(* Title: HOL/Euclidean_Division.thy
Author: Manuel Eberl, TU Muenchen
Author: Florian Haftmann, TU Muenchen
*)
section \<open>Uniquely determined division in euclidean (semi)rings\<close>
theory Euclidean_Division
imports Nat_Transfer
begin
subsection \<open>Quotient and remainder in integral domains\<close>
class semidom_modulo = algebraic_semidom + semiring_modulo
begin
lemma mod_0 [simp]: "0 mod a = 0"
using div_mult_mod_eq [of 0 a] by simp
lemma mod_by_0 [simp]: "a mod 0 = a"
using div_mult_mod_eq [of a 0] by simp
lemma mod_by_1 [simp]:
"a mod 1 = 0"
proof -
from div_mult_mod_eq [of a one] div_by_1 have "a + a mod 1 = a" by simp
then have "a + a mod 1 = a + 0" by simp
then show ?thesis by (rule add_left_imp_eq)
qed
lemma mod_self [simp]:
"a mod a = 0"
using div_mult_mod_eq [of a a] by simp
lemma dvd_imp_mod_0 [simp]:
assumes "a dvd b"
shows "b mod a = 0"
using assms minus_div_mult_eq_mod [of b a] by simp
lemma mod_0_imp_dvd:
assumes "a mod b = 0"
shows "b dvd a"
proof -
have "b dvd ((a div b) * b)" by simp
also have "(a div b) * b = a"
using div_mult_mod_eq [of a b] by (simp add: assms)
finally show ?thesis .
qed
lemma mod_eq_0_iff_dvd:
"a mod b = 0 \<longleftrightarrow> b dvd a"
by (auto intro: mod_0_imp_dvd)
lemma dvd_eq_mod_eq_0 [nitpick_unfold, code]:
"a dvd b \<longleftrightarrow> b mod a = 0"
by (simp add: mod_eq_0_iff_dvd)
lemma dvd_mod_iff:
assumes "c dvd b"
shows "c dvd a mod b \<longleftrightarrow> c dvd a"
proof -
from assms have "(c dvd a mod b) \<longleftrightarrow> (c dvd ((a div b) * b + a mod b))"
by (simp add: dvd_add_right_iff)
also have "(a div b) * b + a mod b = a"
using div_mult_mod_eq [of a b] by simp
finally show ?thesis .
qed
lemma dvd_mod_imp_dvd:
assumes "c dvd a mod b" and "c dvd b"
shows "c dvd a"
using assms dvd_mod_iff [of c b a] by simp
end
class idom_modulo = idom + semidom_modulo
begin
subclass idom_divide ..
lemma div_diff [simp]:
"c dvd a \<Longrightarrow> c dvd b \<Longrightarrow> (a - b) div c = a div c - b div c"
using div_add [of _ _ "- b"] by (simp add: dvd_neg_div)
end
subsection \<open>Euclidean (semi)rings with explicit division and remainder\<close>
class euclidean_semiring = semidom_modulo + normalization_semidom +
fixes euclidean_size :: "'a \<Rightarrow> nat"
assumes size_0 [simp]: "euclidean_size 0 = 0"
assumes mod_size_less:
"b \<noteq> 0 \<Longrightarrow> euclidean_size (a mod b) < euclidean_size b"
assumes size_mult_mono:
"b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (a * b)"
begin
lemma size_mult_mono': "b \<noteq> 0 \<Longrightarrow> euclidean_size a \<le> euclidean_size (b * a)"
by (subst mult.commute) (rule size_mult_mono)
lemma euclidean_size_normalize [simp]:
"euclidean_size (normalize a) = euclidean_size a"
proof (cases "a = 0")
case True
then show ?thesis
by simp
next
case [simp]: False
have "euclidean_size (normalize a) \<le> euclidean_size (normalize a * unit_factor a)"
by (rule size_mult_mono) simp
moreover have "euclidean_size a \<le> euclidean_size (a * (1 div unit_factor a))"
by (rule size_mult_mono) simp
ultimately show ?thesis
by simp
qed
lemma dvd_euclidean_size_eq_imp_dvd:
assumes "a \<noteq> 0" and "euclidean_size a = euclidean_size b"
and "b dvd a"
shows "a dvd b"
proof (rule ccontr)
assume "\<not> a dvd b"
hence "b mod a \<noteq> 0" using mod_0_imp_dvd [of b a] by blast
then have "b mod a \<noteq> 0" by (simp add: mod_eq_0_iff_dvd)
from \<open>b dvd a\<close> have "b dvd b mod a" by (simp add: dvd_mod_iff)
then obtain c where "b mod a = b * c" unfolding dvd_def by blast
with \<open>b mod a \<noteq> 0\<close> have "c \<noteq> 0" by auto
with \<open>b mod a = b * c\<close> have "euclidean_size (b mod a) \<ge> euclidean_size b"
using size_mult_mono by force
moreover from \<open>\<not> a dvd b\<close> and \<open>a \<noteq> 0\<close>
have "euclidean_size (b mod a) < euclidean_size a"
using mod_size_less by blast
ultimately show False using \<open>euclidean_size a = euclidean_size b\<close>
by simp
qed
lemma euclidean_size_times_unit:
assumes "is_unit a"
shows "euclidean_size (a * b) = euclidean_size b"
proof (rule antisym)
from assms have [simp]: "a \<noteq> 0" by auto
thus "euclidean_size (a * b) \<ge> euclidean_size b" by (rule size_mult_mono')
from assms have "is_unit (1 div a)" by simp
hence "1 div a \<noteq> 0" by (intro notI) simp_all
hence "euclidean_size (a * b) \<le> euclidean_size ((1 div a) * (a * b))"
by (rule size_mult_mono')
also from assms have "(1 div a) * (a * b) = b"
by (simp add: algebra_simps unit_div_mult_swap)
finally show "euclidean_size (a * b) \<le> euclidean_size b" .
qed
lemma euclidean_size_unit:
"is_unit a \<Longrightarrow> euclidean_size a = euclidean_size 1"
using euclidean_size_times_unit [of a 1] by simp
lemma unit_iff_euclidean_size:
"is_unit a \<longleftrightarrow> euclidean_size a = euclidean_size 1 \<and> a \<noteq> 0"
proof safe
assume A: "a \<noteq> 0" and B: "euclidean_size a = euclidean_size 1"
show "is_unit a"
by (rule dvd_euclidean_size_eq_imp_dvd [OF A B]) simp_all
qed (auto intro: euclidean_size_unit)
lemma euclidean_size_times_nonunit:
assumes "a \<noteq> 0" "b \<noteq> 0" "\<not> is_unit a"
shows "euclidean_size b < euclidean_size (a * b)"
proof (rule ccontr)
assume "\<not>euclidean_size b < euclidean_size (a * b)"
with size_mult_mono'[OF assms(1), of b]
have eq: "euclidean_size (a * b) = euclidean_size b" by simp
have "a * b dvd b"
by (rule dvd_euclidean_size_eq_imp_dvd [OF _ eq]) (insert assms, simp_all)
hence "a * b dvd 1 * b" by simp
with \<open>b \<noteq> 0\<close> have "is_unit a" by (subst (asm) dvd_times_right_cancel_iff)
with assms(3) show False by contradiction
qed
lemma dvd_imp_size_le:
assumes "a dvd b" "b \<noteq> 0"
shows "euclidean_size a \<le> euclidean_size b"
using assms by (auto elim!: dvdE simp: size_mult_mono)
lemma dvd_proper_imp_size_less:
assumes "a dvd b" "\<not> b dvd a" "b \<noteq> 0"
shows "euclidean_size a < euclidean_size b"
proof -
from assms(1) obtain c where "b = a * c" by (erule dvdE)
hence z: "b = c * a" by (simp add: mult.commute)
from z assms have "\<not>is_unit c" by (auto simp: mult.commute mult_unit_dvd_iff)
with z assms show ?thesis
by (auto intro!: euclidean_size_times_nonunit)
qed
end
class euclidean_ring = idom_modulo + euclidean_semiring
subsection \<open>Uniquely determined division\<close>
class unique_euclidean_semiring = euclidean_semiring +
fixes uniqueness_constraint :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
assumes size_mono_mult:
"b \<noteq> 0 \<Longrightarrow> euclidean_size a < euclidean_size c
\<Longrightarrow> euclidean_size (a * b) < euclidean_size (c * b)"
-- \<open>FIXME justify\<close>
assumes uniqueness_constraint_mono_mult:
"uniqueness_constraint a b \<Longrightarrow> uniqueness_constraint (a * c) (b * c)"
assumes uniqueness_constraint_mod:
"b \<noteq> 0 \<Longrightarrow> \<not> b dvd a \<Longrightarrow> uniqueness_constraint (a mod b) b"
assumes div_bounded:
"b \<noteq> 0 \<Longrightarrow> uniqueness_constraint r b
\<Longrightarrow> euclidean_size r < euclidean_size b
\<Longrightarrow> (q * b + r) div b = q"
begin
lemma divmod_cases [case_names divides remainder by0]:
obtains
(divides) q where "b \<noteq> 0"
and "a div b = q"
and "a mod b = 0"
and "a = q * b"
| (remainder) q r where "b \<noteq> 0" and "r \<noteq> 0"
and "uniqueness_constraint r b"
and "euclidean_size r < euclidean_size b"
and "a div b = q"
and "a mod b = r"
and "a = q * b + r"
| (by0) "b = 0"
proof (cases "b = 0")
case True
then show thesis
by (rule by0)
next
case False
show thesis
proof (cases "b dvd a")
case True
then obtain q where "a = b * q" ..
with \<open>b \<noteq> 0\<close> divides
show thesis
by (simp add: ac_simps)
next
case False
then have "a mod b \<noteq> 0"
by (simp add: mod_eq_0_iff_dvd)
moreover from \<open>b \<noteq> 0\<close> \<open>\<not> b dvd a\<close> have "uniqueness_constraint (a mod b) b"
by (rule uniqueness_constraint_mod)
moreover have "euclidean_size (a mod b) < euclidean_size b"
using \<open>b \<noteq> 0\<close> by (rule mod_size_less)
moreover have "a = a div b * b + a mod b"
by (simp add: div_mult_mod_eq)
ultimately show thesis
using \<open>b \<noteq> 0\<close> by (blast intro: remainder)
qed
qed
lemma div_eqI:
"a div b = q" if "b \<noteq> 0" "uniqueness_constraint r b"
"euclidean_size r < euclidean_size b" "q * b + r = a"
proof -
from that have "(q * b + r) div b = q"
by (auto intro: div_bounded)
with that show ?thesis
by simp
qed
lemma mod_eqI:
"a mod b = r" if "b \<noteq> 0" "uniqueness_constraint r b"
"euclidean_size r < euclidean_size b" "q * b + r = a"
proof -
from that have "a div b = q"
by (rule div_eqI)
moreover have "a div b * b + a mod b = a"
by (fact div_mult_mod_eq)
ultimately have "a div b * b + a mod b = a div b * b + r"
using \<open>q * b + r = a\<close> by simp
then show ?thesis
by simp
qed
end
class unique_euclidean_ring = euclidean_ring + unique_euclidean_semiring
end