src/HOL/Library/Multiset.thy
 author paulson Thu, 21 Dec 2000 10:16:07 +0100 changeset 10714 07f75bf77a33 parent 10392 f27afee8475d child 11464 ddea204de5bc permissions -rw-r--r--
re-orientation of 0=... (no idea why the backslashes have changed too)
```
(*  Title:      HOL/Library/Multiset.thy
ID:         \$Id\$
Author:     Tobias Nipkow, TU Muenchen
Author:     Markus Wenzel, TU Muenchen
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

\title{Multisets}
\author{Tobias Nipkow, Markus Wenzel, and Lawrence C Paulson}
*}

theory Multiset = Accessible_Part:

subsection {* The type of multisets *}

typedef 'a multiset = "{f::'a => nat. finite {x . 0 < f x}}"
proof
show "(\\<lambda>x. 0::nat) \\<in> ?multiset" by simp
qed

lemmas multiset_typedef [simp] =
Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
and [simp] = Rep_multiset_inject [symmetric]

constdefs
Mempty :: "'a multiset"    ("{#}")
"{#} == Abs_multiset (\\<lambda>a. 0)"

single :: "'a => 'a multiset"    ("{#_#}")
"{#a#} == Abs_multiset (\\<lambda>b. if b = a then 1 else 0)"

count :: "'a multiset => 'a => nat"
"count == Rep_multiset"

MCollect :: "'a multiset => ('a => bool) => 'a multiset"
"MCollect M P == Abs_multiset (\\<lambda>x. if P x then Rep_multiset M x else 0)"

syntax
"_Melem" :: "'a => 'a multiset => bool"    ("(_/ :# _)" [50, 51] 50)
"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ : _./ _#})")
translations
"a :# M" == "0 < count M a"
"{#x:M. P#}" == "MCollect M (\\<lambda>x. P)"

constdefs
set_of :: "'a multiset => 'a set"
"set_of M == {x. x :# M}"

instance multiset :: ("term") plus ..
instance multiset :: ("term") minus ..
instance multiset :: ("term") zero ..

union_def: "M + N == Abs_multiset (\\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
diff_def: "M - N == Abs_multiset (\\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
Zero_def [simp]: "0 == {#}"
size_def: "size M == setsum (count M) (set_of M)"

text {*
\medskip Preservation of the representing set @{term multiset}.
*}

lemma const0_in_multiset [simp]: "(\\<lambda>a. 0) \\<in> multiset"
done

lemma only1_in_multiset [simp]: "(\\<lambda>b. if b = a then 1 else 0) \\<in> multiset"
done

lemma union_preserves_multiset [simp]:
"M \\<in> multiset ==> N \\<in> multiset ==> (\\<lambda>a. M a + N a) \\<in> multiset"
apply (unfold multiset_def)
apply simp
apply (drule finite_UnI)
apply assumption
apply (simp del: finite_Un add: Un_def)
done

lemma diff_preserves_multiset [simp]:
"M \\<in> multiset ==> (\\<lambda>a. M a - N a) \\<in> multiset"
apply (unfold multiset_def)
apply simp
apply (rule finite_subset)
prefer 2
apply assumption
apply auto
done

subsection {* Algebraic properties of multisets *}

subsubsection {* Union *}

theorem union_empty [simp]: "M + {#} = M \\<and> {#} + M = M"
done

theorem union_commute: "M + N = N + (M::'a multiset)"
done

theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
done

theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
apply (rule union_commute [THEN trans])
apply (rule union_assoc [THEN trans])
apply (rule union_commute [THEN arg_cong])
done

theorems union_ac = union_assoc union_commute union_lcomm

instance multiset :: ("term") plus_ac0
apply intro_classes
apply (rule union_commute)
apply (rule union_assoc)
apply simp
done

subsubsection {* Difference *}

theorem diff_empty [simp]: "M - {#} = M \\<and> {#} - M = {#}"
done

theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
done

subsubsection {* Count of elements *}

theorem count_empty [simp]: "count {#} a = 0"
done

theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
done

theorem count_union [simp]: "count (M + N) a = count M a + count N a"
done

theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
done

subsubsection {* Set of elements *}

theorem set_of_empty [simp]: "set_of {#} = {}"
done

theorem set_of_single [simp]: "set_of {#b#} = {b}"
done

theorem set_of_union [simp]: "set_of (M + N) = set_of M \\<union> set_of N"
done

theorem set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
apply (auto simp add: set_of_def Mempty_def count_def expand_fun_eq)
done

theorem mem_set_of_iff [simp]: "(x \\<in> set_of M) = (x :# M)"
done

subsubsection {* Size *}

theorem size_empty [simp]: "size {#} = 0"
done

theorem size_single [simp]: "size {#b#} = 1"
done

theorem finite_set_of [iff]: "finite (set_of M)"
apply (cut_tac x = M in Rep_multiset)
apply (simp add: multiset_def set_of_def count_def)
done

theorem setsum_count_Int:
"finite A ==> setsum (count N) (A \\<inter> set_of N) = setsum (count N) A"
apply (erule finite_induct)
apply simp
done

theorem size_union [simp]: "size (M + N::'a multiset) = size M + size N"
apply (unfold size_def)
apply (subgoal_tac "count (M + N) = (\\<lambda>a. count M a + count N a)")
prefer 2
apply (rule ext)
apply simp
apply (subst Int_commute)
done

theorem size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
apply (unfold size_def Mempty_def count_def)
apply auto
apply (simp add: set_of_def count_def expand_fun_eq)
done

theorem size_eq_Suc_imp_elem: "size M = Suc n ==> \\<exists>a. a :# M"
apply (unfold size_def)
apply (drule setsum_SucD)
apply auto
done

subsubsection {* Equality of multisets *}

theorem multiset_eq_conv_count_eq: "(M = N) = (\\<forall>a. count M a = count N a)"
done

theorem single_not_empty [simp]: "{#a#} \\<noteq> {#} \\<and> {#} \\<noteq> {#a#}"
apply (simp add: single_def Mempty_def expand_fun_eq)
done

theorem single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
apply (auto simp add: single_def expand_fun_eq)
done

theorem union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \\<and> N = {#})"
apply (auto simp add: union_def Mempty_def expand_fun_eq)
done

theorem empty_eq_union [iff]: "({#} = M + N) = (M = {#} \\<and> N = {#})"
apply (auto simp add: union_def Mempty_def expand_fun_eq)
done

theorem union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
done

theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
done

theorem union_is_single:
"(M + N = {#a#}) = (M = {#a#} \\<and> N={#} \\<or> M = {#} \\<and> N = {#a#})"
apply (unfold Mempty_def single_def union_def)
apply blast
done

theorem single_is_union:
"({#a#} = M + N) =
({#a#} = M \\<and> N = {#} \\<or> M = {#} \\<and> {#a#} = N)"
apply (unfold Mempty_def single_def union_def)
apply (blast dest: sym)
done

"(M + {#a#} = N + {#b#}) =
(M = N \\<and> a = b \\<or>
M = N - {#a#} + {#b#} \\<and> N = M - {#b#} + {#a#})"
apply (unfold single_def union_def diff_def)
apply (rule conjI)
apply force
apply clarify
apply (rule conjI)
apply blast
apply clarify
apply (rule iffI)
apply (rule conjI)
apply clarify
apply (rule conjI)
apply (simp add: eq_sym_conv)   (* FIXME blast fails !? *)
apply fast
apply simp
apply force
done

(*
val prems = Goal
"[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F";
by (res_inst_tac [("a","F"),("f","\\<lambda>A. if finite A then card A else 0")]
measure_induct 1);
by (Clarify_tac 1);
by (resolve_tac prems 1);
by (assume_tac 1);
by (Clarify_tac 1);
by (subgoal_tac "finite G" 1);
by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2);
by (etac allE 1);
by (etac impE 1);
by (Blast_tac 2);
by (asm_simp_tac (simpset() addsimps [psubset_card]) 1);
no_qed();
val lemma = result();

val prems = Goal
"[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F";
by (rtac (lemma RS mp) 1);
by (REPEAT(ares_tac prems 1));
qed "finite_psubset_induct";

Better: use wf_finite_psubset in WF_Rel
*)

subsection {* Induction over multisets *}

lemma setsum_decr:
"finite F ==> 0 < f a ==>
setsum (f (a := f a - 1)) F = (if a \\<in> F then setsum f F - 1 else setsum f F)"
apply (erule finite_induct)
apply auto
apply (drule_tac a = a in mk_disjoint_insert)
apply auto
done

lemma rep_multiset_induct_aux:
"P (\\<lambda>a. 0) ==> (!!f b. f \\<in> multiset ==> P f ==> P (f (b := f b + 1)))
==> \\<forall>f. f \\<in> multiset --> setsum f {x. 0 < f x} = n --> P f"
proof -
case antecedent
note prems = this [unfolded multiset_def]
show ?thesis
apply (unfold multiset_def)
apply (induct_tac n)
apply simp
apply clarify
apply (subgoal_tac "f = (\\<lambda>a.0)")
apply simp
apply (rule prems)
apply (rule ext)
apply force
apply clarify
apply (frule setsum_SucD)
apply clarify
apply (rename_tac a)
apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}")
prefer 2
apply (rule finite_subset)
prefer 2
apply assumption
apply simp
apply blast
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
prefer 2
apply (rule ext)
apply (simp (no_asm_simp))
apply (erule ssubst, rule prems)
apply blast
apply (erule allE, erule impE, erule_tac [2] mp)
apply blast
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply)
apply (subgoal_tac "{x. x \\<noteq> a --> 0 < f x} = {x. 0 < f x}")
prefer 2
apply blast
apply (subgoal_tac "{x. x \\<noteq> a \\<and> 0 < f x} = {x. 0 < f x} - {a}")
prefer 2
apply blast
done
qed

theorem rep_multiset_induct:
"f \\<in> multiset ==> P (\\<lambda>a. 0) ==>
(!!f b. f \\<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
apply (insert rep_multiset_induct_aux)
apply blast
done

theorem multiset_induct [induct type: multiset]:
"P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M"
proof -
note defns = union_def single_def Mempty_def
assume prem1 [unfolded defns]: "P {#}"
assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})"
show ?thesis
apply (rule Rep_multiset_inverse [THEN subst])
apply (rule Rep_multiset [THEN rep_multiset_induct])
apply (rule prem1)
apply (subgoal_tac "f (b := f b + 1) = (\\<lambda>a. f a + (if a = b then 1 else 0))")
prefer 2
apply (erule ssubst)
apply (erule Abs_multiset_inverse [THEN subst])
apply (erule prem2 [simplified])
done
qed

lemma MCollect_preserves_multiset:
"M \\<in> multiset ==> (\\<lambda>x. if P x then M x else 0) \\<in> multiset"
apply (rule finite_subset)
apply auto
done

theorem count_MCollect [simp]:
"count {# x:M. P x #} a = (if P a then count M a else 0)"
apply (unfold count_def MCollect_def)
done

theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \\<inter> {x. P x}"
done

theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \\<not> P x #}"
apply (subst multiset_eq_conv_count_eq)
apply auto
done

declare Rep_multiset_inject [symmetric, simp del]
declare multiset_typedef [simp del]

"(M + {#a#} = N + {#b#}) =
(M = N \\<and> a = b \\<or> (\\<exists>K. M = K + {#b#} \\<and> N = K + {#a#}))"
done

subsection {* Multiset orderings *}

subsubsection {* Well-foundedness *}

constdefs
mult1 :: "('a \\<times> 'a) set => ('a multiset \\<times> 'a multiset) set"
"mult1 r ==
{(N, M). \\<exists>a M0 K. M = M0 + {#a#} \\<and> N = M0 + K \\<and>
(\\<forall>b. b :# K --> (b, a) \\<in> r)}"

mult :: "('a \\<times> 'a) set => ('a multiset \\<times> 'a multiset) set"
"mult r == (mult1 r)\<^sup>+"

lemma not_less_empty [iff]: "(M, {#}) \\<notin> mult1 r"

lemma less_add: "(N, M0 + {#a#}) \\<in> mult1 r ==>
(\\<exists>M. (M, M0) \\<in> mult1 r \\<and> N = M + {#a#}) \\<or>
(\\<exists>K. (\\<forall>b. b :# K --> (b, a) \\<in> r) \\<and> N = M0 + K)"
(concl is "?case1 (mult1 r) \\<or> ?case2")
proof (unfold mult1_def)
let ?r = "\\<lambda>K a. \\<forall>b. b :# K --> (b, a) \\<in> r"
let ?R = "\\<lambda>N M. \\<exists>a M0 K. M = M0 + {#a#} \\<and> N = M0 + K \\<and> ?r K a"
let ?case1 = "?case1 {(N, M). ?R N M}"

assume "(N, M0 + {#a#}) \\<in> {(N, M). ?R N M}"
hence "\\<exists>a' M0' K.
M0 + {#a#} = M0' + {#a'#} \\<and> N = M0' + K \\<and> ?r K a'" by simp
thus "?case1 \\<or> ?case2"
proof (elim exE conjE)
fix a' M0' K
assume N: "N = M0' + K" and r: "?r K a'"
assume "M0 + {#a#} = M0' + {#a'#}"
hence "M0 = M0' \\<and> a = a' \\<or>
(\\<exists>K'. M0 = K' + {#a'#} \\<and> M0' = K' + {#a#})"
thus ?thesis
proof (elim disjE conjE exE)
assume "M0 = M0'" "a = a'"
with N r have "?r K a \\<and> N = M0 + K" by simp
hence ?case2 .. thus ?thesis ..
next
fix K'
assume "M0' = K' + {#a#}"
with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)

assume "M0 = K' + {#a'#}"
with r have "?R (K' + K) M0" by blast
with n have ?case1 by simp thus ?thesis ..
qed
qed
qed

lemma all_accessible: "wf r ==> \\<forall>M. M \\<in> acc (mult1 r)"
proof
let ?R = "mult1 r"
let ?W = "acc ?R"
{
fix M M0 a
assume M0: "M0 \\<in> ?W"
and wf_hyp: "\\<forall>b. (b, a) \\<in> r --> (\\<forall>M \\<in> ?W. M + {#b#} \\<in> ?W)"
and acc_hyp: "\\<forall>M. (M, M0) \\<in> ?R --> M + {#a#} \\<in> ?W"
have "M0 + {#a#} \\<in> ?W"
proof (rule accI [of "M0 + {#a#}"])
fix N
assume "(N, M0 + {#a#}) \\<in> ?R"
hence "((\\<exists>M. (M, M0) \\<in> ?R \\<and> N = M + {#a#}) \\<or>
(\\<exists>K. (\\<forall>b. b :# K --> (b, a) \\<in> r) \\<and> N = M0 + K))"
thus "N \\<in> ?W"
proof (elim exE disjE conjE)
fix M assume "(M, M0) \\<in> ?R" and N: "N = M + {#a#}"
from acc_hyp have "(M, M0) \\<in> ?R --> M + {#a#} \\<in> ?W" ..
hence "M + {#a#} \\<in> ?W" ..
thus "N \\<in> ?W" by (simp only: N)
next
fix K
assume N: "N = M0 + K"
assume "\\<forall>b. b :# K --> (b, a) \\<in> r"
have "?this --> M0 + K \\<in> ?W" (is "?P K")
proof (induct K)
from M0 have "M0 + {#} \\<in> ?W" by simp
thus "?P {#}" ..

fix K x assume hyp: "?P K"
show "?P (K + {#x#})"
proof
assume a: "\\<forall>b. b :# (K + {#x#}) --> (b, a) \\<in> r"
hence "(x, a) \\<in> r" by simp
with wf_hyp have b: "\\<forall>M \\<in> ?W. M + {#x#} \\<in> ?W" by blast

from a hyp have "M0 + K \\<in> ?W" by simp
with b have "(M0 + K) + {#x#} \\<in> ?W" ..
thus "M0 + (K + {#x#}) \\<in> ?W" by (simp only: union_assoc)
qed
qed
hence "M0 + K \\<in> ?W" ..
thus "N \\<in> ?W" by (simp only: N)
qed
qed
} note tedious_reasoning = this

assume wf: "wf r"
fix M
show "M \\<in> ?W"
proof (induct M)
show "{#} \\<in> ?W"
proof (rule accI)
fix b assume "(b, {#}) \\<in> ?R"
with not_less_empty show "b \\<in> ?W" by contradiction
qed

fix M a assume "M \\<in> ?W"
from wf have "\\<forall>M \\<in> ?W. M + {#a#} \\<in> ?W"
proof induct
fix a
assume "\\<forall>b. (b, a) \\<in> r --> (\\<forall>M \\<in> ?W. M + {#b#} \\<in> ?W)"
show "\\<forall>M \\<in> ?W. M + {#a#} \\<in> ?W"
proof
fix M assume "M \\<in> ?W"
thus "M + {#a#} \\<in> ?W"
by (rule acc_induct) (rule tedious_reasoning)
qed
qed
thus "M + {#a#} \\<in> ?W" ..
qed
qed

theorem wf_mult1: "wf r ==> wf (mult1 r)"
by (rule acc_wfI, rule all_accessible)

theorem wf_mult: "wf r ==> wf (mult r)"
by (unfold mult_def, rule wf_trancl, rule wf_mult1)

subsubsection {* Closure-free presentation *}

(*Badly needed: a linear arithmetic procedure for multisets*)

lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
done

text {* One direction. *}

lemma mult_implies_one_step:
"trans r ==> (M, N) \\<in> mult r ==>
\\<exists>I J K. N = I + J \\<and> M = I + K \\<and> J \\<noteq> {#} \\<and>
(\\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r)"
apply (unfold mult_def mult1_def set_of_def)
apply (erule converse_trancl_induct)
apply clarify
apply (rule_tac x = M0 in exI)
apply simp
apply clarify
apply (case_tac "a :# K")
apply (rule_tac x = I in exI)
apply (simp (no_asm))
apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
apply (simp (no_asm_simp) add: union_assoc [symmetric])
apply (drule_tac f = "\\<lambda>M. M - {#a#}" in arg_cong)
apply blast
apply (subgoal_tac "a :# I")
apply (rule_tac x = "I - {#a#}" in exI)
apply (rule_tac x = "J + {#a#}" in exI)
apply (rule_tac x = "K + Ka" in exI)
apply (rule conjI)
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
apply (rule conjI)
apply (drule_tac f = "\\<lambda>M. M - {#a#}" in arg_cong)
apply simp
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
apply blast
apply (subgoal_tac "a :# (M0 + {#a#})")
apply simp
apply (simp (no_asm))
done

lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
done

lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \\<exists>a N. M = N + {#a#}"
apply (erule size_eq_Suc_imp_elem [THEN exE])
apply (drule elem_imp_eq_diff_union)
apply auto
done

lemma one_step_implies_mult_aux:
"trans r ==>
\\<forall>I J K. (size J = n \\<and> J \\<noteq> {#} \\<and> (\\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r))
--> (I + K, I + J) \\<in> mult r"
apply (induct_tac n)
apply auto
apply (frule size_eq_Suc_imp_eq_union)
apply clarify
apply (rename_tac "J'")
apply simp
apply (erule notE)
apply auto
apply (case_tac "J' = {#}")
apply (rule r_into_trancl)
apply blast
txt {* Now we know @{term "J' \\<noteq> {#}"}. *}
apply (cut_tac M = K and P = "\\<lambda>x. (x, a) \\<in> r" in multiset_partition)
apply (erule_tac P = "\\<forall>k \\<in> set_of K. ?P k" in rev_mp)
apply (erule ssubst)
apply auto
apply (subgoal_tac
"((I + {# x : K. (x, a) \\<in> r #}) + {# x : K. (x, a) \\<notin> r #},
(I + {# x : K. (x, a) \\<in> r #}) + J') \\<in> mult r")
prefer 2
apply force
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
apply (erule trancl_trans)
apply (rule r_into_trancl)
apply (rule_tac x = a in exI)
apply (rule_tac x = "I + J'" in exI)
done

theorem one_step_implies_mult:
"trans r ==> J \\<noteq> {#} ==> \\<forall>k \\<in> set_of K. \\<exists>j \\<in> set_of J. (k, j) \\<in> r
==> (I + K, I + J) \\<in> mult r"
apply (insert one_step_implies_mult_aux)
apply blast
done

subsubsection {* Partial-order properties *}

instance multiset :: ("term") ord ..

less_multiset_def: "M' < M == (M', M) \\<in> mult {(x', x). x' < x}"
le_multiset_def: "M' <= M == M' = M \\<or> M' < (M::'a multiset)"

lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
apply (unfold trans_def)
apply (blast intro: order_less_trans)
done

text {*
\medskip Irreflexivity.
*}

lemma mult_irrefl_aux:
"finite A ==> (\\<forall>x \\<in> A. \\<exists>y \\<in> A. x < (y::'a::order)) --> A = {}"
apply (erule finite_induct)
apply (auto intro: order_less_trans)
done

theorem mult_less_not_refl: "\\<not> M < (M::'a::order multiset)"
apply (unfold less_multiset_def)
apply auto
apply (drule trans_base_order [THEN mult_implies_one_step])
apply auto
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
done

lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
apply (insert mult_less_not_refl)
apply blast
done

text {* Transitivity. *}

theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
apply (unfold less_multiset_def mult_def)
apply (blast intro: trancl_trans)
done

text {* Asymmetry. *}

theorem mult_less_not_sym: "M < N ==> \\<not> N < (M::'a::order multiset)"
apply auto
apply (rule mult_less_not_refl [THEN notE])
apply (erule mult_less_trans)
apply assumption
done

theorem mult_less_asym:
"M < N ==> (\\<not> P ==> N < (M::'a::order multiset)) ==> P"
apply (insert mult_less_not_sym)
apply blast
done

theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
apply (unfold le_multiset_def)
apply auto
done

text {* Anti-symmetry. *}

theorem mult_le_antisym:
"M <= N ==> N <= M ==> M = (N::'a::order multiset)"
apply (unfold le_multiset_def)
apply (blast dest: mult_less_not_sym)
done

text {* Transitivity. *}

theorem mult_le_trans:
"K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
apply (unfold le_multiset_def)
apply (blast intro: mult_less_trans)
done

theorem mult_less_le: "M < N = (M <= N \\<and> M \\<noteq> (N::'a::order multiset))"
apply (unfold le_multiset_def)
apply auto
done

text {* Partial order. *}

instance multiset :: (order) order
apply intro_classes
apply (rule mult_le_refl)
apply (erule mult_le_trans)
apply assumption
apply (erule mult_le_antisym)
apply assumption
apply (rule mult_less_le)
done

subsubsection {* Monotonicity of multiset union *}

theorem mult1_union:
"(B, D) \\<in> mult1 r ==> trans r ==> (C + B, C + D) \\<in> mult1 r"
apply (unfold mult1_def)
apply auto
apply (rule_tac x = a in exI)
apply (rule_tac x = "C + M0" in exI)
done

lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
apply (unfold less_multiset_def mult_def)
apply (erule trancl_induct)
apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
done

lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
apply (subst union_commute [of B C])
apply (subst union_commute [of D C])
apply (erule union_less_mono2)
done

theorem union_less_mono:
"A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
done

theorem union_le_mono:
"A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
apply (unfold le_multiset_def)
apply (blast intro: union_less_mono union_less_mono1 union_less_mono2)
done

theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)"
apply (unfold le_multiset_def less_multiset_def)
apply (case_tac "M = {#}")
prefer 2
apply (subgoal_tac "({#} + {#}, {#} + M) \\<in> mult (Collect (split op <))")
prefer 2
apply (rule one_step_implies_mult)
apply (simp only: trans_def)
apply auto
apply (blast intro: order_less_trans)
done

theorem union_upper1: "A <= A + (B::'a::order multiset)"
apply (subgoal_tac "A + {#} <= A + B")
prefer 2
apply (rule union_le_mono)
apply auto
done

theorem union_upper2: "B <= A + (B::'a::order multiset)"
apply (subst union_commute, rule union_upper1)
done

end
```