(*  Title:      HOL/Library/Multiset.thy

    ID:         $Id$

    Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker

*)

header {* Multisets *}

theory Multiset

imports Plain "~~/src/HOL/List"

begin

subsection {* The type of multisets *}

typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"

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]

definition

  Mempty :: "'a multiset"  ("{#}") where

  "{#} = Abs_multiset (\<lambda>a. 0)"

definition

  single :: "'a => 'a multiset" where

  "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"

declare

  Mempty_def[code func del] single_def[code func del]

definition

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

  "count = Rep_multiset"

definition

  MCollect :: "'a multiset => ('a => bool) => 'a multiset" where

  "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"

abbreviation

  Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where

  "a :# M == 0 < count M a"

notation (xsymbols)

  Melem (infix "\<in>#" 50)

syntax

  "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")

translations

  "{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"

definition

  set_of :: "'a multiset => 'a set" where

  "set_of M = {x. x :# M}"

instantiation multiset :: (type) "{plus, minus, zero, size}" 

begin

definition

  union_def[code func del]:

  "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"

definition

  diff_def: "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"

definition

  Zero_multiset_def [simp]: "0 = {#}"

definition

  size_def[code func del]: "size M = setsum (count M) (set_of M)"

instance ..

end

definition

  multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where

  "multiset_inter A B = A - (A - B)"

text {* Multiset Enumeration *}

syntax

  "_multiset" :: "args => 'a multiset"    ("{#(_)#}")

translations

  "{#x, xs#}" == "{#x#} + {#xs#}"

  "{#x#}" == "CONST single x"

text {*

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

*}

lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"

by (simp add: multiset_def)

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

by (simp add: multiset_def)

lemma union_preserves_multiset:

  "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"

apply (simp add: multiset_def)

apply (drule (1) finite_UnI)

apply (simp del: finite_Un add: Un_def)

done

lemma diff_preserves_multiset:

  "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"

apply (simp add: multiset_def)

apply (rule finite_subset)

 apply auto

done

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, auto)

done

lemmas in_multiset = const0_in_multiset only1_in_multiset

  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset

subsection {* Algebraic properties *}

subsubsection {* Union *}

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

by (simp add: union_def Mempty_def in_multiset)

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

by (simp add: union_def add_ac in_multiset)

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

by (simp add: union_def add_ac in_multiset)

lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"

proof -

  have "M + (N + K) = (N + K) + M" by (rule union_commute)

  also have "\<dots> = N + (K + M)" by (rule union_assoc)

  also have "K + M = M + K" by (rule union_commute)

  finally show ?thesis .

qed

lemmas union_ac = union_assoc union_commute union_lcomm

instance multiset :: (type) comm_monoid_add

proof

  fix a b c :: "'a multiset"

  show "(a + b) + c = a + (b + c)" by (rule union_assoc)

  show "a + b = b + a" by (rule union_commute)

  show "0 + a = a" by simp

qed

subsubsection {* Difference *}

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

by (simp add: Mempty_def diff_def in_multiset)

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

by (simp add: union_def diff_def in_multiset)

lemma diff_cancel: "A - A = {#}"

by (simp add: diff_def Mempty_def)

subsubsection {* Count of elements *}

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

by (simp add: count_def Mempty_def in_multiset)

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

by (simp add: count_def single_def in_multiset)

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

by (simp add: count_def union_def in_multiset)

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

by (simp add: count_def diff_def in_multiset)

lemma count_MCollect [simp]:

  "count {# x:#M. P x #} a = (if P a then count M a else 0)"

by (simp add: count_def MCollect_def in_multiset)

subsubsection {* Set of elements *}

lemma set_of_empty [simp]: "set_of {#} = {}"

by (simp add: set_of_def)

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

by (simp add: set_of_def)

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

by (auto simp add: set_of_def)

lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"

by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])

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

by (auto simp add: set_of_def)

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

by (auto simp add: set_of_def)

subsubsection {* Size *}

lemma size_empty [simp,code func]: "size {#} = 0"

by (simp add: size_def)

lemma size_single [simp,code func]: "size {#b#} = 1"

by (simp add: size_def)

lemma finite_set_of [iff]: "finite (set_of M)"

using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)

lemma setsum_count_Int:

  "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"

apply (induct rule: finite_induct)

 apply simp

apply (simp add: Int_insert_left set_of_def)

done

lemma size_union[simp,code func]: "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, simp)

apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)

apply (subst Int_commute)

apply (simp (no_asm_simp) add: setsum_count_Int)

done

lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"

apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)

apply (simp add: set_of_def count_def in_multiset expand_fun_eq)

done

lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"

by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)

lemma 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 *}

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

by (simp add: count_def expand_fun_eq)

lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"

by (simp add: single_def Mempty_def in_multiset expand_fun_eq)

lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"

by (auto simp add: single_def in_multiset expand_fun_eq)

lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"

by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)

lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"

by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)

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

by (simp add: union_def in_multiset expand_fun_eq)

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

by (simp add: union_def in_multiset expand_fun_eq)

lemma union_is_single:

  "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"

apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)

apply blast

done

lemma single_is_union:

  "({#a#} = M + N) \<longleftrightarrow> ({#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 in_multiset expand_fun_eq)

apply (blast dest: sym)

done

lemma add_eq_conv_diff:

  "(M + {#a#} = N + {#b#}) =

   (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"

using [[simproc del: neq]]

apply (unfold single_def union_def diff_def)

apply (simp (no_asm) add: in_multiset expand_fun_eq)

apply (rule conjI, force, safe, simp_all)

apply (simp add: eq_sym_conv)

done

declare Rep_multiset_inject [symmetric, simp del]

instance multiset :: (type) cancel_ab_semigroup_add

proof

  fix a b c :: "'a multiset"

  show "a + b = a + c \<Longrightarrow> b = c" by simp

qed

lemma insert_DiffM:

  "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"

by (clarsimp simp: multiset_eq_conv_count_eq)

lemma insert_DiffM2[simp]:

  "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"

by (clarsimp simp: multiset_eq_conv_count_eq)

lemma multi_union_self_other_eq: 

  "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"

by (induct A arbitrary: X Y) auto

lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"

by (metis single_not_empty union_empty union_left_cancel)

lemma insert_noteq_member: 

  assumes BC: "B + {#b#} = C + {#c#}"

   and bnotc: "b \<noteq> c"

  shows "c \<in># B"

proof -

  have "c \<in># C + {#c#}" by simp

  have nc: "\<not> c \<in># {#b#}" using bnotc by simp

  then have "c \<in># B + {#b#}" using BC by simp

  then show "c \<in># B" using nc by simp

qed

lemma add_eq_conv_ex:

  "(M + {#a#} = N + {#b#}) =

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

by (auto simp add: add_eq_conv_diff)

lemma empty_multiset_count:

  "(\<forall>x. count A x = 0) = (A = {#})"

by (metis count_empty multiset_eq_conv_count_eq)

subsubsection {* Intersection *}

lemma multiset_inter_count:

  "count (A #\<inter> B) x = min (count A x) (count B x)"

by (simp add: multiset_inter_def min_def)

lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"

by (simp add: multiset_eq_conv_count_eq multiset_inter_count

    min_max.inf_commute)

lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"

by (simp add: multiset_eq_conv_count_eq multiset_inter_count

    min_max.inf_assoc)

lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"

by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)

lemmas multiset_inter_ac =

  multiset_inter_commute

  multiset_inter_assoc

  multiset_inter_left_commute

lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"

by (simp add: multiset_eq_conv_count_eq multiset_inter_count)

lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"

apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def

    split: split_if_asm)

apply clarsimp

apply (erule_tac x = a in allE)

apply auto

done

subsubsection {* Comprehension (filter) *}

lemma MCollect_empty[simp, code func]: "MCollect {#} P = {#}"

by (simp add: MCollect_def Mempty_def Abs_multiset_inject

    in_multiset expand_fun_eq)

lemma MCollect_single[simp, code func]:

  "MCollect {#x#} P = (if P x then {#x#} else {#})"

by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject

    in_multiset expand_fun_eq)

lemma MCollect_union[simp, code func]:

  "MCollect (M+N) f = MCollect M f + MCollect N f"

by (simp add: MCollect_def union_def Abs_multiset_inject

    in_multiset expand_fun_eq)

subsection {* Induction and case splits *}

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 (induct rule: finite_induct)

 apply auto

apply (drule_tac a = a in mk_disjoint_insert, auto)

done

lemma rep_multiset_induct_aux:

assumes 1: "P (\<lambda>a. (0::nat))"

  and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"

shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"

apply (unfold multiset_def)

apply (induct_tac n, simp, clarify)

 apply (subgoal_tac "f = (\<lambda>a.0)")

  apply simp

  apply (rule 1)

 apply (rule ext, force, clarify)

apply (frule setsum_SucD, clarify)

apply (rename_tac a)

apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")

 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 2 [unfolded multiset_def], blast)

apply (erule allE, erule impE, erule_tac [2] mp, blast)

apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)

apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")

 prefer 2

 apply blast

apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")

 prefer 2

 apply blast

apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)

done

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"

using rep_multiset_induct_aux by blast

theorem multiset_induct [case_names empty add, induct type: multiset]:

assumes empty: "P {#}"

  and add: "!!M x. P M ==> P (M + {#x#})"

shows "P M"

proof -

  note defns = union_def single_def Mempty_def

  show ?thesis

    apply (rule Rep_multiset_inverse [THEN subst])

    apply (rule Rep_multiset [THEN rep_multiset_induct])

     apply (rule empty [unfolded defns])

    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 (drule add [unfolded defns, simplified])

    apply(simp add:in_multiset)

    done

qed

lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"

by (induct M) auto

lemma multiset_cases [cases type, case_names empty add]:

assumes em:  "M = {#} \<Longrightarrow> P"

assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"

shows "P"

proof (cases "M = {#}")

  assume "M = {#}" then show ?thesis using em by simp

next

  assume "M \<noteq> {#}"

  then obtain M' m where "M = M' + {#m#}" 

    by (blast dest: multi_nonempty_split)

  then show ?thesis using add by simp

qed

lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"

apply (cases M)

 apply simp

apply (rule_tac x="M - {#x#}" in exI, simp)

done

lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"

apply (subst multiset_eq_conv_count_eq)

apply auto

done

declare multiset_typedef [simp del]

lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"

by (cases "B = {#}") (auto dest: multi_member_split)

subsection {* Orderings *}

subsubsection {* Well-foundedness *}

definition

  mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where

  [code func del]:"mult1 r =

    {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>

      (\<forall>b. b :# K --> (b, a) \<in> r)}"

definition

  mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where

  "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)"

  (is "_ \<Longrightarrow> ?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}"

  then have "\<exists>a' M0' K.

      M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp

  then show "?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'#}"

    then have "M0 = M0' \<and> a = a' \<or>

        (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"

      by (simp only: add_eq_conv_ex)

    then show ?thesis

    proof (elim disjE conjE exE)

      assume "M0 = M0'" "a = a'"

      with N r have "?r K a \<and> N = M0 + K" by simp

      then have ?case2 .. then show ?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 then show ?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: "!!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"

      then have "((\<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)

      then show "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" ..

        from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..

        then show "N \<in> ?W" by (simp only: N)

      next

        fix K

        assume N: "N = M0 + K"

        assume "\<forall>b. b :# K --> (b, a) \<in> r"

        then have "M0 + K \<in> ?W"

        proof (induct K)

          case empty

          from M0 show "M0 + {#} \<in> ?W" by simp

        next

          case (add K x)

          from add.prems have "(x, a) \<in> r" by simp

          with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast

          moreover from add have "M0 + K \<in> ?W" by simp

          ultimately have "(M0 + K) + {#x#} \<in> ?W" ..

          then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)

        qed

        then show "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 r: "!!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"

        then show "M + {#a#} \<in> ?W"

          by (rule acc_induct) (rule tedious_reasoning [OF _ r])

      qed

    qed

    from this and `M \<in> ?W` show "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)"

unfolding mult_def by (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#})"

by (simp add: multiset_eq_conv_count_eq)

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, clarify)

 apply (rule_tac x = M0 in exI, simp, 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, 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#}"

by (simp add: multiset_eq_conv_count_eq)

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, 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, auto)

apply (frule size_eq_Suc_imp_eq_union, clarify)

apply (rename_tac "J'", simp)

apply (erule notE, auto)

apply (case_tac "J' = {#}")

 apply (simp add: mult_def)

 apply (rule r_into_trancl)

 apply (simp add: mult1_def set_of_def, 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, 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

lemma 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"

using one_step_implies_mult_aux by blast

subsubsection {* Partial-order properties *}

instantiation multiset :: (order) order

begin

definition

  less_multiset_def [code func del]: "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"

definition

  le_multiset_def [code func del]: "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"

lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"

unfolding trans_def by (blast intro: order_less_trans)

text {*

 \medskip Irreflexivity.

*}

lemma mult_irrefl_aux:

  "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"

by (induct rule: finite_induct) (auto intro: order_less_trans)

lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"

apply (unfold less_multiset_def, auto)

apply (drule trans_base_order [THEN mult_implies_one_step], 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"

using insert mult_less_not_refl by fast

text {* Transitivity. *}

theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"

unfolding less_multiset_def mult_def by (blast intro: trancl_trans)

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, assumption)

done

theorem mult_less_asym:

  "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"

using mult_less_not_sym by blast

theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"

unfolding le_multiset_def by auto

text {* Anti-symmetry. *}

theorem mult_le_antisym:

  "M <= N ==> N <= M ==> M = (N::'a::order multiset)"

unfolding le_multiset_def by (blast dest: mult_less_not_sym)

text {* Transitivity. *}

theorem mult_le_trans:

  "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"

unfolding le_multiset_def by (blast intro: mult_less_trans)

theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"

unfolding le_multiset_def by auto

instance

apply intro_classes

   apply (rule mult_less_le)

  apply (rule mult_le_refl)

 apply (erule mult_le_trans, assumption)

apply (erule mult_le_antisym, assumption)

done

end

subsubsection {* Monotonicity of multiset union *}

lemma 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

lemma union_less_mono:

  "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"

by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)

lemma union_le_mono:

  "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"

unfolding le_multiset_def

by (blast intro: union_less_mono union_less_mono1 union_less_mono2)

lemma 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

done

lemma union_upper1: "A <= A + (B::'a::order multiset)"

proof -

  have "A + {#} <= A + B" by (blast intro: union_le_mono)

  then show ?thesis by simp

qed

lemma union_upper2: "B <= A + (B::'a::order multiset)"

by (subst union_commute) (rule union_upper1)

instance multiset :: (order) pordered_ab_semigroup_add

apply intro_classes

apply (erule union_le_mono[OF mult_le_refl])

done

subsection {* Link with lists *}

primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where

  "multiset_of [] = {#}" |

  "multiset_of (a # x) = multiset_of x + {# a #}"

lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"

by (induct x) auto

lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"

by (induct x) auto

lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"

by (induct x) auto

lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"

by (induct xs) auto

lemma multiset_of_append [simp]:

  "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"

by (induct xs arbitrary: ys) (auto simp: union_ac)

lemma surj_multiset_of: "surj multiset_of"

apply (unfold surj_def)

apply (rule allI)

apply (rule_tac M = y in multiset_induct)

 apply auto

apply (rule_tac x = "x # xa" in exI)

apply auto

done

lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"

by (induct x) auto

lemma distinct_count_atmost_1:

  "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"

apply (induct x, simp, rule iffI, simp_all)

apply (rule conjI)

apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)

apply (erule_tac x = a in allE, simp, clarify)

apply (erule_tac x = aa in allE, simp)

done

lemma multiset_of_eq_setD:

  "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"

by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)

lemma set_eq_iff_multiset_of_eq_distinct:

  "distinct x \<Longrightarrow> distinct y \<Longrightarrow>

    (set x = set y) = (multiset_of x = multiset_of y)"

by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)

lemma set_eq_iff_multiset_of_remdups_eq:

   "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"

apply (rule iffI)

apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])

apply (drule distinct_remdups [THEN distinct_remdups

      [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])

apply simp

done

lemma multiset_of_compl_union [simp]:

  "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"

by (induct xs) (auto simp: union_ac)

lemma count_filter:

  "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"

by (induct xs) auto

lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"

apply (induct ls arbitrary: i)

 apply simp

apply (case_tac i)

 apply auto

done

lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"

by (induct xs) (auto simp add: multiset_eq_conv_count_eq)

lemma multiset_of_eq_length:

assumes "multiset_of xs = multiset_of ys"

shows "length xs = length ys"

using assms

proof (induct arbitrary: ys rule: length_induct)

  case (1 xs ys)

  show ?case

  proof (cases xs)

    case Nil with "1.prems" show ?thesis by simp

  next

    case (Cons x xs')

    note xCons = Cons

    show ?thesis

    proof (cases ys)

      case Nil

      with "1.prems" Cons show ?thesis by simp

    next

      case (Cons y ys')

      have x_in_ys: "x = y \<or> x \<in> set ys'"

      proof (cases "x = y")

	case True then show ?thesis ..

      next

	case False

	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp

	with False show ?thesis by (simp add: mem_set_multiset_eq)

      qed

      from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>

	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast

      from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"

	apply -

	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)

	apply fastsimp

	done

      with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp

      from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto

      with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)

    qed

  qed

qed

text {*

  This lemma shows which properties suffice to show that a function

  @{text "f"} with @{text "f xs = ys"} behaves like sort.

*}

lemma properties_for_sort:

  "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"

proof (induct xs arbitrary: ys)

  case Nil then show ?case by simp

next

  case (Cons x xs)

  then have "x \<in> set ys"

    by (auto simp add:  mem_set_multiset_eq intro!: ccontr)

  with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case

    by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)

qed

subsection {* Pointwise ordering induced by count *}

definition

  mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where

  [code func del]: "(A \<le># B) = (\<forall>a. count A a \<le> count B a)"

definition

  mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where