(*  Title:      HOL/Library/Multiset.thy

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

*)

header {* (Finite) multisets *}

theory Multiset

imports Main

begin

subsection {* The type of multisets *}

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

  morphisms count Abs_multiset

proof

  show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp

qed

lemmas multiset_typedef = Abs_multiset_inverse count_inverse count

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

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

notation (xsymbols)

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

lemma multiset_ext_iff:

  "M = N \<longleftrightarrow> (\<forall>a. count M a = count N a)"

  by (simp only: count_inject [symmetric] ext_iff)

lemma multiset_ext:

  "(\<And>x. count A x = count B x) \<Longrightarrow> A = B"

  using multiset_ext_iff by auto

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 n else 0) \<in> multiset"

  by (simp add: multiset_def)

lemma union_preserves_multiset:

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

  by (simp add: multiset_def)

lemma diff_preserves_multiset:

  assumes "M \<in> multiset"

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

proof -

  have "{x. N x < M x} \<subseteq> {x. 0 < M x}"

    by auto

  with assms show ?thesis

    by (auto simp add: multiset_def intro: finite_subset)

qed

lemma MCollect_preserves_multiset:

  assumes "M \<in> multiset"

  shows "(\<lambda>x. if P x then M x else 0) \<in> multiset"

proof -

  have "{x. (P x \<longrightarrow> 0 < M x) \<and> P x} \<subseteq> {x. 0 < M x}"

    by auto

  with assms show ?thesis

    by (auto simp add: multiset_def intro: finite_subset)

qed

lemmas in_multiset = const0_in_multiset only1_in_multiset

  union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset

subsection {* Representing multisets *}

text {* Multiset comprehension *}

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

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

syntax

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

translations

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

text {* Multiset enumeration *}

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

begin

definition Mempty_def:

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

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

  "Mempty \<equiv> 0"

definition union_def:

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

instance ..

end

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

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

syntax

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

translations

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

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

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

  by (simp add: Mempty_def in_multiset multiset_typedef)

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

  by (simp add: single_def in_multiset multiset_typedef)

subsection {* Basic operations *}

subsubsection {* Union *}

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

  by (simp add: union_def in_multiset multiset_typedef)

instance multiset :: (type) cancel_comm_monoid_add proof

qed (simp_all add: multiset_ext_iff)

subsubsection {* Difference *}

instantiation multiset :: (type) minus

begin

definition diff_def:

  "M - N = Abs_multiset (\<lambda>a. count M a - count N a)"

instance ..

end

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

  by (simp add: diff_def in_multiset multiset_typedef)

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

by(simp add: multiset_ext_iff)

lemma diff_cancel[simp]: "A - A = {#}"

by (rule multiset_ext) simp

lemma diff_union_cancelR [simp]: "M + N - N = (M::'a multiset)"

by(simp add: multiset_ext_iff)

lemma diff_union_cancelL [simp]: "N + M - N = (M::'a multiset)"

by(simp add: multiset_ext_iff)

lemma insert_DiffM:

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

  by (clarsimp simp: multiset_ext_iff)

lemma insert_DiffM2 [simp]:

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

  by (clarsimp simp: multiset_ext_iff)

lemma diff_right_commute:

  "(M::'a multiset) - N - Q = M - Q - N"

  by (auto simp add: multiset_ext_iff)

lemma diff_add:

  "(M::'a multiset) - (N + Q) = M - N - Q"

by (simp add: multiset_ext_iff)

lemma diff_union_swap:

  "a \<noteq> b \<Longrightarrow> M - {#a#} + {#b#} = M + {#b#} - {#a#}"

  by (auto simp add: multiset_ext_iff)

lemma diff_union_single_conv:

  "a \<in># J \<Longrightarrow> I + J - {#a#} = I + (J - {#a#})"

  by (simp add: multiset_ext_iff)

subsubsection {* Equality of multisets *}

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

  by (simp add: multiset_ext_iff)

lemma single_eq_single [simp]: "{#a#} = {#b#} \<longleftrightarrow> a = b"

  by (auto simp add: multiset_ext_iff)

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

  by (auto simp add: multiset_ext_iff)

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

  by (auto simp add: multiset_ext_iff)

lemma multi_self_add_other_not_self [simp]: "M = M + {#x#} \<longleftrightarrow> False"

  by (auto simp add: multiset_ext_iff)

lemma diff_single_trivial:

  "\<not> x \<in># M \<Longrightarrow> M - {#x#} = M"

  by (auto simp add: multiset_ext_iff)

lemma diff_single_eq_union:

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

  by auto

lemma union_single_eq_diff:

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

  by (auto dest: sym)

lemma union_single_eq_member:

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

  by auto

lemma union_is_single:

  "M + N = {#a#} \<longleftrightarrow> M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#}" (is "?lhs = ?rhs")proof

  assume ?rhs then show ?lhs by auto

next

  assume ?lhs thus ?rhs

    by(simp add: multiset_ext_iff split:if_splits) (metis add_is_1)

qed

lemma single_is_union:

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

  by (auto simp add: eq_commute [of "{#a#}" "M + N"] union_is_single)

lemma add_eq_conv_diff:

  "M + {#a#} = N + {#b#} \<longleftrightarrow> M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#}"  (is "?lhs = ?rhs")

(* shorter: by (simp add: multiset_ext_iff) fastsimp *)

proof

  assume ?rhs then show ?lhs

  by (auto simp add: add_assoc add_commute [of "{#b#}"])

    (drule sym, simp add: add_assoc [symmetric])

next

  assume ?lhs

  show ?rhs

  proof (cases "a = b")

    case True with `?lhs` show ?thesis by simp

  next

    case False

    from `?lhs` have "a \<in># N + {#b#}" by (rule union_single_eq_member)

    with False have "a \<in># N" by auto

    moreover from `?lhs` have "M = N + {#b#} - {#a#}" by (rule union_single_eq_diff)

    moreover note False

    ultimately show ?thesis by (auto simp add: diff_right_commute [of _ "{#a#}"] diff_union_swap)

  qed

qed

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)

subsubsection {* Pointwise ordering induced by count *}

instantiation multiset :: (type) ordered_ab_semigroup_add_imp_le

begin

definition less_eq_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where

  mset_le_def: "A \<le> B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"

definition less_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" where

  mset_less_def: "(A::'a multiset) < B \<longleftrightarrow> A \<le> B \<and> A \<noteq> B"

instance proof

qed (auto simp add: mset_le_def mset_less_def multiset_ext_iff intro: order_trans antisym)

end

lemma mset_less_eqI:

  "(\<And>x. count A x \<le> count B x) \<Longrightarrow> A \<le> B"

  by (simp add: mset_le_def)

lemma mset_le_exists_conv:

  "(A::'a multiset) \<le> B \<longleftrightarrow> (\<exists>C. B = A + C)"

apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)

apply (auto intro: multiset_ext_iff [THEN iffD2])

done

lemma mset_le_mono_add_right_cancel [simp]:

  "(A::'a multiset) + C \<le> B + C \<longleftrightarrow> A \<le> B"

  by (fact add_le_cancel_right)

lemma mset_le_mono_add_left_cancel [simp]:

  "C + (A::'a multiset) \<le> C + B \<longleftrightarrow> A \<le> B"

  by (fact add_le_cancel_left)

lemma mset_le_mono_add:

  "(A::'a multiset) \<le> B \<Longrightarrow> C \<le> D \<Longrightarrow> A + C \<le> B + D"

  by (fact add_mono)

lemma mset_le_add_left [simp]:

  "(A::'a multiset) \<le> A + B"

  unfolding mset_le_def by auto

lemma mset_le_add_right [simp]:

  "B \<le> (A::'a multiset) + B"

  unfolding mset_le_def by auto

lemma mset_le_single:

  "a :# B \<Longrightarrow> {#a#} \<le> B"

  by (simp add: mset_le_def)

lemma multiset_diff_union_assoc:

  "C \<le> B \<Longrightarrow> (A::'a multiset) + B - C = A + (B - C)"

  by (simp add: multiset_ext_iff mset_le_def)

lemma mset_le_multiset_union_diff_commute:

  "B \<le> A \<Longrightarrow> (A::'a multiset) - B + C = A + C - B"

by (simp add: multiset_ext_iff mset_le_def)

lemma diff_le_self[simp]: "(M::'a multiset) - N \<le> M"

by(simp add: mset_le_def)

lemma mset_lessD: "A < B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"

apply (clarsimp simp: mset_le_def mset_less_def)

apply (erule_tac x=x in allE)

apply auto

done

lemma mset_leD: "A \<le> B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"

apply (clarsimp simp: mset_le_def mset_less_def)

apply (erule_tac x = x in allE)

apply auto

done

lemma mset_less_insertD: "(A + {#x#} < B) \<Longrightarrow> (x \<in># B \<and> A < B)"

apply (rule conjI)

 apply (simp add: mset_lessD)

apply (clarsimp simp: mset_le_def mset_less_def)

apply safe

 apply (erule_tac x = a in allE)

 apply (auto split: split_if_asm)

done

lemma mset_le_insertD: "(A + {#x#} \<le> B) \<Longrightarrow> (x \<in># B \<and> A \<le> B)"

apply (rule conjI)

 apply (simp add: mset_leD)

apply (force simp: mset_le_def mset_less_def split: split_if_asm)

done

lemma mset_less_of_empty[simp]: "A < {#} \<longleftrightarrow> False"

  by (auto simp add: mset_less_def mset_le_def multiset_ext_iff)

lemma multi_psub_of_add_self[simp]: "A < A + {#x#}"

  by (auto simp: mset_le_def mset_less_def)

lemma multi_psub_self[simp]: "(A::'a multiset) < A = False"

  by simp

lemma mset_less_add_bothsides:

  "T + {#x#} < S + {#x#} \<Longrightarrow> T < S"

  by (fact add_less_imp_less_right)

lemma mset_less_empty_nonempty:

  "{#} < S \<longleftrightarrow> S \<noteq> {#}"

  by (auto simp: mset_le_def mset_less_def)

lemma mset_less_diff_self:

  "c \<in># B \<Longrightarrow> B - {#c#} < B"

  by (auto simp: mset_le_def mset_less_def multiset_ext_iff)

subsubsection {* Intersection *}

instantiation multiset :: (type) semilattice_inf

begin

definition inf_multiset :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" where

  multiset_inter_def: "inf_multiset A B = A - (A - B)"

instance proof -

  have aux: "\<And>m n q :: nat. m \<le> n \<Longrightarrow> m \<le> q \<Longrightarrow> m \<le> n - (n - q)" by arith

  show "OFCLASS('a multiset, semilattice_inf_class)" proof

  qed (auto simp add: multiset_inter_def mset_le_def aux)

qed

end

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

  "multiset_inter \<equiv> inf"

lemma multiset_inter_count:

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

  by (simp add: multiset_inter_def multiset_typedef)

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

  by (rule multiset_ext) (auto simp add: multiset_inter_count)

lemma multiset_union_diff_commute:

  assumes "B #\<inter> C = {#}"

  shows "A + B - C = A - C + B"

proof (rule multiset_ext)

  fix x

  from assms have "min (count B x) (count C x) = 0"

    by (auto simp add: multiset_inter_count multiset_ext_iff)

  then have "count B x = 0 \<or> count C x = 0"

    by auto

  then show "count (A + B - C) x = count (A - C + B) x"

    by auto

qed

subsubsection {* Comprehension (filter) *}

lemma count_MCollect [simp]:

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

  by (simp add: MCollect_def in_multiset multiset_typedef)

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

  by (rule multiset_ext) simp

lemma MCollect_single [simp]:

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

  by (rule multiset_ext) simp

lemma MCollect_union [simp]:

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

  by (rule multiset_ext) simp

subsubsection {* Set of elements *}

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

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

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 add: set_of_def multiset_ext_iff)

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)

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

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

subsubsection {* Size *}

instantiation multiset :: (type) size

begin

definition size_def:

  "size M = setsum (count M) (set_of M)"

instance ..

end

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

by (simp add: size_def)

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

by (simp add: size_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]: "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 = {#})"

by (auto simp add: size_def multiset_ext_iff)

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

lemma size_eq_Suc_imp_eq_union:

  assumes "size M = Suc n"

  shows "\<exists>a N. M = N + {#a#}"

proof -

  from assms obtain a where "a \<in># M"

    by (erule size_eq_Suc_imp_elem [THEN exE])

  then have "M = M - {#a#} + {#a#}" by simp

  then show ?thesis by blast

qed

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

  note add' = add [unfolded defns, simplified]

  have aux: "\<And>a::'a. count (Abs_multiset (\<lambda>b. if b = a then 1 else 0)) =

    (\<lambda>b. if b = a then 1 else 0)" by (simp add: Abs_multiset_inverse in_multiset) 

  show ?thesis

    apply (rule count_inverse [THEN subst])

    apply (rule count [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: ext_iff)

    apply (erule ssubst)

    apply (erule Abs_multiset_inverse [THEN subst])

    apply (drule add')

    apply (simp add: aux)

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

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

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

apply (subst multiset_ext_iff)

apply auto

done

lemma mset_less_size: "(A::'a multiset) < B \<Longrightarrow> size A < size B"

proof (induct A arbitrary: B)

  case (empty M)

  then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)

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

    by (blast dest: multi_nonempty_split)

  then show ?case by simp

next

  case (add S x T)

  have IH: "\<And>B. S < B \<Longrightarrow> size S < size B" by fact

  have SxsubT: "S + {#x#} < T" by fact

  then have "x \<in># T" and "S < T" by (auto dest: mset_less_insertD)

  then obtain T' where T: "T = T' + {#x#}" 

    by (blast dest: multi_member_split)

  then have "S < T'" using SxsubT 

    by (blast intro: mset_less_add_bothsides)

  then have "size S < size T'" using IH by simp

  then show ?case using T by simp

qed

subsubsection {* Strong induction and subset induction for multisets *}

text {* Well-foundedness of proper subset operator: *}

text {* proper multiset subset *}

definition

  mset_less_rel :: "('a multiset * 'a multiset) set" where

  "mset_less_rel = {(A,B). A < B}"

lemma multiset_add_sub_el_shuffle: 

  assumes "c \<in># B" and "b \<noteq> c" 

  shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"

proof -

  from `c \<in># B` obtain A where B: "B = A + {#c#}" 

    by (blast dest: multi_member_split)

  have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp

  then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 

    by (simp add: add_ac)

  then show ?thesis using B by simp

qed

lemma wf_mset_less_rel: "wf mset_less_rel"

apply (unfold mset_less_rel_def)

apply (rule wf_measure [THEN wf_subset, where f1=size])

apply (clarsimp simp: measure_def inv_image_def mset_less_size)

done

text {* The induction rules: *}

lemma full_multiset_induct [case_names less]:

assumes ih: "\<And>B. \<forall>(A::'a multiset). A < B \<longrightarrow> P A \<Longrightarrow> P B"

shows "P B"

apply (rule wf_mset_less_rel [THEN wf_induct])

apply (rule ih, auto simp: mset_less_rel_def)

done

lemma multi_subset_induct [consumes 2, case_names empty add]:

assumes "F \<le> A"

  and empty: "P {#}"

  and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"

shows "P F"

proof -

  from `F \<le> A`

  show ?thesis

  proof (induct F)

    show "P {#}" by fact

  next

    fix x F

    assume P: "F \<le> A \<Longrightarrow> P F" and i: "F + {#x#} \<le> A"

    show "P (F + {#x#})"

    proof (rule insert)

      from i show "x \<in># A" by (auto dest: mset_le_insertD)

      from i have "F \<le> A" by (auto dest: mset_le_insertD)

      with P show "P F" .

    qed

  qed

qed

subsection {* Alternative representations *}

subsubsection {* Lists *}

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

  "multiset_of [] = {#}" |

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

lemma in_multiset_in_set:

  "x \<in># multiset_of xs \<longleftrightarrow> x \<in> set xs"

  by (induct xs) simp_all

lemma count_multiset_of:

  "count (multiset_of xs) x = length (filter (\<lambda>y. x = y) xs)"

  by (induct xs) simp_all

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: add_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_ext_iff 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_ext_iff 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: add_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[simp]:

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

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

lemma multiset_of_eq_length:

  assumes "multiset_of xs = multiset_of ys"

  shows "length xs = length ys"

using assms proof (induct xs arbitrary: ys)

  case Nil then show ?case by simp

next

  case (Cons x xs)

  then have "x \<in># multiset_of ys" by (simp add: union_single_eq_member)

  then have "x \<in> set ys" by (simp add: in_multiset_in_set)

  from Cons.prems [symmetric] have "multiset_of xs = multiset_of (remove1 x ys)"

    by simp

  with Cons.hyps have "length xs = length (remove1 x ys)" .

  with `x \<in> set ys` show ?case

    by (auto simp add: length_remove1 dest: length_pos_if_in_set)

qed

lemma (in linorder) multiset_of_insort [simp]:

  "multiset_of (insort x xs) = {#x#} + multiset_of xs"

  by (induct xs) (simp_all add: ac_simps)

lemma (in linorder) multiset_of_sort [simp]:

  "multiset_of (sort xs) = multiset_of xs"

  by (induct xs) (simp_all add: ac_simps)

text {*

  This lemma shows which properties suffice to show that a function

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

*}

lemma (in linorder) 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

lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le> multiset_of xs"

  by (induct xs) (auto intro: order_trans)

lemma multiset_of_update:

  "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"

proof (induct ls arbitrary: i)

  case Nil then show ?case by simp

next

  case (Cons x xs)

  show ?case

  proof (cases i)

    case 0 then show ?thesis by simp

  next

    case (Suc i')

    with Cons show ?thesis

      apply simp

      apply (subst add_assoc)

      apply (subst add_commute [of "{#v#}" "{#x#}"])

      apply (subst add_assoc [symmetric])

      apply simp

      apply (rule mset_le_multiset_union_diff_commute)

      apply (simp add: mset_le_single nth_mem_multiset_of)

      done

  qed

qed

lemma multiset_of_swap:

  "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>

    multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"

  by (cases "i = j") (simp_all add: multiset_of_update nth_mem_multiset_of)

subsubsection {* Association lists -- including rudimentary code generation *}

definition count_of :: "('a \<times> nat) list \<Rightarrow> 'a \<Rightarrow> nat" where

  "count_of xs x = (case map_of xs x of None \<Rightarrow> 0 | Some n \<Rightarrow> n)"

lemma count_of_multiset:

  "count_of xs \<in> multiset"

proof -

  let ?A = "{x::'a. 0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)}"

  have "?A \<subseteq> dom (map_of xs)"

  proof

    fix x

    assume "x \<in> ?A"

    then have "0 < (case map_of xs x of None \<Rightarrow> 0\<Colon>nat | Some (n\<Colon>nat) \<Rightarrow> n)" by simp

    then have "map_of xs x \<noteq> None" by (cases "map_of xs x") auto

    then show "x \<in> dom (map_of xs)" by auto

  qed

  with finite_dom_map_of [of xs] have "finite ?A"

    by (auto intro: finite_subset)

  then show ?thesis

    by (simp add: count_of_def ext_iff multiset_def)

qed

lemma count_simps [simp]:

  "count_of [] = (\<lambda>_. 0)"

  "count_of ((x, n) # xs) = (\<lambda>y. if x = y then n else count_of xs y)"

  by (simp_all add: count_of_def ext_iff)

lemma count_of_empty:

  "x \<notin> fst ` set xs \<Longrightarrow> count_of xs x = 0"

  by (induct xs) (simp_all add: count_of_def)

lemma count_of_filter:

  "count_of (filter (P \<circ> fst) xs) x = (if P x then count_of xs x else 0)"

  by (induct xs) auto

definition Bag :: "('a \<times> nat) list \<Rightarrow> 'a multiset" where

  "Bag xs = Abs_multiset (count_of xs)"

code_datatype Bag

lemma count_Bag [simp, code]:

  "count (Bag xs) = count_of xs"

  by (simp add: Bag_def count_of_multiset Abs_multiset_inverse)

lemma Mempty_Bag [code]:

  "{#} = Bag []"

  by (simp add: multiset_ext_iff)

lemma single_Bag [code]:

  "{#x#} = Bag [(x, 1)]"

  by (simp add: multiset_ext_iff)

lemma MCollect_Bag [code]:

  "MCollect (Bag xs) P = Bag (filter (P \<circ> fst) xs)"

  by (simp add: multiset_ext_iff count_of_filter)

lemma mset_less_eq_Bag [code]:

  "Bag xs \<le> A \<longleftrightarrow> (\<forall>(x, n) \<in> set xs. count_of xs x \<le> count A x)"

    (is "?lhs \<longleftrightarrow> ?rhs")

proof

  assume ?lhs then show ?rhs

    by (auto simp add: mset_le_def count_Bag)

next

  assume ?rhs

  show ?lhs

  proof (rule mset_less_eqI)

    fix x

    from `?rhs` have "count_of xs x \<le> count A x"

      by (cases "x \<in> fst ` set xs") (auto simp add: count_of_empty)

    then show "count (Bag xs) x \<le> count A x"

      by (simp add: mset_le_def count_Bag)

  qed

qed

instantiation multiset :: (equal) equal

begin

definition

  "HOL.equal A B \<longleftrightarrow> (A::'a multiset) \<le> B \<and> B \<le> A"

instance proof

qed (simp add: equal_multiset_def eq_iff)

end

lemma [code nbe]:

  "HOL.equal (A :: 'a::equal multiset) A \<longleftrightarrow> True"

  by (fact equal_refl)

definition (in term_syntax)

  bagify :: "('a\<Colon>typerep \<times> nat) list \<times> (unit \<Rightarrow> Code_Evaluation.term)

    \<Rightarrow> 'a multiset \<times> (unit \<Rightarrow> Code_Evaluation.term)" where

  [code_unfold]: "bagify xs = Code_Evaluation.valtermify Bag {\<cdot>} xs"

notation fcomp (infixl "\<circ>>" 60)

notation scomp (infixl "\<circ>\<rightarrow>" 60)

instantiation multiset :: (random) random

begin

definition

  "Quickcheck.random i = Quickcheck.random i \<circ>\<rightarrow> (\<lambda>xs. Pair (bagify xs))"

instance ..

end

no_notation fcomp (infixl "\<circ>>" 60)

no_notation scomp (infixl "\<circ>\<rightarrow>" 60)

hide_const (open) bagify

subsection {* The multiset order *}

subsubsection {* Well-foundedness *}

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

  "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"