(* Title: HOL/Library/Multiset_Order.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, Inria, LORIA, MPII
*)
section \<open>More Theorems about the Multiset Order\<close>
theory Multiset_Order
imports Multiset
begin
subsection \<open>Alternative Characterizations\<close>
subsubsection \<open>The Dershowitz--Manna Ordering\<close>
definition multp\<^sub>D\<^sub>M where
"multp\<^sub>D\<^sub>M r M N \<longleftrightarrow>
(\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = (N - X) + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> r k a)))"
lemma multp\<^sub>D\<^sub>M_imp_multp:
"multp\<^sub>D\<^sub>M r M N \<Longrightarrow> multp r M N"
proof -
assume "multp\<^sub>D\<^sub>M r M N"
then obtain X Y where
"X \<noteq> {#}" and "X \<subseteq># N" and "M = N - X + Y" and "\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> r k a)"
unfolding multp\<^sub>D\<^sub>M_def by blast
then have "multp r (N - X + Y) (N - X + X)"
by (intro one_step_implies_multp) (auto simp: Bex_def trans_def)
with \<open>M = N - X + Y\<close> \<open>X \<subseteq># N\<close> show "multp r M N"
by (metis subset_mset.diff_add)
qed
subsubsection \<open>The Huet--Oppen Ordering\<close>
definition multp\<^sub>H\<^sub>O where
"multp\<^sub>H\<^sub>O r M N \<longleftrightarrow> M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. r y x \<and> count M x < count N x))"
lemma multp_imp_multp\<^sub>H\<^sub>O:
assumes "asymp r" and "transp r"
shows "multp r M N \<Longrightarrow> multp\<^sub>H\<^sub>O r M N"
unfolding multp_def mult_def
proof (induction rule: trancl_induct)
case (base P)
then show ?case
using \<open>asymp r\<close>
by (auto elim!: mult1_lessE simp: count_eq_zero_iff multp\<^sub>H\<^sub>O_def split: if_splits
dest!: Suc_lessD)
next
case (step N P)
from step(3) have "M \<noteq> N" and
**: "\<And>y. count N y < count M y \<Longrightarrow> (\<exists>x. r y x \<and> count M x < count N x)"
by (simp_all add: multp\<^sub>H\<^sub>O_def)
from step(2) obtain M0 a K where
*: "P = add_mset a M0" "N = M0 + K" "a \<notin># K" "\<And>b. b \<in># K \<Longrightarrow> r b a"
using \<open>asymp r\<close> by (auto elim: mult1_lessE)
from \<open>M \<noteq> N\<close> ** *(1,2,3) have "M \<noteq> P"
using *(4) \<open>asymp r\<close>
by (metis asymp.cases add_cancel_right_right add_diff_cancel_left' add_mset_add_single count_inI
count_union diff_diff_add_mset diff_single_trivial in_diff_count multi_member_last)
moreover
{ assume "count P a \<le> count M a"
with \<open>a \<notin># K\<close> have "count N a < count M a" unfolding *(1,2)
by (auto simp add: not_in_iff)
with ** obtain z where z: "r a z" "count M z < count N z"
by blast
with * have "count N z \<le> count P z"
using \<open>asymp r\<close>
by (metis add_diff_cancel_left' add_mset_add_single asymp.cases diff_diff_add_mset
diff_single_trivial in_diff_count not_le_imp_less)
with z have "\<exists>z. r a z \<and> count M z < count P z" by auto
} note count_a = this
{ fix y
assume count_y: "count P y < count M y"
have "\<exists>x. r y x \<and> count M x < count P x"
proof (cases "y = a")
case True
with count_y count_a show ?thesis by auto
next
case False
show ?thesis
proof (cases "y \<in># K")
case True
with *(4) have "r y a" by simp
then show ?thesis
by (cases "count P a \<le> count M a") (auto dest: count_a intro: \<open>transp r\<close>[THEN transpD])
next
case False
with \<open>y \<noteq> a\<close> have "count P y = count N y" unfolding *(1,2)
by (simp add: not_in_iff)
with count_y ** obtain z where z: "r y z" "count M z < count N z" by auto
show ?thesis
proof (cases "z \<in># K")
case True
with *(4) have "r z a" by simp
with z(1) show ?thesis
by (cases "count P a \<le> count M a") (auto dest!: count_a intro: \<open>transp r\<close>[THEN transpD])
next
case False
with \<open>a \<notin># K\<close> have "count N z \<le> count P z" unfolding *
by (auto simp add: not_in_iff)
with z show ?thesis by auto
qed
qed
qed
}
ultimately show ?case unfolding multp\<^sub>H\<^sub>O_def by blast
qed
lemma multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M: "multp\<^sub>H\<^sub>O r M N \<Longrightarrow> multp\<^sub>D\<^sub>M r M N"
unfolding multp\<^sub>D\<^sub>M_def
proof (intro iffI exI conjI)
assume "multp\<^sub>H\<^sub>O r M N"
then obtain z where z: "count M z < count N z"
unfolding multp\<^sub>H\<^sub>O_def by (auto simp: multiset_eq_iff nat_neq_iff)
define X where "X = N - M"
define Y where "Y = M - N"
from z show "X \<noteq> {#}" unfolding X_def by (auto simp: multiset_eq_iff not_less_eq_eq Suc_le_eq)
from z show "X \<subseteq># N" unfolding X_def by auto
show "M = (N - X) + Y" unfolding X_def Y_def multiset_eq_iff count_union count_diff by force
show "\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> r k a)"
proof (intro allI impI)
fix k
assume "k \<in># Y"
then have "count N k < count M k" unfolding Y_def
by (auto simp add: in_diff_count)
with \<open>multp\<^sub>H\<^sub>O r M N\<close> obtain a where "r k a" and "count M a < count N a"
unfolding multp\<^sub>H\<^sub>O_def by blast
then show "\<exists>a. a \<in># X \<and> r k a" unfolding X_def
by (auto simp add: in_diff_count)
qed
qed
lemma multp_eq_multp\<^sub>D\<^sub>M: "asymp r \<Longrightarrow> transp r \<Longrightarrow> multp r = multp\<^sub>D\<^sub>M r"
using multp\<^sub>D\<^sub>M_imp_multp multp_imp_multp\<^sub>H\<^sub>O[THEN multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M]
by blast
lemma multp_eq_multp\<^sub>H\<^sub>O: "asymp r \<Longrightarrow> transp r \<Longrightarrow> multp r = multp\<^sub>H\<^sub>O r"
using multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M[THEN multp\<^sub>D\<^sub>M_imp_multp] multp_imp_multp\<^sub>H\<^sub>O
by blast
subsubsection \<open>Properties of Preorders\<close>
context preorder
begin
lemma order_mult: "class.order
(\<lambda>M N. (M, N) \<in> mult {(x, y). x < y} \<or> M = N)
(\<lambda>M N. (M, N) \<in> mult {(x, y). x < y})"
(is "class.order ?le ?less")
proof -
have irrefl: "\<And>M :: 'a multiset. \<not> ?less M M"
proof
fix M :: "'a multiset"
have "trans {(x'::'a, x). x' < x}"
by (rule transI) (blast intro: less_trans)
moreover
assume "(M, M) \<in> mult {(x, y). x < y}"
ultimately have "\<exists>I J K. M = I + J \<and> M = I + K
\<and> J \<noteq> {#} \<and> (\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})"
by (rule mult_implies_one_step)
then obtain I J K where "M = I + J" and "M = I + K"
and "J \<noteq> {#}" and "(\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset J. (k, j) \<in> {(x, y). x < y})" by blast
then have aux1: "K \<noteq> {#}" and aux2: "\<forall>k\<in>set_mset K. \<exists>j\<in>set_mset K. k < j" by auto
have "finite (set_mset K)" by simp
moreover note aux2
ultimately have "set_mset K = {}"
by (induct rule: finite_induct)
(simp, metis (mono_tags) insert_absorb insert_iff insert_not_empty less_irrefl less_trans)
with aux1 show False by simp
qed
have trans: "\<And>K M N :: 'a multiset. ?less K M \<Longrightarrow> ?less M N \<Longrightarrow> ?less K N"
unfolding mult_def by (blast intro: trancl_trans)
show "class.order ?le ?less"
by standard (auto simp add: less_eq_multiset_def irrefl dest: trans)
qed
text \<open>The Dershowitz--Manna ordering:\<close>
definition less_multiset\<^sub>D\<^sub>M where
"less_multiset\<^sub>D\<^sub>M M N \<longleftrightarrow>
(\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = (N - X) + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
text \<open>The Huet--Oppen ordering:\<close>
definition less_multiset\<^sub>H\<^sub>O where
"less_multiset\<^sub>H\<^sub>O M N \<longleftrightarrow> M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. y < x \<and> count M x < count N x))"
lemma mult_imp_less_multiset\<^sub>H\<^sub>O:
"(M, N) \<in> mult {(x, y). x < y} \<Longrightarrow> less_multiset\<^sub>H\<^sub>O M N"
unfolding multp_def[of "(<)", symmetric]
using multp_imp_multp\<^sub>H\<^sub>O[of "(<)"]
by (simp add: less_multiset\<^sub>H\<^sub>O_def multp\<^sub>H\<^sub>O_def)
lemma less_multiset\<^sub>D\<^sub>M_imp_mult:
"less_multiset\<^sub>D\<^sub>M M N \<Longrightarrow> (M, N) \<in> mult {(x, y). x < y}"
unfolding multp_def[of "(<)", symmetric]
by (rule multp\<^sub>D\<^sub>M_imp_multp[of "(<)" M N]) (simp add: less_multiset\<^sub>D\<^sub>M_def multp\<^sub>D\<^sub>M_def)
lemma less_multiset\<^sub>H\<^sub>O_imp_less_multiset\<^sub>D\<^sub>M: "less_multiset\<^sub>H\<^sub>O M N \<Longrightarrow> less_multiset\<^sub>D\<^sub>M M N"
unfolding less_multiset\<^sub>D\<^sub>M_def less_multiset\<^sub>H\<^sub>O_def
unfolding multp\<^sub>D\<^sub>M_def[symmetric] multp\<^sub>H\<^sub>O_def[symmetric]
by (rule multp\<^sub>H\<^sub>O_imp_multp\<^sub>D\<^sub>M)
lemma mult_less_multiset\<^sub>D\<^sub>M: "(M, N) \<in> mult {(x, y). x < y} \<longleftrightarrow> less_multiset\<^sub>D\<^sub>M M N"
unfolding multp_def[of "(<)", symmetric]
using multp_eq_multp\<^sub>D\<^sub>M[of "(<)", simplified]
by (simp add: multp\<^sub>D\<^sub>M_def less_multiset\<^sub>D\<^sub>M_def)
lemma mult_less_multiset\<^sub>H\<^sub>O: "(M, N) \<in> mult {(x, y). x < y} \<longleftrightarrow> less_multiset\<^sub>H\<^sub>O M N"
unfolding multp_def[of "(<)", symmetric]
using multp_eq_multp\<^sub>H\<^sub>O[of "(<)", simplified]
by (simp add: multp\<^sub>H\<^sub>O_def less_multiset\<^sub>H\<^sub>O_def)
lemmas mult\<^sub>D\<^sub>M = mult_less_multiset\<^sub>D\<^sub>M[unfolded less_multiset\<^sub>D\<^sub>M_def]
lemmas mult\<^sub>H\<^sub>O = mult_less_multiset\<^sub>H\<^sub>O[unfolded less_multiset\<^sub>H\<^sub>O_def]
end
lemma less_multiset_less_multiset\<^sub>H\<^sub>O: "M < N \<longleftrightarrow> less_multiset\<^sub>H\<^sub>O M N"
unfolding less_multiset_def multp_def mult\<^sub>H\<^sub>O less_multiset\<^sub>H\<^sub>O_def ..
lemma less_multiset\<^sub>D\<^sub>M:
"M < N \<longleftrightarrow> (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> k < a)))"
by (rule mult\<^sub>D\<^sub>M[folded multp_def less_multiset_def])
lemma less_multiset\<^sub>H\<^sub>O:
"M < N \<longleftrightarrow> M \<noteq> N \<and> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x>y. count M x < count N x))"
by (rule mult\<^sub>H\<^sub>O[folded multp_def less_multiset_def])
lemma subset_eq_imp_le_multiset:
shows "M \<subseteq># N \<Longrightarrow> M \<le> N"
unfolding less_eq_multiset_def less_multiset\<^sub>H\<^sub>O
by (simp add: less_le_not_le subseteq_mset_def)
(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_right_total: "M < add_mset x M"
unfolding less_eq_multiset_def less_multiset\<^sub>H\<^sub>O by simp
lemma less_eq_multiset_empty_left[simp]:
shows "{#} \<le> M"
by (simp add: subset_eq_imp_le_multiset)
lemma ex_gt_imp_less_multiset: "(\<exists>y. y \<in># N \<and> (\<forall>x. x \<in># M \<longrightarrow> x < y)) \<Longrightarrow> M < N"
unfolding less_multiset\<^sub>H\<^sub>O
by (metis count_eq_zero_iff count_greater_zero_iff less_le_not_le)
lemma less_eq_multiset_empty_right[simp]: "M \<noteq> {#} \<Longrightarrow> \<not> M \<le> {#}"
by (metis less_eq_multiset_empty_left antisym)
(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_left[simp]: "M \<noteq> {#} \<Longrightarrow> {#} < M"
by (simp add: less_multiset\<^sub>H\<^sub>O)
(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_right[simp]: "\<not> M < {#}"
using subset_mset.le_zero_eq less_multiset_def multp_def less_multiset\<^sub>D\<^sub>M by blast
(* FIXME: "le" should be "less" in this and other names *)
lemma union_le_diff_plus: "P \<subseteq># M \<Longrightarrow> N < P \<Longrightarrow> M - P + N < M"
by (drule subset_mset.diff_add[symmetric]) (metis union_le_mono2)
instantiation multiset :: (preorder) ordered_ab_semigroup_monoid_add_imp_le
begin
lemma less_eq_multiset\<^sub>H\<^sub>O:
"M \<le> N \<longleftrightarrow> (\<forall>y. count N y < count M y \<longrightarrow> (\<exists>x. y < x \<and> count M x < count N x))"
by (auto simp: less_eq_multiset_def less_multiset\<^sub>H\<^sub>O)
instance by standard (auto simp: less_eq_multiset\<^sub>H\<^sub>O)
lemma
fixes M N :: "'a multiset"
shows
less_eq_multiset_plus_left: "N \<le> (M + N)" and
less_eq_multiset_plus_right: "M \<le> (M + N)"
by simp_all
lemma
fixes M N :: "'a multiset"
shows
le_multiset_plus_left_nonempty: "M \<noteq> {#} \<Longrightarrow> N < M + N" and
le_multiset_plus_right_nonempty: "N \<noteq> {#} \<Longrightarrow> M < M + N"
by simp_all
end
lemma all_lt_Max_imp_lt_mset: "N \<noteq> {#} \<Longrightarrow> (\<forall>a \<in># M. a < Max (set_mset N)) \<Longrightarrow> M < N"
by (meson Max_in[OF finite_set_mset] ex_gt_imp_less_multiset set_mset_eq_empty_iff)
lemma lt_imp_ex_count_lt: "M < N \<Longrightarrow> \<exists>y. count M y < count N y"
by (meson less_eq_multiset\<^sub>H\<^sub>O less_le_not_le)
lemma subset_imp_less_mset: "A \<subset># B \<Longrightarrow> A < B"
by (simp add: order.not_eq_order_implies_strict subset_eq_imp_le_multiset)
lemma image_mset_strict_mono:
assumes
mono_f: "\<forall>x \<in> set_mset M. \<forall>y \<in> set_mset N. x < y \<longrightarrow> f x < f y" and
less: "M < N"
shows "image_mset f M < image_mset f N"
proof -
obtain Y X where
y_nemp: "Y \<noteq> {#}" and y_sub_N: "Y \<subseteq># N" and M_eq: "M = N - Y + X" and
ex_y: "\<forall>x. x \<in># X \<longrightarrow> (\<exists>y. y \<in># Y \<and> x < y)"
using less[unfolded less_multiset\<^sub>D\<^sub>M] by blast
have x_sub_M: "X \<subseteq># M"
using M_eq by simp
let ?fY = "image_mset f Y"
let ?fX = "image_mset f X"
show ?thesis
unfolding less_multiset\<^sub>D\<^sub>M
proof (intro exI conjI)
show "image_mset f M = image_mset f N - ?fY + ?fX"
using M_eq[THEN arg_cong, of "image_mset f"] y_sub_N
by (metis image_mset_Diff image_mset_union)
next
obtain y where y: "\<forall>x. x \<in># X \<longrightarrow> y x \<in># Y \<and> x < y x"
using ex_y by moura
show "\<forall>fx. fx \<in># ?fX \<longrightarrow> (\<exists>fy. fy \<in># ?fY \<and> fx < fy)"
proof (intro allI impI)
fix fx
assume "fx \<in># ?fX"
then obtain x where fx: "fx = f x" and x_in: "x \<in># X"
by auto
hence y_in: "y x \<in># Y" and y_gt: "x < y x"
using y[rule_format, OF x_in] by blast+
hence "f (y x) \<in># ?fY \<and> f x < f (y x)"
using mono_f y_sub_N x_sub_M x_in
by (metis image_eqI in_image_mset mset_subset_eqD)
thus "\<exists>fy. fy \<in># ?fY \<and> fx < fy"
unfolding fx by auto
qed
qed (auto simp: y_nemp y_sub_N image_mset_subseteq_mono)
qed
lemma image_mset_mono:
assumes
mono_f: "\<forall>x \<in> set_mset M. \<forall>y \<in> set_mset N. x < y \<longrightarrow> f x < f y" and
less: "M \<le> N"
shows "image_mset f M \<le> image_mset f N"
by (metis eq_iff image_mset_strict_mono less less_imp_le mono_f order.not_eq_order_implies_strict)
lemma mset_lt_single_right_iff[simp]: "M < {#y#} \<longleftrightarrow> (\<forall>x \<in># M. x < y)" for y :: "'a::linorder"
proof (rule iffI)
assume M_lt_y: "M < {#y#}"
show "\<forall>x \<in># M. x < y"
proof
fix x
assume x_in: "x \<in># M"
hence M: "M - {#x#} + {#x#} = M"
by (meson insert_DiffM2)
hence "\<not> {#x#} < {#y#} \<Longrightarrow> x < y"
using x_in M_lt_y
by (metis diff_single_eq_union le_multiset_empty_left less_add_same_cancel2 mset_le_trans)
also have "\<not> {#y#} < M"
using M_lt_y mset_le_not_sym by blast
ultimately show "x < y"
by (metis (no_types) Max_ge all_lt_Max_imp_lt_mset empty_iff finite_set_mset insertE
less_le_trans linorder_less_linear mset_le_not_sym set_mset_add_mset_insert
set_mset_eq_empty_iff x_in)
qed
next
assume y_max: "\<forall>x \<in># M. x < y"
show "M < {#y#}"
by (rule all_lt_Max_imp_lt_mset) (auto intro!: y_max)
qed
lemma mset_le_single_right_iff[simp]:
"M \<le> {#y#} \<longleftrightarrow> M = {#y#} \<or> (\<forall>x \<in># M. x < y)" for y :: "'a::linorder"
by (meson less_eq_multiset_def mset_lt_single_right_iff)
subsection \<open>Simprocs\<close>
lemma mset_le_add_iff1:
"j \<le> (i::nat) \<Longrightarrow> (repeat_mset i u + m \<le> repeat_mset j u + n) = (repeat_mset (i-j) u + m \<le> n)"
proof -
assume "j \<le> i"
then have "j + (i - j) = i"
using le_add_diff_inverse by blast
then show ?thesis
by (metis (no_types) add_le_cancel_left left_add_mult_distrib_mset)
qed
lemma mset_le_add_iff2:
"i \<le> (j::nat) \<Longrightarrow> (repeat_mset i u + m \<le> repeat_mset j u + n) = (m \<le> repeat_mset (j-i) u + n)"
proof -
assume "i \<le> j"
then have "i + (j - i) = j"
using le_add_diff_inverse by blast
then show ?thesis
by (metis (no_types) add_le_cancel_left left_add_mult_distrib_mset)
qed
simproc_setup msetless_cancel
("(l::'a::preorder multiset) + m < n" | "(l::'a multiset) < m + n" |
"add_mset a m < n" | "m < add_mset a n" |
"replicate_mset p a < n" | "m < replicate_mset p a" |
"repeat_mset p m < n" | "m < repeat_mset p n") =
\<open>fn phi => Cancel_Simprocs.less_cancel\<close>
simproc_setup msetle_cancel
("(l::'a::preorder multiset) + m \<le> n" | "(l::'a multiset) \<le> m + n" |
"add_mset a m \<le> n" | "m \<le> add_mset a n" |
"replicate_mset p a \<le> n" | "m \<le> replicate_mset p a" |
"repeat_mset p m \<le> n" | "m \<le> repeat_mset p n") =
\<open>fn phi => Cancel_Simprocs.less_eq_cancel\<close>
subsection \<open>Additional facts and instantiations\<close>
lemma ex_gt_count_imp_le_multiset:
"(\<forall>y :: 'a :: order. y \<in># M + N \<longrightarrow> y \<le> x) \<Longrightarrow> count M x < count N x \<Longrightarrow> M < N"
unfolding less_multiset\<^sub>H\<^sub>O
by (metis count_greater_zero_iff le_imp_less_or_eq less_imp_not_less not_gr_zero union_iff)
lemma mset_lt_single_iff[iff]: "{#x#} < {#y#} \<longleftrightarrow> x < y"
unfolding less_multiset\<^sub>H\<^sub>O by simp
lemma mset_le_single_iff[iff]: "{#x#} \<le> {#y#} \<longleftrightarrow> x \<le> y" for x y :: "'a::order"
unfolding less_eq_multiset\<^sub>H\<^sub>O by force
instance multiset :: (linorder) linordered_cancel_ab_semigroup_add
by standard (metis less_eq_multiset\<^sub>H\<^sub>O not_less_iff_gr_or_eq)
lemma less_eq_multiset_total:
fixes M N :: "'a :: linorder multiset"
shows "\<not> M \<le> N \<Longrightarrow> N \<le> M"
by simp
instantiation multiset :: (wellorder) wellorder
begin
lemma wf_less_multiset: "wf {(M :: 'a multiset, N). M < N}"
unfolding less_multiset_def multp_def by (auto intro: wf_mult wf)
instance by standard (metis less_multiset_def multp_def wf wf_def wf_mult)
end
instantiation multiset :: (preorder) order_bot
begin
definition bot_multiset :: "'a multiset" where "bot_multiset = {#}"
instance by standard (simp add: bot_multiset_def)
end
instance multiset :: (preorder) no_top
proof standard
fix x :: "'a multiset"
obtain a :: 'a where True by simp
have "x < x + (x + {#a#})"
by simp
then show "\<exists>y. x < y"
by blast
qed
instance multiset :: (preorder) ordered_cancel_comm_monoid_add
by standard
instantiation multiset :: (linorder) distrib_lattice
begin
definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
"inf_multiset A B = (if A < B then A else B)"
definition sup_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where
"sup_multiset A B = (if B > A then B else A)"
instance
by intro_classes (auto simp: inf_multiset_def sup_multiset_def)
end
end