int" | "- 1 mod 1 :: int" |
haftmann@60868
  2281
   "- 1 div - 1 :: int" | "- 1 mod - 1 :: int" | "- 1 div numeral b :: int" | "- 1 mod numeral b :: int" |
haftmann@60868
  2282
   "- 1 div - numeral b :: int" | "- 1 mod - numeral b :: int" |
haftmann@66806
  2283
   "numeral a div 0 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 0 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@66806
  2284
   "numeral a div 1 :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod 1 :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  2285
   "numeral a div - 1 :: int" | "numeral a mod - 1 :: int" |
haftmann@66806
  2286
   "numeral a div numeral b :: 'a :: unique_euclidean_semiring_numeral" | "numeral a mod numeral b :: 'a :: unique_euclidean_semiring_numeral" |
haftmann@60868
  2287
   "numeral a div - numeral b :: int" | "numeral a mod - numeral b :: int" |
haftmann@60868
  2288
   "- numeral a div 0 :: int" | "- numeral a mod 0 :: int" |
haftmann@60868
  2289
   "- numeral a div 1 :: int" | "- numeral a mod 1 :: int" |
haftmann@60868
  2290
   "- numeral a div - 1 :: int" | "- numeral a mod - 1 :: int" |
haftmann@60868
  2291
   "- numeral a div numeral b :: int" | "- numeral a mod numeral b :: int" |
haftmann@60868
  2292
   "- numeral a div - numeral b :: int" | "- numeral a mod - numeral b :: int") =
haftmann@60868
  2293
\<open> let
haftmann@60868
  2294
    val if_cong = the (Code.get_case_cong @{theory} @{const_name If});
haftmann@60868
  2295
    fun successful_rewrite ctxt ct =
haftmann@60868
  2296
      let
haftmann@60868
  2297
        val thm = Simplifier.rewrite ctxt ct
haftmann@60868
  2298
      in if Thm.is_reflexive thm then NONE else SOME thm end;
haftmann@60868
  2299
  in fn phi =>
haftmann@60868
  2300
    let
haftmann@60868
  2301
      val simps = Morphism.fact phi (@{thms div_0 mod_0 div_by_0 mod_by_0 div_by_1 mod_by_1
haftmann@60868
  2302
        one_div_numeral one_mod_numeral minus_one_div_numeral minus_one_mod_numeral
haftmann@60868
  2303
        one_div_minus_numeral one_mod_minus_numeral
haftmann@60868
  2304
        numeral_div_numeral numeral_mod_numeral minus_numeral_div_numeral minus_numeral_mod_numeral
haftmann@60868
  2305
        numeral_div_minus_numeral numeral_mod_minus_numeral
haftmann@60930
  2306
        div_minus_minus mod_minus_minus Divides.adjust_div_eq of_bool_eq one_neq_zero
haftmann@60868
  2307
        numeral_neq_zero neg_equal_0_iff_equal arith_simps arith_special divmod_trivial
haftmann@60868
  2308
        divmod_cancel divmod_steps divmod_step_eq fst_conv snd_conv numeral_One
haftmann@60930
  2309
        case_prod_beta rel_simps Divides.adjust_mod_def div_minus1_right mod_minus1_right
haftmann@60868
  2310
        minus_minus numeral_times_numeral mult_zero_right mult_1_right}
haftmann@60868
  2311
        @ [@{lemma "0 = 0 \<longleftrightarrow> True" by simp}]);
haftmann@60868
  2312
      fun prepare_simpset ctxt = HOL_ss |> Simplifier.simpset_map ctxt
haftmann@60868
  2313
        (Simplifier.add_cong if_cong #> fold Simplifier.add_simp simps)
haftmann@60868
  2314
    in fn ctxt => successful_rewrite (Simplifier.put_simpset (prepare_simpset ctxt) ctxt) end
haftmann@60868
  2315
  end;
haftmann@60868
  2316
\<close>
blanchet@34126
  2317
haftmann@35673
  2318
wenzelm@60758
  2319
subsubsection \<open>Code generation\<close>
haftmann@33361
  2320
haftmann@60868
  2321
lemma [code]:
haftmann@60868
  2322
  fixes k :: int
haftmann@60868
  2323
  shows 
haftmann@60868
  2324
    "k div 0 = 0"
haftmann@60868
  2325
    "k mod 0 = k"
haftmann@60868
  2326
    "0 div k = 0"
haftmann@60868
  2327
    "0 mod k = 0"
haftmann@60868
  2328
    "k div Int.Pos Num.One = k"
haftmann@60868
  2329
    "k mod Int.Pos Num.One = 0"
haftmann@60868
  2330
    "k div Int.Neg Num.One = - k"
haftmann@60868
  2331
    "k mod Int.Neg Num.One = 0"
haftmann@60868
  2332
    "Int.Pos m div Int.Pos n = (fst (divmod m n) :: int)"
haftmann@60868
  2333
    "Int.Pos m mod Int.Pos n = (snd (divmod m n) :: int)"
haftmann@60930
  2334
    "Int.Neg m div Int.Pos n = - (Divides.adjust_div (divmod m n) :: int)"
haftmann@60930
  2335
    "Int.Neg m mod Int.Pos n = Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
haftmann@60930
  2336
    "Int.Pos m div Int.Neg n = - (Divides.adjust_div (divmod m n) :: int)"
haftmann@60930
  2337
    "Int.Pos m mod Int.Neg n = - Divides.adjust_mod (Int.Pos n) (snd (divmod m n) :: int)"
haftmann@60868
  2338
    "Int.Neg m div Int.Neg n = (fst (divmod m n) :: int)"
haftmann@60868
  2339
    "Int.Neg m mod Int.Neg n = - (snd (divmod m n) :: int)"
haftmann@60868
  2340
  by simp_all
haftmann@53069
  2341
haftmann@52435
  2342
code_identifier
haftmann@52435
  2343
  code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  2344
haftmann@60868
  2345
lemma dvd_eq_mod_eq_0_numeral:
haftmann@66806
  2346
  "numeral x dvd (numeral y :: 'a) \<longleftrightarrow> numeral y mod numeral x = (0 :: 'a::semidom_modulo)"
haftmann@60868
  2347
  by (fact dvd_eq_mod_eq_0)
haftmann@60868
  2348
haftmann@64246
  2349
declare minus_div_mult_eq_mod [symmetric, nitpick_unfold]
haftmann@64246
  2350
haftmann@33361
  2351
end