(*  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> ?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 ..

 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_multiset_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

 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

 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 = {#}"

   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: add_is_1 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 safe

   apply (simp_all add: eq_sym_conv)

   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::nat) < 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::nat)) ==> (!!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 rule_context

   note premises = 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 premises)

      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 premises)

      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 One_nat_def)

     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 Rep_multiset_inject [symmetric, 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)\<^sup>+"

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

   by (simp add: mult1_def)

 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 ..

 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

 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)

   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

 end