--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Multiset.thy Wed Oct 18 23:28:33 2000 +0200
@@ -0,0 +1,854 @@
+(* 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
+*)
+
+header {*
+ \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> {f. finite {x. 0 < f x}}"
+ by simp
+qed
+
+lemmas multiset_typedef [simp] =
+ Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
+
+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 by intro_classes
+instance multiset :: ("term") minus by intro_classes
+instance multiset :: ("term") zero by intro_classes
+
+defs (overloaded)
+ 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"
+ apply (simp add: multiset_def)
+ done
+
+lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
+ apply (simp add: multiset_def)
+ 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
+
+text {*
+ \medskip Injectivity of @{term Rep_multiset}.
+*} (* FIXME typedef package (!?) *)
+
+lemma multiset_eq_conv_Rep_eq [simp]:
+ "(M = N) = (Rep_multiset M = Rep_multiset N)"
+ apply (rule iffI)
+ apply simp
+ apply (drule_tac f = Abs_multiset in arg_cong)
+ apply simp
+ done
+
+(* FIXME
+Goal
+ "[| f : multiset; g : multiset |] ==> \
+\ (Abs_multiset f = Abs_multiset g) = (!x. f x = g x)";
+by (rtac iffI 1);
+ by (dres_inst_tac [("f","Rep_multiset")] arg_cong 1);
+ by (Asm_full_simp_tac 1);
+by (subgoal_tac "f = g" 1);
+ by (Asm_simp_tac 1);
+by (rtac ext 1);
+by (Blast_tac 1);
+qed "Abs_multiset_eq";
+Addsimps [Abs_multiset_eq];
+*)
+
+
+subsection {* Algebraic properties of multisets *}
+
+subsubsection {* Union *}
+
+theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
+ apply (simp add: union_def Mempty_def)
+ done
+
+theorem union_commute: "M + N = N + (M::'a multiset)"
+ apply (simp add: union_def add_ac)
+ done
+
+theorem union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
+ apply (simp add: union_def add_ac)
+ 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
+
+
+subsubsection {* Difference *}
+
+theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
+ apply (simp add: Mempty_def diff_def)
+ done
+
+theorem diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
+ apply (simp add: union_def diff_def)
+ done
+
+
+subsubsection {* Count of elements *}
+
+theorem count_empty [simp]: "count {#} a = 0"
+ apply (simp add: count_def Mempty_def)
+ done
+
+theorem count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
+ apply (simp add: count_def single_def)
+ done
+
+theorem count_union [simp]: "count (M + N) a = count M a + count N a"
+ apply (simp add: count_def union_def)
+ done
+
+theorem count_diff [simp]: "count (M - N) a = count M a - count N a"
+ apply (simp add: count_def diff_def)
+ done
+
+
+subsubsection {* Set of elements *}
+
+theorem set_of_empty [simp]: "set_of {#} = {}"
+ apply (simp add: set_of_def)
+ done
+
+theorem set_of_single [simp]: "set_of {#b#} = {b}"
+ apply (simp add: set_of_def)
+ done
+
+theorem set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
+ apply (auto simp add: set_of_def)
+ 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)"
+ apply (auto simp add: set_of_def)
+ done
+
+
+subsubsection {* Size *}
+
+theorem size_empty [simp]: "size {#} = 0"
+ apply (simp add: size_def)
+ done
+
+theorem size_single [simp]: "size {#b#} = 1"
+ apply (simp add: size_def)
+ 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
+ apply (simp add: Int_insert_left set_of_def)
+ 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 (simp (no_asm_simp) add: setsum_Un setsum_addf setsum_count_Int)
+ apply (subst Int_commute)
+ apply (simp (no_asm_simp) add: setsum_count_Int)
+ 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)"
+ apply (simp add: count_def expand_fun_eq)
+ 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))"
+ apply (simp add: union_def expand_fun_eq)
+ done
+
+theorem union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
+ apply (simp add: union_def expand_fun_eq)
+ 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 (simp add: add_is_1 expand_fun_eq)
+ 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 (simp add: one_is_add expand_fun_eq)
+ apply (blast dest: sym)
+ done
+
+theorem add_eq_conv_diff:
+ "(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 (simp (no_asm) add: expand_fun_eq)
+ 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
+ apply (simp add: le_imp_diff_is_add setsum_diff1 cong: conj_cong)
+ 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 (simp add: expand_fun_eq)
+ 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 (simp add: multiset_def)
+ 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)
+ apply (simp add: MCollect_preserves_multiset)
+ done
+
+theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}"
+ apply (auto simp add: set_of_def)
+ 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 multiset_eq_conv_Rep_eq [simp del]
+declare multiset_typedef [simp del]
+
+theorem add_eq_conv_ex:
+ "(M + {#a#} = N + {#b#}) =
+ (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
+ apply (auto simp add: add_eq_conv_diff)
+ 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)^+"
+
+lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
+ apply (simp add: mult1_def)
+ done
+
+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#})"
+ by (simp only: add_eq_conv_ex)
+ 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))"
+ by (rule less_add)
+ 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#})"
+ apply (simp add: multiset_eq_conv_count_eq)
+ 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 (simp add: diff_union_single_conv)
+ apply (simp (no_asm_use) add: trans_def)
+ 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 (simp (no_asm_use) add: trans_def)
+ 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#}"
+ apply (simp add: multiset_eq_conv_count_eq)
+ 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 (simp add: mult_def)
+ apply (rule r_into_trancl)
+ apply (simp add: mult1_def set_of_def)
+ 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 (simp add: Ball_def)
+ 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 (simp add: mult1_def set_of_def)
+ apply (rule_tac x = a in exI)
+ apply (rule_tac x = "I + J'" in exI)
+ apply (simp add: union_ac)
+ 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 by intro_classes
+
+defs (overloaded)
+ 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)]])
+ apply (simp add: set_of_eq_empty_iff)
+ 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
+
+
+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)
+ apply (simp add: union_assoc)
+ 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
+
+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
+
+instance multiset :: ("term") plus_ac0
+ apply intro_classes
+ apply (rule union_commute)
+ apply (rule union_assoc)
+ apply simp
+ done
+
+end