(* Title: HOL/Divides.thy
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1999 University of Cambridge
*)
section \<open>More on quotient and remainder\<close>
theory Divides
imports Parity
begin
subsection \<open>More on division\<close>
subsubsection \<open>Monotonicity in the First Argument (Dividend)\<close>
lemma unique_quotient_lemma:
assumes "b * q' + r' \<le> b * q + r" "0 \<le> r'" "r' < b" "r < b" shows "q' \<le> (q::int)"
proof -
have "r' + b * (q'-q) \<le> r"
using assms by (simp add: right_diff_distrib)
moreover have "0 < b * (1 + q - q') "
using assms by (simp add: right_diff_distrib distrib_left)
moreover have "b * q' < b * (1 + q)"
using assms by (simp add: right_diff_distrib distrib_left)
ultimately show ?thesis
using assms by (simp add: mult_less_cancel_left)
qed
lemma unique_quotient_lemma_neg:
"b * q' + r' \<le> b*q + r \<Longrightarrow> r \<le> 0 \<Longrightarrow> b < r \<Longrightarrow> b < r' \<Longrightarrow> q \<le> (q'::int)"
using unique_quotient_lemma[where b = "-b" and r = "-r'" and r'="-r"] by auto
lemma zdiv_mono1:
\<open>a div b \<le> a' div b\<close>
if \<open>a \<le> a'\<close> \<open>0 < b\<close>
for a b b' :: int
proof (rule unique_quotient_lemma)
show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
using \<open>a \<le> a'\<close> by auto
qed (use that in auto)
lemma zdiv_mono1_neg:
fixes b::int
assumes "a \<le> a'" "b < 0" shows "a' div b \<le> a div b"
proof (rule unique_quotient_lemma_neg)
show "b * (a div b) + a mod b \<le> b * (a' div b) + a' mod b"
using assms(1) by auto
qed (use assms in auto)
subsubsection \<open>Monotonicity in the Second Argument (Divisor)\<close>
lemma q_pos_lemma:
fixes q'::int
assumes "0 \<le> b'*q' + r'" "r' < b'" "0 < b'"
shows "0 \<le> q'"
proof -
have "0 < b'* (q' + 1)"
using assms by (simp add: distrib_left)
with assms show ?thesis
by (simp add: zero_less_mult_iff)
qed
lemma zdiv_mono2_lemma:
fixes q'::int
assumes eq: "b*q + r = b'*q' + r'" and le: "0 \<le> b'*q' + r'" and "r' < b'" "0 \<le> r" "0 < b'" "b' \<le> b"
shows "q \<le> q'"
proof -
have "0 \<le> q'"
using q_pos_lemma le \<open>r' < b'\<close> \<open>0 < b'\<close> by blast
moreover have "b*q = r' - r + b'*q'"
using eq by linarith
ultimately have "b*q < b* (q' + 1)"
using mult_right_mono assms unfolding distrib_left by fastforce
with assms show ?thesis
by (simp add: mult_less_cancel_left_pos)
qed
lemma zdiv_mono2:
fixes a::int
assumes "0 \<le> a" "0 < b'" "b' \<le> b" shows "a div b \<le> a div b'"
proof (rule zdiv_mono2_lemma)
have "b \<noteq> 0"
using assms by linarith
show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
by simp
qed (use assms in auto)
lemma zdiv_mono2_neg_lemma:
fixes q'::int
assumes "b*q + r = b'*q' + r'" "b'*q' + r' < 0" "r < b" "0 \<le> r'" "0 < b'" "b' \<le> b"
shows "q' \<le> q"
proof -
have "b'*q' < 0"
using assms by linarith
with assms have "q' \<le> 0"
by (simp add: mult_less_0_iff)
have "b*q' \<le> b'*q'"
by (simp add: \<open>q' \<le> 0\<close> assms(6) mult_right_mono_neg)
then have "b*q' < b* (q + 1)"
using assms by (simp add: distrib_left)
then show ?thesis
using assms by (simp add: mult_less_cancel_left)
qed
lemma zdiv_mono2_neg:
fixes a::int
assumes "a < 0" "0 < b'" "b' \<le> b" shows "a div b' \<le> a div b"
proof (rule zdiv_mono2_neg_lemma)
have "b \<noteq> 0"
using assms by linarith
show "b * (a div b) + a mod b = b' * (a div b') + a mod b'"
by simp
qed (use assms in auto)
subsubsection \<open>Quotients of Signs\<close>
lemma div_eq_minus1: "0 < b \<Longrightarrow> - 1 div b = - 1" for b :: int
by (simp add: divide_int_def)
lemma zmod_minus1: "0 < b \<Longrightarrow> - 1 mod b = b - 1" for b :: int
by (auto simp add: modulo_int_def)
lemma minus_mod_int_eq:
\<open>- k mod l = l - 1 - (k - 1) mod l\<close> if \<open>l \<ge> 0\<close> for k l :: int
proof (cases \<open>l = 0\<close>)
case True
then show ?thesis
by simp
next
case False
with that have \<open>l > 0\<close>
by simp
then show ?thesis
proof (cases \<open>l dvd k\<close>)
case True
then obtain j where \<open>k = l * j\<close> ..
moreover have \<open>(l * j mod l - 1) mod l = l - 1\<close>
using \<open>l > 0\<close> by (simp add: zmod_minus1)
then have \<open>(l * j - 1) mod l = l - 1\<close>
by (simp only: mod_simps)
ultimately show ?thesis
by simp
next
case False
moreover have 1: \<open>0 < k mod l\<close>
using \<open>0 < l\<close> False le_less by fastforce
moreover have 2: \<open>k mod l < 1 + l\<close>
using \<open>0 < l\<close> pos_mod_bound[of l k] by linarith
from 1 2 \<open>l > 0\<close> have \<open>(k mod l - 1) mod l = k mod l - 1\<close>
by (simp add: zmod_trivial_iff)
ultimately show ?thesis
by (simp only: zmod_zminus1_eq_if)
(simp add: mod_eq_0_iff_dvd algebra_simps mod_simps)
qed
qed
lemma div_neg_pos_less0:
fixes a::int
assumes "a < 0" "0 < b"
shows "a div b < 0"
proof -
have "a div b \<le> - 1 div b"
using zdiv_mono1 assms by auto
also have "... \<le> -1"
by (simp add: assms(2) div_eq_minus1)
finally show ?thesis
by force
qed
lemma div_nonneg_neg_le0: "[| (0::int) \<le> a; b < 0 |] ==> a div b \<le> 0"
by (drule zdiv_mono1_neg, auto)
lemma div_nonpos_pos_le0: "[| (a::int) \<le> 0; b > 0 |] ==> a div b \<le> 0"
by (drule zdiv_mono1, auto)
text\<open>Now for some equivalences of the form \<open>a div b >=< 0 \<longleftrightarrow> \<dots>\<close>
conditional upon the sign of \<open>a\<close> or \<open>b\<close>. There are many more.
They should all be simp rules unless that causes too much search.\<close>
lemma pos_imp_zdiv_nonneg_iff:
fixes a::int
assumes "0 < b"
shows "(0 \<le> a div b) = (0 \<le> a)"
proof
show "0 \<le> a div b \<Longrightarrow> 0 \<le> a"
using assms
by (simp add: linorder_not_less [symmetric]) (blast intro: div_neg_pos_less0)
next
assume "0 \<le> a"
then have "0 div b \<le> a div b"
using zdiv_mono1 assms by blast
then show "0 \<le> a div b"
by auto
qed
lemma pos_imp_zdiv_pos_iff:
"0<k \<Longrightarrow> 0 < (i::int) div k \<longleftrightarrow> k \<le> i"
using pos_imp_zdiv_nonneg_iff[of k i] zdiv_eq_0_iff[of i k] by arith
lemma neg_imp_zdiv_nonneg_iff:
fixes a::int
assumes "b < 0"
shows "(0 \<le> a div b) = (a \<le> 0)"
using assms by (simp add: div_minus_minus [of a, symmetric] pos_imp_zdiv_nonneg_iff del: div_minus_minus)
(*But not (a div b \<le> 0 iff a\<le>0); consider a=1, b=2 when a div b = 0.*)
lemma pos_imp_zdiv_neg_iff: "(0::int) < b ==> (a div b < 0) = (a < 0)"
by (simp add: linorder_not_le [symmetric] pos_imp_zdiv_nonneg_iff)
(*Again the law fails for \<le>: consider a = -1, b = -2 when a div b = 0*)
lemma neg_imp_zdiv_neg_iff: "b < (0::int) ==> (a div b < 0) = (0 < a)"
by (simp add: linorder_not_le [symmetric] neg_imp_zdiv_nonneg_iff)
lemma nonneg1_imp_zdiv_pos_iff:
fixes a::int
assumes "0 \<le> a"
shows "a div b > 0 \<longleftrightarrow> a \<ge> b \<and> b>0"
proof -
have "0 < a div b \<Longrightarrow> b \<le> a"
using div_pos_pos_trivial[of a b] assms by arith
moreover have "0 < a div b \<Longrightarrow> b > 0"
using assms div_nonneg_neg_le0[of a b] by(cases "b=0"; force)
moreover have "b \<le> a \<and> 0 < b \<Longrightarrow> 0 < a div b"
using int_one_le_iff_zero_less[of "a div b"] zdiv_mono1[of b a b] by simp
ultimately show ?thesis
by blast
qed
lemma zmod_le_nonneg_dividend: "(m::int) \<ge> 0 \<Longrightarrow> m mod k \<le> m"
by (rule split_zmod[THEN iffD2]) (fastforce dest: q_pos_lemma intro: split_mult_pos_le)
lemma sgn_div_eq_sgn_mult:
\<open>sgn (k div l) = of_bool (k div l \<noteq> 0) * sgn (k * l)\<close>
for k l :: int
proof (cases \<open>k div l = 0\<close>)
case True
then show ?thesis
by simp
next
case False
have \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close>
by (cases \<open>l = 0\<close>) (simp_all add: pos_imp_zdiv_nonneg_iff)
then have \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> 0 < \<bar>k\<bar> div \<bar>l\<bar>\<close>
by (simp add: less_le)
also have \<open>\<dots> \<longleftrightarrow> \<bar>k\<bar> \<ge> \<bar>l\<bar>\<close>
using False nonneg1_imp_zdiv_pos_iff by auto
finally have *: \<open>\<bar>k\<bar> div \<bar>l\<bar> \<noteq> 0 \<longleftrightarrow> \<bar>l\<bar> \<le> \<bar>k\<bar>\<close> .
show ?thesis
using \<open>0 \<le> \<bar>k\<bar> div \<bar>l\<bar>\<close> False
by (auto simp add: div_eq_div_abs [of k l] div_eq_sgn_abs [of k l]
sgn_mult sgn_1_pos sgn_1_neg sgn_eq_0_iff nonneg1_imp_zdiv_pos_iff * dest: sgn_not_eq_imp)
qed
lemma
fixes a b q r :: int
assumes \<open>a = b * q + r\<close> \<open>0 \<le> r\<close> \<open>r < b\<close>
shows int_div_pos_eq:
\<open>a div b = q\<close> (is ?Q)
and int_mod_pos_eq:
\<open>a mod b = r\<close> (is ?R)
proof -
from assms have \<open>(a div b, a mod b) = (q, r)\<close>
by (cases b q r a rule: euclidean_relation_intI)
(auto simp add: ac_simps dvd_add_left_iff sgn_1_pos le_less dest: zdvd_imp_le)
then show ?Q and ?R
by simp_all
qed
lemma int_div_neg_eq:
\<open>a div b = q\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
using that int_div_pos_eq [of a \<open>- b\<close> \<open>- q\<close> \<open>- r\<close>] by simp_all
lemma int_mod_neg_eq:
\<open>a mod b = r\<close> if \<open>a = b * q + r\<close> \<open>r \<le> 0\<close> \<open>b < r\<close> for a b q r :: int
using that int_div_neg_eq [of a b q r] by simp
subsubsection \<open>Further properties\<close>
lemma div_int_pos_iff:
"k div l \<ge> 0 \<longleftrightarrow> k = 0 \<or> l = 0 \<or> k \<ge> 0 \<and> l \<ge> 0
\<or> k < 0 \<and> l < 0"
for k l :: int
proof (cases "k = 0 \<or> l = 0")
case False
then have *: "k \<noteq> 0" "l \<noteq> 0"
by auto
then have "0 \<le> k div l \<Longrightarrow> \<not> k < 0 \<Longrightarrow> 0 \<le> l"
by (meson neg_imp_zdiv_neg_iff not_le not_less_iff_gr_or_eq)
then show ?thesis
using * by (auto simp add: pos_imp_zdiv_nonneg_iff neg_imp_zdiv_nonneg_iff)
qed auto
lemma mod_int_pos_iff:
"k mod l \<ge> 0 \<longleftrightarrow> l dvd k \<or> l = 0 \<and> k \<ge> 0 \<or> l > 0"
for k l :: int
proof (cases "l > 0")
case False
then show ?thesis
by (simp add: dvd_eq_mod_eq_0) (use neg_mod_sign [of l k] in \<open>auto simp add: le_less not_less\<close>)
qed auto
text \<open>Simplify expressions in which div and mod combine numerical constants\<close>
lemma abs_div: "(y::int) dvd x \<Longrightarrow> \<bar>x div y\<bar> = \<bar>x\<bar> div \<bar>y\<bar>"
unfolding dvd_def by (cases "y=0") (auto simp add: abs_mult)
text\<open>Suggested by Matthias Daum\<close>
lemma int_power_div_base:
fixes k :: int
assumes "0 < m" "0 < k"
shows "k ^ m div k = (k::int) ^ (m - Suc 0)"
proof -
have eq: "k ^ m = k ^ ((m - Suc 0) + Suc 0)"
by (simp add: assms)
show ?thesis
using assms by (simp only: power_add eq) auto
qed
text\<open>Suggested by Matthias Daum\<close>
lemma int_div_less_self:
fixes x::int
assumes "0 < x" "1 < k"
shows "x div k < x"
proof -
have "nat x div nat k < nat x"
by (simp add: assms)
with assms show ?thesis
by (simp add: nat_div_distrib [symmetric])
qed
lemma mod_eq_dvd_iff_nat:
"m mod q = n mod q \<longleftrightarrow> q dvd m - n" if "m \<ge> n" for m n q :: nat
proof -
have "int m mod int q = int n mod int q \<longleftrightarrow> int q dvd int m - int n"
by (simp add: mod_eq_dvd_iff)
with that have "int (m mod q) = int (n mod q) \<longleftrightarrow> int q dvd int (m - n)"
by (simp only: of_nat_mod of_nat_diff)
then show ?thesis
by simp
qed
lemma mod_eq_nat1E:
fixes m n q :: nat
assumes "m mod q = n mod q" and "m \<ge> n"
obtains s where "m = n + q * s"
proof -
from assms have "q dvd m - n"
by (simp add: mod_eq_dvd_iff_nat)
then obtain s where "m - n = q * s" ..
with \<open>m \<ge> n\<close> have "m = n + q * s"
by simp
with that show thesis .
qed
lemma mod_eq_nat2E:
fixes m n q :: nat
assumes "m mod q = n mod q" and "n \<ge> m"
obtains s where "n = m + q * s"
using assms mod_eq_nat1E [of n q m] by (auto simp add: ac_simps)
lemma nat_mod_eq_lemma:
assumes "(x::nat) mod n = y mod n" and "y \<le> x"
shows "\<exists>q. x = y + n * q"
using assms by (rule mod_eq_nat1E) (rule exI)
lemma nat_mod_eq_iff: "(x::nat) mod n = y mod n \<longleftrightarrow> (\<exists>q1 q2. x + n * q1 = y + n * q2)"
(is "?lhs = ?rhs")
proof
assume H: "x mod n = y mod n"
{assume xy: "x \<le> y"
from H have th: "y mod n = x mod n" by simp
from nat_mod_eq_lemma[OF th xy] have ?rhs
proof
fix q
assume "y = x + n * q"
then have "x + n * q = y + n * 0"
by simp
then show "\<exists>q1 q2. x + n * q1 = y + n * q2"
by blast
qed}
moreover
{assume xy: "y \<le> x"
from nat_mod_eq_lemma[OF H xy] have ?rhs
proof
fix q
assume "x = y + n * q"
then have "x + n * 0 = y + n * q"
by simp
then show "\<exists>q1 q2. x + n * q1 = y + n * q2"
by blast
qed}
ultimately show ?rhs using linear[of x y] by blast
next
assume ?rhs then obtain q1 q2 where q12: "x + n * q1 = y + n * q2" by blast
hence "(x + n * q1) mod n = (y + n * q2) mod n" by simp
thus ?lhs by simp
qed
subsubsection \<open>Computing \<open>div\<close> and \<open>mod\<close> with shifting\<close>
lemma div_pos_geq:
fixes k l :: int
assumes "0 < l" and "l \<le> k"
shows "k div l = (k - l) div l + 1"
proof -
have "k = (k - l) + l" by simp
then obtain j where k: "k = j + l" ..
with assms show ?thesis by (simp add: div_add_self2)
qed
lemma mod_pos_geq:
fixes k l :: int
assumes "0 < l" and "l \<le> k"
shows "k mod l = (k - l) mod l"
proof -
have "k = (k - l) + l" by simp
then obtain j where k: "k = j + l" ..
with assms show ?thesis by simp
qed
text\<open>computing div by shifting\<close>
lemma pos_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = b div a\<close> (is ?Q)
and pos_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 1 + 2 * (b mod a)\<close> (is ?R)
if \<open>0 \<le> a\<close> for a b :: int
proof -
have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = (b div a, 1 + 2 * (b mod a))\<close>
proof (cases \<open>2 * a\<close> \<open>b div a\<close> \<open>1 + 2 * (b mod a)\<close> \<open>1 + 2 * b\<close> rule: euclidean_relation_intI)
case by0
then show ?case
by simp
next
case divides
have \<open>even (2 * a)\<close>
by simp
then have \<open>even (1 + 2 * b)\<close>
using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
then show ?case
by simp
next
case euclidean_relation
with that have \<open>a > 0\<close>
by simp
moreover have \<open>b mod a < a\<close>
using \<open>a > 0\<close> by simp
then have \<open>1 + 2 * (b mod a) < 2 * a\<close>
by simp
moreover have \<open>2 * (b mod a) + a * (2 * (b div a)) = 2 * (b div a * a + b mod a)\<close>
by (simp only: algebra_simps)
moreover have \<open>0 \<le> 2 * (b mod a)\<close>
using \<open>a > 0\<close> by simp
ultimately show ?case
by (simp add: algebra_simps)
qed
then show ?Q and ?R
by simp_all
qed
lemma neg_zdiv_mult_2: \<open>(1 + 2 * b) div (2 * a) = (b + 1) div a\<close> (is ?Q)
and neg_zmod_mult_2: \<open>(1 + 2 * b) mod (2 * a) = 2 * ((b + 1) mod a) - 1\<close> (is ?R)
if \<open>a \<le> 0\<close> for a b :: int
proof -
have \<open>((1 + 2 * b) div (2 * a), (1 + 2 * b) mod (2 * a)) = ((b + 1) div a, 2 * ((b + 1) mod a) - 1)\<close>
proof (cases \<open>2 * a\<close> \<open>(b + 1) div a\<close> \<open>2 * ((b + 1) mod a) - 1\<close> \<open>1 + 2 * b\<close> rule: euclidean_relation_intI)
case by0
then show ?case
by simp
next
case divides
have \<open>even (2 * a)\<close>
by simp
then have \<open>even (1 + 2 * b)\<close>
using \<open>2 * a dvd 1 + 2 * b\<close> by (rule dvd_trans)
then show ?case
by simp
next
case euclidean_relation
with that have \<open>a < 0\<close>
by simp
moreover have \<open>(b + 1) mod a > a\<close>
using \<open>a < 0\<close> by simp
then have \<open>2 * ((b + 1) mod a) > 1 + 2 * a\<close>
by simp
moreover have \<open>((1 + b) mod a) \<le> 0\<close>
using \<open>a < 0\<close> by simp
then have \<open>2 * ((1 + b) mod a) \<le> 0\<close>
by simp
moreover have \<open>2 * ((1 + b) mod a) + a * (2 * ((1 + b) div a)) =
2 * ((1 + b) div a * a + (1 + b) mod a)\<close>
by (simp only: algebra_simps)
ultimately show ?case
by (simp add: algebra_simps sgn_mult abs_mult)
qed
then show ?Q and ?R
by simp_all
qed
lemma zdiv_numeral_Bit0 [simp]:
"numeral (Num.Bit0 v) div numeral (Num.Bit0 w) =
numeral v div (numeral w :: int)"
unfolding numeral.simps unfolding mult_2 [symmetric]
by (rule div_mult_mult1, simp)
lemma zdiv_numeral_Bit1 [simp]:
"numeral (Num.Bit1 v) div numeral (Num.Bit0 w) =
(numeral v div (numeral w :: int))"
unfolding numeral.simps
unfolding mult_2 [symmetric] add.commute [of _ 1]
by (rule pos_zdiv_mult_2, simp)
lemma zmod_numeral_Bit0 [simp]:
"numeral (Num.Bit0 v) mod numeral (Num.Bit0 w) =
(2::int) * (numeral v mod numeral w)"
unfolding numeral_Bit0 [of v] numeral_Bit0 [of w]
unfolding mult_2 [symmetric] by (rule mod_mult_mult1)
lemma zmod_numeral_Bit1 [simp]:
"numeral (Num.Bit1 v) mod numeral (Num.Bit0 w) =
2 * (numeral v mod numeral w) + (1::int)"
unfolding numeral_Bit1 [of v] numeral_Bit0 [of w]
unfolding mult_2 [symmetric] add.commute [of _ 1]
by (rule pos_zmod_mult_2, simp)
code_identifier
code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
subsection \<open>Lemmas of doubtful value\<close>
class unique_euclidean_semiring_numeral = unique_euclidean_semiring_with_nat + linordered_semidom +
assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
begin
lemma divmod_digit_1:
assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
and "a mod (2 * b) - b = a mod b" (is "?Q")
proof -
from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
by (auto intro: trans)
with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
then have [simp]: "1 \<le> a div b" by (simp add: discrete)
with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
define w where "w = a div b mod 2"
then have w_exhaust: "w = 0 \<or> w = 1" by auto
have mod_w: "a mod (2 * b) = a mod b + b * w"
by (simp add: w_def mod_mult2_eq ac_simps)
from assms w_exhaust have "w = 1"
using mod_less by (auto simp add: mod_w)
with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
have "2 * (a div (2 * b)) = a div b - w"
by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
then show ?P and ?Q
by (simp_all add: div mod add_implies_diff [symmetric])
qed
lemma divmod_digit_0:
assumes "0 < b" and "a mod (2 * b) < b"
shows "2 * (a div (2 * b)) = a div b" (is "?P")
and "a mod (2 * b) = a mod b" (is "?Q")
proof -
define w where "w = a div b mod 2"
then have w_exhaust: "w = 0 \<or> w = 1" by auto
have mod_w: "a mod (2 * b) = a mod b + b * w"
by (simp add: w_def mod_mult2_eq ac_simps)
moreover have "b \<le> a mod b + b"
proof -
from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
then show ?thesis by simp
qed
moreover note assms w_exhaust
ultimately have "w = 0" by auto
with mod_w have mod: "a mod (2 * b) = a mod b" by simp
have "2 * (a div (2 * b)) = a div b - w"
by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
then show ?P and ?Q
by (simp_all add: div mod)
qed
lemma mod_double_modulus:
assumes "m > 0" "x \<ge> 0"
shows "x mod (2 * m) = x mod m \<or> x mod (2 * m) = x mod m + m"
proof (cases "x mod (2 * m) < m")
case True
thus ?thesis using assms using divmod_digit_0(2)[of m x] by auto
next
case False
hence *: "x mod (2 * m) - m = x mod m"
using assms by (intro divmod_digit_1) auto
hence "x mod (2 * m) = x mod m + m"
by (subst * [symmetric], subst le_add_diff_inverse2) (use False in auto)
thus ?thesis by simp
qed
end
hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
instance nat :: unique_euclidean_semiring_numeral
by standard
(auto simp add: div_greater_zero_iff div_mult2_eq mod_mult2_eq)
instance int :: unique_euclidean_semiring_numeral
by standard (auto intro: zmod_le_nonneg_dividend simp add:
pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
lemma div_geq: "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
lemma mod_geq: "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
lemma mod_eq_0D: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: nat
using that by (auto simp add: mod_eq_0_iff_dvd)
lemma pos_mod_conj: "0 < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b" for a b :: int
by simp
lemma neg_mod_conj: "b < 0 \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b" for a b :: int
by simp
lemma zmod_eq_0_iff: "m mod d = 0 \<longleftrightarrow> (\<exists>q. m = d * q)" for m d :: int
by (auto simp add: mod_eq_0_iff_dvd)
(* REVISIT: should this be generalized to all semiring_div types? *)
lemma zmod_eq_0D [dest!]: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
using that by auto
lemma div_positive_int:
"k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
using that by (simp add: nonneg1_imp_zdiv_pos_iff)
end