wenzelm@10249: (*  Title:      HOL/Library/Multiset.thy
wenzelm@10249:     ID:         $Id$
paulson@15072:     Author:     Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker
wenzelm@10249: *)
wenzelm@10249: 
wenzelm@14706: header {* Multisets *}
wenzelm@10249: 
nipkow@15131: theory Multiset
haftmann@27487: imports Plain "~~/src/HOL/List"
nipkow@15131: begin
wenzelm@10249: 
wenzelm@10249: subsection {* The type of multisets *}
wenzelm@10249: 
nipkow@25162: typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}"
wenzelm@10249: proof
nipkow@11464:   show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp
wenzelm@10249: qed
wenzelm@10249: 
wenzelm@10249: lemmas multiset_typedef [simp] =
wenzelm@10277:     Abs_multiset_inverse Rep_multiset_inverse Rep_multiset
wenzelm@10277:   and [simp] = Rep_multiset_inject [symmetric]
wenzelm@10249: 
haftmann@28708: definition Mempty :: "'a multiset"  ("{#}") where
haftmann@28708:   [code del]: "{#} = Abs_multiset (\<lambda>a. 0)"
wenzelm@10249: 
haftmann@28708: definition single :: "'a => 'a multiset" where
haftmann@28708:   [code del]: "single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)"
wenzelm@10249: 
haftmann@28708: definition count :: "'a multiset => 'a => nat" where
wenzelm@19086:   "count = Rep_multiset"
wenzelm@10249: 
haftmann@28708: definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where
wenzelm@19086:   "MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)"
wenzelm@19086: 
haftmann@28708: abbreviation Melem :: "'a => 'a multiset => bool"  ("(_/ :# _)" [50, 51] 50) where
kleing@25610:   "a :# M == 0 < count M a"
kleing@25610: 
wenzelm@26145: notation (xsymbols)
wenzelm@26145:   Melem (infix "\<in>#" 50)
wenzelm@10249: 
wenzelm@10249: syntax
nipkow@26033:   "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset"    ("(1{# _ :# _./ _#})")
wenzelm@10249: translations
nipkow@26033:   "{#x :# M. P#}" == "CONST MCollect M (\<lambda>x. P)"
wenzelm@10249: 
haftmann@28708: definition set_of :: "'a multiset => 'a set" where
wenzelm@19086:   "set_of M = {x. x :# M}"
wenzelm@10249: 
haftmann@25571: instantiation multiset :: (type) "{plus, minus, zero, size}" 
haftmann@25571: begin
haftmann@25571: 
haftmann@28708: definition union_def [code del]:
wenzelm@26145:   "M + N = Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)"
haftmann@25571: 
haftmann@28708: definition diff_def [code del]:
haftmann@28708:   "M - N = Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)"
haftmann@25571: 
haftmann@28708: definition Zero_multiset_def [simp]:
haftmann@28708:   "0 = {#}"
haftmann@25571: 
haftmann@28708: definition size_def:
haftmann@28708:   "size M = setsum (count M) (set_of M)"
haftmann@25571: 
haftmann@25571: instance ..
haftmann@25571: 
haftmann@25571: end
wenzelm@10249: 
wenzelm@19086: definition
wenzelm@21404:   multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset"  (infixl "#\<inter>" 70) where
wenzelm@19086:   "multiset_inter A B = A - (A - B)"
kleing@15869: 
wenzelm@26145: text {* Multiset Enumeration *}
wenzelm@26145: syntax
wenzelm@26176:   "_multiset" :: "args => 'a multiset"    ("{#(_)#}")
nipkow@25507: translations
nipkow@25507:   "{#x, xs#}" == "{#x#} + {#xs#}"
nipkow@25507:   "{#x#}" == "CONST single x"
nipkow@25507: 
wenzelm@10249: 
wenzelm@10249: text {*
wenzelm@10249:  \medskip Preservation of the representing set @{term multiset}.
wenzelm@10249: *}
wenzelm@10249: 
nipkow@26016: lemma const0_in_multiset: "(\<lambda>a. 0) \<in> multiset"
nipkow@26178: by (simp add: multiset_def)
wenzelm@10249: 
nipkow@26016: lemma only1_in_multiset: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset"
nipkow@26178: by (simp add: multiset_def)
wenzelm@10249: 
nipkow@26016: lemma union_preserves_multiset:
nipkow@26178:   "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
nipkow@26178: apply (simp add: multiset_def)
nipkow@26178: apply (drule (1) finite_UnI)
nipkow@26178: apply (simp del: finite_Un add: Un_def)
nipkow@26178: done
wenzelm@10249: 
nipkow@26016: lemma diff_preserves_multiset:
nipkow@26178:   "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"
nipkow@26178: apply (simp add: multiset_def)
nipkow@26178: apply (rule finite_subset)
nipkow@26178:  apply auto
nipkow@26178: done
wenzelm@10249: 
nipkow@26016: lemma MCollect_preserves_multiset:
nipkow@26178:   "M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset"
nipkow@26178: apply (simp add: multiset_def)
nipkow@26178: apply (rule finite_subset, auto)
nipkow@26178: done
wenzelm@10249: 
nipkow@26016: lemmas in_multiset = const0_in_multiset only1_in_multiset
nipkow@26016:   union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset
nipkow@26016: 
wenzelm@26145: 
nipkow@26016: subsection {* Algebraic properties *}
wenzelm@10249: 
wenzelm@10249: subsubsection {* Union *}
wenzelm@10249: 
wenzelm@17161: lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M"
nipkow@26178: by (simp add: union_def Mempty_def in_multiset)
wenzelm@10249: 
wenzelm@17161: lemma union_commute: "M + N = N + (M::'a multiset)"
nipkow@26178: by (simp add: union_def add_ac in_multiset)
wenzelm@17161: 
wenzelm@17161: lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))"
nipkow@26178: by (simp add: union_def add_ac in_multiset)
wenzelm@10249: 
wenzelm@17161: lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))"
wenzelm@17161: proof -
nipkow@26178:   have "M + (N + K) = (N + K) + M" by (rule union_commute)
nipkow@26178:   also have "\<dots> = N + (K + M)" by (rule union_assoc)
nipkow@26178:   also have "K + M = M + K" by (rule union_commute)
wenzelm@17161:   finally show ?thesis .
wenzelm@17161: qed
wenzelm@10249: 
wenzelm@17161: lemmas union_ac = union_assoc union_commute union_lcomm
wenzelm@10249: 
obua@14738: instance multiset :: (type) comm_monoid_add
wenzelm@17200: proof
obua@14722:   fix a b c :: "'a multiset"
obua@14722:   show "(a + b) + c = a + (b + c)" by (rule union_assoc)
obua@14722:   show "a + b = b + a" by (rule union_commute)
obua@14722:   show "0 + a = a" by simp
obua@14722: qed
wenzelm@10277: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Difference *}
wenzelm@10249: 
wenzelm@17161: lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}"
nipkow@26178: by (simp add: Mempty_def diff_def in_multiset)
wenzelm@10249: 
wenzelm@17161: lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M"
nipkow@26178: by (simp add: union_def diff_def in_multiset)
wenzelm@10249: 
bulwahn@26143: lemma diff_cancel: "A - A = {#}"
nipkow@26178: by (simp add: diff_def Mempty_def)
bulwahn@26143: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Count of elements *}
wenzelm@10249: 
wenzelm@17161: lemma count_empty [simp]: "count {#} a = 0"
nipkow@26178: by (simp add: count_def Mempty_def in_multiset)
wenzelm@10249: 
wenzelm@17161: lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)"
nipkow@26178: by (simp add: count_def single_def in_multiset)
wenzelm@10249: 
wenzelm@17161: lemma count_union [simp]: "count (M + N) a = count M a + count N a"
nipkow@26178: by (simp add: count_def union_def in_multiset)
wenzelm@10249: 
wenzelm@17161: lemma count_diff [simp]: "count (M - N) a = count M a - count N a"
nipkow@26178: by (simp add: count_def diff_def in_multiset)
nipkow@26016: 
nipkow@26016: lemma count_MCollect [simp]:
nipkow@26178:   "count {# x:#M. P x #} a = (if P a then count M a else 0)"
nipkow@26178: by (simp add: count_def MCollect_def in_multiset)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Set of elements *}
wenzelm@10249: 
wenzelm@17161: lemma set_of_empty [simp]: "set_of {#} = {}"
nipkow@26178: by (simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_single [simp]: "set_of {#b#} = {b}"
nipkow@26178: by (simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N"
nipkow@26178: by (auto simp add: set_of_def)
wenzelm@10249: 
wenzelm@17161: lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})"
berghofe@26818: by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"])
wenzelm@10249: 
wenzelm@17161: lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)"
nipkow@26178: by (auto simp add: set_of_def)
nipkow@26016: 
nipkow@26033: lemma set_of_MCollect [simp]: "set_of {# x:#M. P x #} = set_of M \<inter> {x. P x}"
nipkow@26178: by (auto simp add: set_of_def)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Size *}
wenzelm@10249: 
haftmann@28708: lemma size_empty [simp]: "size {#} = 0"
nipkow@26178: by (simp add: size_def)
wenzelm@10249: 
haftmann@28708: lemma size_single [simp]: "size {#b#} = 1"
nipkow@26178: by (simp add: size_def)
wenzelm@10249: 
wenzelm@17161: lemma finite_set_of [iff]: "finite (set_of M)"
nipkow@26178: using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def)
wenzelm@10249: 
wenzelm@17161: lemma setsum_count_Int:
nipkow@26178:   "finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A"
nipkow@26178: apply (induct rule: finite_induct)
nipkow@26178:  apply simp
nipkow@26178: apply (simp add: Int_insert_left set_of_def)
nipkow@26178: done
wenzelm@10249: 
haftmann@28708: lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N"
nipkow@26178: apply (unfold size_def)
nipkow@26178: apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)")
nipkow@26178:  prefer 2
nipkow@26178:  apply (rule ext, simp)
nipkow@26178: apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int)
nipkow@26178: apply (subst Int_commute)
nipkow@26178: apply (simp (no_asm_simp) add: setsum_count_Int)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})"
nipkow@26178: apply (unfold size_def Mempty_def count_def, auto simp: in_multiset)
nipkow@26178: apply (simp add: set_of_def count_def in_multiset expand_fun_eq)
nipkow@26178: done
nipkow@26016: 
nipkow@26016: lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)"
nipkow@26178: by (metis gr0I gr_implies_not0 size_empty size_eq_0_iff_empty)
wenzelm@10249: 
wenzelm@17161: lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M"
nipkow@26178: apply (unfold size_def)
nipkow@26178: apply (drule setsum_SucD)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
wenzelm@26145: 
wenzelm@10249: subsubsection {* Equality of multisets *}
wenzelm@10249: 
wenzelm@17161: lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)"
nipkow@26178: by (simp add: count_def expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}"
nipkow@26178: by (simp add: single_def Mempty_def in_multiset expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)"
nipkow@26178: by (auto simp add: single_def in_multiset expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})"
nipkow@26178: by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})"
nipkow@26178: by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))"
nipkow@26178: by (simp add: union_def in_multiset expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))"
nipkow@26178: by (simp add: union_def in_multiset expand_fun_eq)
wenzelm@10249: 
wenzelm@17161: lemma union_is_single:
nipkow@26178:   "(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})"
nipkow@26178: apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq)
nipkow@26178: apply blast
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma single_is_union:
nipkow@26178:   "({#a#} = M + N) \<longleftrightarrow> ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)"
nipkow@26178: apply (unfold Mempty_def single_def union_def)
nipkow@26178: apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq)
nipkow@26178: apply (blast dest: sym)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma add_eq_conv_diff:
wenzelm@10249:   "(M + {#a#} = N + {#b#}) =
paulson@15072:    (M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})"
nipkow@26178: using [[simproc del: neq]]
nipkow@26178: apply (unfold single_def union_def diff_def)
nipkow@26178: apply (simp (no_asm) add: in_multiset expand_fun_eq)
nipkow@26178: apply (rule conjI, force, safe, simp_all)
nipkow@26178: apply (simp add: eq_sym_conv)
nipkow@26178: done
wenzelm@10249: 
kleing@15869: declare Rep_multiset_inject [symmetric, simp del]
kleing@15869: 
nipkow@23611: instance multiset :: (type) cancel_ab_semigroup_add
nipkow@23611: proof
nipkow@23611:   fix a b c :: "'a multiset"
nipkow@23611:   show "a + b = a + c \<Longrightarrow> b = c" by simp
nipkow@23611: qed
kleing@15869: 
kleing@25610: lemma insert_DiffM:
kleing@25610:   "x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M"
nipkow@26178: by (clarsimp simp: multiset_eq_conv_count_eq)
kleing@25610: 
kleing@25610: lemma insert_DiffM2[simp]:
kleing@25610:   "x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M"
nipkow@26178: by (clarsimp simp: multiset_eq_conv_count_eq)
kleing@25610: 
kleing@25610: lemma multi_union_self_other_eq: 
kleing@25610:   "(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y"
nipkow@26178: by (induct A arbitrary: X Y) auto
kleing@25610: 
kleing@25610: lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False"
nipkow@26178: by (metis single_not_empty union_empty union_left_cancel)
kleing@25610: 
kleing@25610: lemma insert_noteq_member: 
kleing@25610:   assumes BC: "B + {#b#} = C + {#c#}"
kleing@25610:    and bnotc: "b \<noteq> c"
kleing@25610:   shows "c \<in># B"
kleing@25610: proof -
kleing@25610:   have "c \<in># C + {#c#}" by simp
kleing@25610:   have nc: "\<not> c \<in># {#b#}" using bnotc by simp
wenzelm@26145:   then have "c \<in># B + {#b#}" using BC by simp
wenzelm@26145:   then show "c \<in># B" using nc by simp
kleing@25610: qed
kleing@25610: 
kleing@25610: 
nipkow@26016: lemma add_eq_conv_ex:
nipkow@26016:   "(M + {#a#} = N + {#b#}) =
nipkow@26016:     (M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))"
nipkow@26178: by (auto simp add: add_eq_conv_diff)
nipkow@26016: 
nipkow@26016: 
nipkow@26016: lemma empty_multiset_count:
nipkow@26016:   "(\<forall>x. count A x = 0) = (A = {#})"
nipkow@26178: by (metis count_empty multiset_eq_conv_count_eq)
nipkow@26016: 
nipkow@26016: 
kleing@15869: subsubsection {* Intersection *}
kleing@15869: 
kleing@15869: lemma multiset_inter_count:
nipkow@26178:   "count (A #\<inter> B) x = min (count A x) (count B x)"
nipkow@26178: by (simp add: multiset_inter_def min_def)
kleing@15869: 
kleing@15869: lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214:     min_max.inf_commute)
kleing@15869: 
kleing@15869: lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq multiset_inter_count
haftmann@21214:     min_max.inf_assoc)
kleing@15869: 
kleing@15869: lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def)
kleing@15869: 
wenzelm@17161: lemmas multiset_inter_ac =
wenzelm@17161:   multiset_inter_commute
wenzelm@17161:   multiset_inter_assoc
wenzelm@17161:   multiset_inter_left_commute
kleing@15869: 
bulwahn@26143: lemma multiset_inter_single: "a \<noteq> b \<Longrightarrow> {#a#} #\<inter> {#b#} = {#}"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq multiset_inter_count)
bulwahn@26143: 
kleing@15869: lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B"
nipkow@26178: apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def
wenzelm@17161:     split: split_if_asm)
nipkow@26178: apply clarsimp
nipkow@26178: apply (erule_tac x = a in allE)
nipkow@26178: apply auto
nipkow@26178: done
kleing@15869: 
wenzelm@10249: 
nipkow@26016: subsubsection {* Comprehension (filter) *}
nipkow@26016: 
haftmann@28708: lemma MCollect_empty [simp]: "MCollect {#} P = {#}"
nipkow@26178: by (simp add: MCollect_def Mempty_def Abs_multiset_inject
wenzelm@26145:     in_multiset expand_fun_eq)
nipkow@26016: 
haftmann@28708: lemma MCollect_single [simp]:
nipkow@26178:   "MCollect {#x#} P = (if P x then {#x#} else {#})"
nipkow@26178: by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject
wenzelm@26145:     in_multiset expand_fun_eq)
nipkow@26016: 
haftmann@28708: lemma MCollect_union [simp]:
nipkow@26016:   "MCollect (M+N) f = MCollect M f + MCollect N f"
nipkow@26178: by (simp add: MCollect_def union_def Abs_multiset_inject
wenzelm@26145:     in_multiset expand_fun_eq)
nipkow@26016: 
nipkow@26016: 
nipkow@26016: subsection {* Induction and case splits *}
wenzelm@10249: 
wenzelm@10249: lemma setsum_decr:
wenzelm@11701:   "finite F ==> (0::nat) < f a ==>
paulson@15072:     setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)"
nipkow@26178: apply (induct rule: finite_induct)
nipkow@26178:  apply auto
nipkow@26178: apply (drule_tac a = a in mk_disjoint_insert, auto)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10313: lemma rep_multiset_induct_aux:
nipkow@26178: assumes 1: "P (\<lambda>a. (0::nat))"
nipkow@26178:   and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))"
nipkow@26178: shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f"
nipkow@26178: apply (unfold multiset_def)
nipkow@26178: apply (induct_tac n, simp, clarify)
nipkow@26178:  apply (subgoal_tac "f = (\<lambda>a.0)")
nipkow@26178:   apply simp
nipkow@26178:   apply (rule 1)
nipkow@26178:  apply (rule ext, force, clarify)
nipkow@26178: apply (frule setsum_SucD, clarify)
nipkow@26178: apply (rename_tac a)
nipkow@26178: apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}")
nipkow@26178:  prefer 2
nipkow@26178:  apply (rule finite_subset)
nipkow@26178:   prefer 2
nipkow@26178:   apply assumption
nipkow@26178:  apply simp
nipkow@26178:  apply blast
nipkow@26178: apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)")
nipkow@26178:  prefer 2
nipkow@26178:  apply (rule ext)
nipkow@26178:  apply (simp (no_asm_simp))
nipkow@26178:  apply (erule ssubst, rule 2 [unfolded multiset_def], blast)
nipkow@26178: apply (erule allE, erule impE, erule_tac [2] mp, blast)
nipkow@26178: apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def)
nipkow@26178: apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}")
nipkow@26178:  prefer 2
nipkow@26178:  apply blast
nipkow@26178: apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}")
nipkow@26178:  prefer 2
nipkow@26178:  apply blast
nipkow@26178: apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10313: theorem rep_multiset_induct:
nipkow@11464:   "f \<in> multiset ==> P (\<lambda>a. 0) ==>
wenzelm@11701:     (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f"
nipkow@26178: using rep_multiset_induct_aux by blast
wenzelm@10249: 
wenzelm@18258: theorem multiset_induct [case_names empty add, induct type: multiset]:
nipkow@26178: assumes empty: "P {#}"
nipkow@26178:   and add: "!!M x. P M ==> P (M + {#x#})"
nipkow@26178: shows "P M"
wenzelm@10249: proof -
wenzelm@10249:   note defns = union_def single_def Mempty_def
wenzelm@10249:   show ?thesis
wenzelm@10249:     apply (rule Rep_multiset_inverse [THEN subst])
wenzelm@10313:     apply (rule Rep_multiset [THEN rep_multiset_induct])
wenzelm@18258:      apply (rule empty [unfolded defns])
paulson@15072:     apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))")
wenzelm@10249:      prefer 2
wenzelm@10249:      apply (simp add: expand_fun_eq)
wenzelm@10249:     apply (erule ssubst)
wenzelm@17200:     apply (erule Abs_multiset_inverse [THEN subst])
nipkow@26016:     apply (drule add [unfolded defns, simplified])
nipkow@26016:     apply(simp add:in_multiset)
wenzelm@10249:     done
wenzelm@10249: qed
wenzelm@10249: 
kleing@25610: lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}"
nipkow@26178: by (induct M) auto
kleing@25610: 
kleing@25610: lemma multiset_cases [cases type, case_names empty add]:
nipkow@26178: assumes em:  "M = {#} \<Longrightarrow> P"
nipkow@26178: assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P"
nipkow@26178: shows "P"
kleing@25610: proof (cases "M = {#}")
wenzelm@26145:   assume "M = {#}" then show ?thesis using em by simp
kleing@25610: next
kleing@25610:   assume "M \<noteq> {#}"
kleing@25610:   then obtain M' m where "M = M' + {#m#}" 
kleing@25610:     by (blast dest: multi_nonempty_split)
wenzelm@26145:   then show ?thesis using add by simp
kleing@25610: qed
kleing@25610: 
kleing@25610: lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}"
nipkow@26178: apply (cases M)
nipkow@26178:  apply simp
nipkow@26178: apply (rule_tac x="M - {#x#}" in exI, simp)
nipkow@26178: done
kleing@25610: 
nipkow@26033: lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. \<not> P x #}"
nipkow@26178: apply (subst multiset_eq_conv_count_eq)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
kleing@15869: declare multiset_typedef [simp del]
wenzelm@10249: 
kleing@25610: lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B"
nipkow@26178: by (cases "B = {#}") (auto dest: multi_member_split)
wenzelm@26145: 
wenzelm@17161: 
nipkow@26016: subsection {* Orderings *}
wenzelm@10249: 
wenzelm@10249: subsubsection {* Well-foundedness *}
wenzelm@10249: 
haftmann@28708: definition mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
haftmann@28708:   [code del]: "mult1 r = {(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and>
berghofe@23751:       (\<forall>b. b :# K --> (b, a) \<in> r)}"
wenzelm@10249: 
haftmann@28708: definition mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where
berghofe@23751:   "mult r = (mult1 r)\<^sup>+"
wenzelm@10249: 
berghofe@23751: lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r"
nipkow@26178: by (simp add: mult1_def)
wenzelm@10249: 
berghofe@23751: lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==>
berghofe@23751:     (\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or>
berghofe@23751:     (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)"
wenzelm@19582:   (is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2")
wenzelm@10249: proof (unfold mult1_def)
berghofe@23751:   let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r"
nipkow@11464:   let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a"
berghofe@23751:   let ?case1 = "?case1 {(N, M). ?R N M}"
wenzelm@10249: 
berghofe@23751:   assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}"
wenzelm@18258:   then have "\<exists>a' M0' K.
nipkow@11464:       M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp
wenzelm@18258:   then show "?case1 \<or> ?case2"
wenzelm@10249:   proof (elim exE conjE)
wenzelm@10249:     fix a' M0' K
wenzelm@10249:     assume N: "N = M0' + K" and r: "?r K a'"
wenzelm@10249:     assume "M0 + {#a#} = M0' + {#a'#}"
wenzelm@18258:     then have "M0 = M0' \<and> a = a' \<or>
nipkow@11464:         (\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})"
wenzelm@10249:       by (simp only: add_eq_conv_ex)
wenzelm@18258:     then show ?thesis
wenzelm@10249:     proof (elim disjE conjE exE)
wenzelm@10249:       assume "M0 = M0'" "a = a'"
nipkow@11464:       with N r have "?r K a \<and> N = M0 + K" by simp
wenzelm@18258:       then have ?case2 .. then show ?thesis ..
wenzelm@10249:     next
wenzelm@10249:       fix K'
wenzelm@10249:       assume "M0' = K' + {#a#}"
wenzelm@10249:       with N have n: "N = K' + K + {#a#}" by (simp add: union_ac)
wenzelm@10249: 
wenzelm@10249:       assume "M0 = K' + {#a'#}"
wenzelm@10249:       with r have "?R (K' + K) M0" by blast
wenzelm@18258:       with n have ?case1 by simp then show ?thesis ..
wenzelm@10249:     qed
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
berghofe@23751: lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)"
wenzelm@10249: proof
wenzelm@10249:   let ?R = "mult1 r"
wenzelm@10249:   let ?W = "acc ?R"
wenzelm@10249:   {
wenzelm@10249:     fix M M0 a
berghofe@23751:     assume M0: "M0 \<in> ?W"
berghofe@23751:       and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751:       and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W"
berghofe@23751:     have "M0 + {#a#} \<in> ?W"
berghofe@23751:     proof (rule accI [of "M0 + {#a#}"])
wenzelm@10249:       fix N
berghofe@23751:       assume "(N, M0 + {#a#}) \<in> ?R"
berghofe@23751:       then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or>
berghofe@23751:           (\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))"
wenzelm@10249:         by (rule less_add)
berghofe@23751:       then show "N \<in> ?W"
wenzelm@10249:       proof (elim exE disjE conjE)
berghofe@23751:         fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}"
berghofe@23751:         from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" ..
berghofe@23751:         from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" ..
berghofe@23751:         then show "N \<in> ?W" by (simp only: N)
wenzelm@10249:       next
wenzelm@10249:         fix K
wenzelm@10249:         assume N: "N = M0 + K"
berghofe@23751:         assume "\<forall>b. b :# K --> (b, a) \<in> r"
berghofe@23751:         then have "M0 + K \<in> ?W"
wenzelm@10249:         proof (induct K)
wenzelm@18730:           case empty
berghofe@23751:           from M0 show "M0 + {#} \<in> ?W" by simp
wenzelm@18730:         next
wenzelm@18730:           case (add K x)
berghofe@23751:           from add.prems have "(x, a) \<in> r" by simp
berghofe@23751:           with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast
berghofe@23751:           moreover from add have "M0 + K \<in> ?W" by simp
berghofe@23751:           ultimately have "(M0 + K) + {#x#} \<in> ?W" ..
berghofe@23751:           then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc)
wenzelm@10249:         qed
berghofe@23751:         then show "N \<in> ?W" by (simp only: N)
wenzelm@10249:       qed
wenzelm@10249:     qed
wenzelm@10249:   } note tedious_reasoning = this
wenzelm@10249: 
berghofe@23751:   assume wf: "wf r"
wenzelm@10249:   fix M
berghofe@23751:   show "M \<in> ?W"
wenzelm@10249:   proof (induct M)
berghofe@23751:     show "{#} \<in> ?W"
wenzelm@10249:     proof (rule accI)
berghofe@23751:       fix b assume "(b, {#}) \<in> ?R"
berghofe@23751:       with not_less_empty show "b \<in> ?W" by contradiction
wenzelm@10249:     qed
wenzelm@10249: 
berghofe@23751:     fix M a assume "M \<in> ?W"
berghofe@23751:     from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:     proof induct
wenzelm@10249:       fix a
berghofe@23751:       assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)"
berghofe@23751:       show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W"
wenzelm@10249:       proof
berghofe@23751:         fix M assume "M \<in> ?W"
berghofe@23751:         then show "M + {#a#} \<in> ?W"
wenzelm@23373:           by (rule acc_induct) (rule tedious_reasoning [OF _ r])
wenzelm@10249:       qed
wenzelm@10249:     qed
berghofe@23751:     from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" ..
wenzelm@10249:   qed
wenzelm@10249: qed
wenzelm@10249: 
berghofe@23751: theorem wf_mult1: "wf r ==> wf (mult1 r)"
nipkow@26178: by (rule acc_wfI) (rule all_accessible)
wenzelm@10249: 
berghofe@23751: theorem wf_mult: "wf r ==> wf (mult r)"
nipkow@26178: unfolding mult_def by (rule wf_trancl) (rule wf_mult1)
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Closure-free presentation *}
wenzelm@10249: 
wenzelm@10249: (*Badly needed: a linear arithmetic procedure for multisets*)
wenzelm@10249: 
wenzelm@10249: lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249: 
wenzelm@10249: text {* One direction. *}
wenzelm@10249: 
wenzelm@10249: lemma mult_implies_one_step:
berghofe@23751:   "trans r ==> (M, N) \<in> mult r ==>
nipkow@11464:     \<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and>
berghofe@23751:     (\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)"
nipkow@26178: apply (unfold mult_def mult1_def set_of_def)
nipkow@26178: apply (erule converse_trancl_induct, clarify)
nipkow@26178:  apply (rule_tac x = M0 in exI, simp, clarify)
nipkow@26178: apply (case_tac "a :# K")
nipkow@26178:  apply (rule_tac x = I in exI)
nipkow@26178:  apply (simp (no_asm))
nipkow@26178:  apply (rule_tac x = "(K - {#a#}) + Ka" in exI)
nipkow@26178:  apply (simp (no_asm_simp) add: union_assoc [symmetric])
nipkow@26178:  apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong)
nipkow@26178:  apply (simp add: diff_union_single_conv)
nipkow@26178:  apply (simp (no_asm_use) add: trans_def)
nipkow@26178:  apply blast
nipkow@26178: apply (subgoal_tac "a :# I")
nipkow@26178:  apply (rule_tac x = "I - {#a#}" in exI)
nipkow@26178:  apply (rule_tac x = "J + {#a#}" in exI)
nipkow@26178:  apply (rule_tac x = "K + Ka" in exI)
nipkow@26178:  apply (rule conjI)
nipkow@26178:   apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow@26178:  apply (rule conjI)
nipkow@26178:   apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp)
nipkow@26178:   apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split)
nipkow@26178:  apply (simp (no_asm_use) add: trans_def)
nipkow@26178:  apply blast
nipkow@26178: apply (subgoal_tac "a :# (M0 + {#a#})")
nipkow@26178:  apply simp
nipkow@26178: apply (simp (no_asm))
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq)
wenzelm@10249: 
nipkow@11464: lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}"
nipkow@26178: apply (erule size_eq_Suc_imp_elem [THEN exE])
nipkow@26178: apply (drule elem_imp_eq_diff_union, auto)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma one_step_implies_mult_aux:
berghofe@23751:   "trans r ==>
berghofe@23751:     \<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))
berghofe@23751:       --> (I + K, I + J) \<in> mult r"
nipkow@26178: apply (induct_tac n, auto)
nipkow@26178: apply (frule size_eq_Suc_imp_eq_union, clarify)
nipkow@26178: apply (rename_tac "J'", simp)
nipkow@26178: apply (erule notE, auto)
nipkow@26178: apply (case_tac "J' = {#}")
nipkow@26178:  apply (simp add: mult_def)
nipkow@26178:  apply (rule r_into_trancl)
nipkow@26178:  apply (simp add: mult1_def set_of_def, blast)
nipkow@26178: txt {* Now we know @{term "J' \<noteq> {#}"}. *}
nipkow@26178: apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition)
nipkow@26178: apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp)
nipkow@26178: apply (erule ssubst)
nipkow@26178: apply (simp add: Ball_def, auto)
nipkow@26178: apply (subgoal_tac
nipkow@26178:   "((I + {# x :# K. (x, a) \<in> r #}) + {# x :# K. (x, a) \<notin> r #},
nipkow@26178:     (I + {# x :# K. (x, a) \<in> r #}) + J') \<in> mult r")
nipkow@26178:  prefer 2
nipkow@26178:  apply force
nipkow@26178: apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def)
nipkow@26178: apply (erule trancl_trans)
nipkow@26178: apply (rule r_into_trancl)
nipkow@26178: apply (simp add: mult1_def set_of_def)
nipkow@26178: apply (rule_tac x = a in exI)
nipkow@26178: apply (rule_tac x = "I + J'" in exI)
nipkow@26178: apply (simp add: union_ac)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma one_step_implies_mult:
berghofe@23751:   "trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r
berghofe@23751:     ==> (I + K, I + J) \<in> mult r"
nipkow@26178: using one_step_implies_mult_aux by blast
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Partial-order properties *}
wenzelm@10249: 
haftmann@26567: instantiation multiset :: (order) order
haftmann@26567: begin
wenzelm@10249: 
haftmann@28708: definition less_multiset_def [code del]:
haftmann@28708:   "M' < M \<longleftrightarrow> (M', M) \<in> mult {(x', x). x' < x}"
haftmann@26567: 
haftmann@28708: definition le_multiset_def [code del]:
haftmann@28708:   "M' <= M \<longleftrightarrow> M' = M \<or> M' < (M::'a multiset)"
wenzelm@10249: 
berghofe@23751: lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}"
nipkow@26178: unfolding trans_def by (blast intro: order_less_trans)
wenzelm@10249: 
wenzelm@10249: text {*
wenzelm@10249:  \medskip Irreflexivity.
wenzelm@10249: *}
wenzelm@10249: 
wenzelm@10249: lemma mult_irrefl_aux:
nipkow@26178:   "finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}"
nipkow@26178: by (induct rule: finite_induct) (auto intro: order_less_trans)
wenzelm@10249: 
wenzelm@17161: lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)"
nipkow@26178: apply (unfold less_multiset_def, auto)
nipkow@26178: apply (drule trans_base_order [THEN mult_implies_one_step], auto)
nipkow@26178: apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]])
nipkow@26178: apply (simp add: set_of_eq_empty_iff)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R"
nipkow@26178: using insert mult_less_not_refl by fast
wenzelm@10249: 
wenzelm@10249: 
wenzelm@10249: text {* Transitivity. *}
wenzelm@10249: 
wenzelm@10249: theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)"
nipkow@26178: unfolding less_multiset_def mult_def by (blast intro: trancl_trans)
wenzelm@10249: 
wenzelm@10249: text {* Asymmetry. *}
wenzelm@10249: 
nipkow@11464: theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)"
nipkow@26178: apply auto
nipkow@26178: apply (rule mult_less_not_refl [THEN notE])
nipkow@26178: apply (erule mult_less_trans, assumption)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: theorem mult_less_asym:
nipkow@26178:   "M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P"
nipkow@26178: using mult_less_not_sym by blast
wenzelm@10249: 
wenzelm@10249: theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)"
nipkow@26178: unfolding le_multiset_def by auto
wenzelm@10249: 
wenzelm@10249: text {* Anti-symmetry. *}
wenzelm@10249: 
wenzelm@10249: theorem mult_le_antisym:
nipkow@26178:   "M <= N ==> N <= M ==> M = (N::'a::order multiset)"
nipkow@26178: unfolding le_multiset_def by (blast dest: mult_less_not_sym)
wenzelm@10249: 
wenzelm@10249: text {* Transitivity. *}
wenzelm@10249: 
wenzelm@10249: theorem mult_le_trans:
nipkow@26178:   "K <= M ==> M <= N ==> K <= (N::'a::order multiset)"
nipkow@26178: unfolding le_multiset_def by (blast intro: mult_less_trans)
wenzelm@10249: 
wenzelm@11655: theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))"
nipkow@26178: unfolding le_multiset_def by auto
wenzelm@10249: 
haftmann@27682: instance proof
haftmann@27682: qed (auto simp add: mult_less_le dest: mult_le_antisym elim: mult_le_trans)
wenzelm@10277: 
haftmann@26567: end
haftmann@26567: 
wenzelm@10249: 
wenzelm@10249: subsubsection {* Monotonicity of multiset union *}
wenzelm@10249: 
wenzelm@17161: lemma mult1_union:
nipkow@26178:   "(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r"
nipkow@26178: apply (unfold mult1_def)
nipkow@26178: apply auto
nipkow@26178: apply (rule_tac x = a in exI)
nipkow@26178: apply (rule_tac x = "C + M0" in exI)
nipkow@26178: apply (simp add: union_assoc)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)"
nipkow@26178: apply (unfold less_multiset_def mult_def)
nipkow@26178: apply (erule trancl_induct)
nipkow@26178:  apply (blast intro: mult1_union transI order_less_trans r_into_trancl)
nipkow@26178: apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans)
nipkow@26178: done
wenzelm@10249: 
wenzelm@10249: lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)"
nipkow@26178: apply (subst union_commute [of B C])
nipkow@26178: apply (subst union_commute [of D C])
nipkow@26178: apply (erule union_less_mono2)
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma union_less_mono:
nipkow@26178:   "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)"
nipkow@26178: by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans)
wenzelm@10249: 
wenzelm@17161: lemma union_le_mono:
nipkow@26178:   "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)"
nipkow@26178: unfolding le_multiset_def
nipkow@26178: by (blast intro: union_less_mono union_less_mono1 union_less_mono2)
wenzelm@10249: 
wenzelm@17161: lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)"
nipkow@26178: apply (unfold le_multiset_def less_multiset_def)
nipkow@26178: apply (case_tac "M = {#}")
nipkow@26178:  prefer 2
nipkow@26178:  apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))")
nipkow@26178:   prefer 2
nipkow@26178:   apply (rule one_step_implies_mult)
nipkow@26178:     apply (simp only: trans_def)
nipkow@26178:     apply auto
nipkow@26178: done
wenzelm@10249: 
wenzelm@17161: lemma union_upper1: "A <= A + (B::'a::order multiset)"
paulson@15072: proof -
wenzelm@17200:   have "A + {#} <= A + B" by (blast intro: union_le_mono)
wenzelm@18258:   then show ?thesis by simp
paulson@15072: qed
paulson@15072: 
wenzelm@17161: lemma union_upper2: "B <= A + (B::'a::order multiset)"
nipkow@26178: by (subst union_commute) (rule union_upper1)
paulson@15072: 
nipkow@23611: instance multiset :: (order) pordered_ab_semigroup_add
nipkow@26178: apply intro_classes
nipkow@26178: apply (erule union_le_mono[OF mult_le_refl])
nipkow@26178: done
wenzelm@26145: 
paulson@15072: 
wenzelm@17200: subsection {* Link with lists *}
paulson@15072: 
nipkow@26016: primrec multiset_of :: "'a list \<Rightarrow> 'a multiset" where
wenzelm@26145:   "multiset_of [] = {#}" |
wenzelm@26145:   "multiset_of (a # x) = multiset_of x + {# a #}"
paulson@15072: 
paulson@15072: lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])"
nipkow@26178: by (induct x) auto
paulson@15072: 
paulson@15072: lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])"
nipkow@26178: by (induct x) auto
paulson@15072: 
paulson@15072: lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x"
nipkow@26178: by (induct x) auto
kleing@15867: 
kleing@15867: lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)"
nipkow@26178: by (induct xs) auto
paulson@15072: 
wenzelm@18258: lemma multiset_of_append [simp]:
nipkow@26178:   "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys"
nipkow@26178: by (induct xs arbitrary: ys) (auto simp: union_ac)
wenzelm@18730: 
paulson@15072: lemma surj_multiset_of: "surj multiset_of"
nipkow@26178: apply (unfold surj_def)
nipkow@26178: apply (rule allI)
nipkow@26178: apply (rule_tac M = y in multiset_induct)
nipkow@26178:  apply auto
nipkow@26178: apply (rule_tac x = "x # xa" in exI)
nipkow@26178: apply auto
nipkow@26178: done
wenzelm@10249: 
nipkow@25162: lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}"
nipkow@26178: by (induct x) auto
paulson@15072: 
wenzelm@17200: lemma distinct_count_atmost_1:
nipkow@26178:   "distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))"
nipkow@26178: apply (induct x, simp, rule iffI, simp_all)
nipkow@26178: apply (rule conjI)
nipkow@26178: apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of)
nipkow@26178: apply (erule_tac x = a in allE, simp, clarify)
nipkow@26178: apply (erule_tac x = aa in allE, simp)
nipkow@26178: done
paulson@15072: 
wenzelm@17200: lemma multiset_of_eq_setD:
kleing@15867:   "multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys"
nipkow@26178: by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0)
kleing@15867: 
wenzelm@17200: lemma set_eq_iff_multiset_of_eq_distinct:
wenzelm@26145:   "distinct x \<Longrightarrow> distinct y \<Longrightarrow>
wenzelm@26145:     (set x = set y) = (multiset_of x = multiset_of y)"
nipkow@26178: by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1)
paulson@15072: 
wenzelm@17200: lemma set_eq_iff_multiset_of_remdups_eq:
paulson@15072:    "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))"
nipkow@26178: apply (rule iffI)
nipkow@26178: apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1])
nipkow@26178: apply (drule distinct_remdups [THEN distinct_remdups
wenzelm@26145:       [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]])
nipkow@26178: apply simp
nipkow@26178: done
wenzelm@10249: 
wenzelm@18258: lemma multiset_of_compl_union [simp]:
nipkow@26178:   "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs"
nipkow@26178: by (induct xs) (auto simp: union_ac)
paulson@15072: 
wenzelm@17200: lemma count_filter:
nipkow@26178:   "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]"
nipkow@26178: by (induct xs) auto
kleing@15867: 
bulwahn@26143: lemma nth_mem_multiset_of: "i < length ls \<Longrightarrow> (ls ! i) :# multiset_of ls"
nipkow@26178: apply (induct ls arbitrary: i)
nipkow@26178:  apply simp
nipkow@26178: apply (case_tac i)
nipkow@26178:  apply auto
nipkow@26178: done
bulwahn@26143: 
bulwahn@26143: lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}"
nipkow@26178: by (induct xs) (auto simp add: multiset_eq_conv_count_eq)
bulwahn@26143: 
bulwahn@26143: lemma multiset_of_eq_length:
nipkow@26178: assumes "multiset_of xs = multiset_of ys"
nipkow@26178: shows "length xs = length ys"
nipkow@26178: using assms
bulwahn@26143: proof (induct arbitrary: ys rule: length_induct)
bulwahn@26143:   case (1 xs ys)
bulwahn@26143:   show ?case
bulwahn@26143:   proof (cases xs)
wenzelm@26145:     case Nil with "1.prems" show ?thesis by simp
bulwahn@26143:   next
bulwahn@26143:     case (Cons x xs')
bulwahn@26143:     note xCons = Cons
bulwahn@26143:     show ?thesis
bulwahn@26143:     proof (cases ys)
bulwahn@26143:       case Nil
wenzelm@26145:       with "1.prems" Cons show ?thesis by simp
bulwahn@26143:     next
bulwahn@26143:       case (Cons y ys')
bulwahn@26143:       have x_in_ys: "x = y \<or> x \<in> set ys'"
bulwahn@26143:       proof (cases "x = y")
wenzelm@26145: 	case True then show ?thesis ..
bulwahn@26143:       next
bulwahn@26143: 	case False
wenzelm@26145: 	from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp
bulwahn@26143: 	with False show ?thesis by (simp add: mem_set_multiset_eq)
bulwahn@26143:       qed
wenzelm@26145:       from "1.hyps" have IH: "length xs' < length xs \<longrightarrow>
wenzelm@26145: 	(\<forall>x. multiset_of xs' = multiset_of x \<longrightarrow> length xs' = length x)" by blast
wenzelm@26145:       from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))"
bulwahn@26143: 	apply -
bulwahn@26143: 	apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff)
bulwahn@26143: 	apply fastsimp
bulwahn@26143: 	done
wenzelm@26145:       with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp
wenzelm@26145:       from x_in_ys have "x \<noteq> y \<Longrightarrow> length ys' > 0" by auto
bulwahn@26143:       with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1)
bulwahn@26143:     qed
bulwahn@26143:   qed
bulwahn@26143: qed
bulwahn@26143: 
wenzelm@26145: text {*
wenzelm@26145:   This lemma shows which properties suffice to show that a function
wenzelm@26145:   @{text "f"} with @{text "f xs = ys"} behaves like sort.
wenzelm@26145: *}
wenzelm@26145: lemma properties_for_sort:
wenzelm@26145:   "multiset_of ys = multiset_of xs \<Longrightarrow> sorted ys \<Longrightarrow> sort xs = ys"
bulwahn@26143: proof (induct xs arbitrary: ys)
wenzelm@26145:   case Nil then show ?case by simp
bulwahn@26143: next
bulwahn@26143:   case (Cons x xs)
wenzelm@26145:   then have "x \<in> set ys"
wenzelm@26145:     by (auto simp add:  mem_set_multiset_eq intro!: ccontr)
bulwahn@26143:   with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case
bulwahn@26143:     by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1)
bulwahn@26143: qed
bulwahn@26143: 
kleing@15867: 
paulson@15072: subsection {* Pointwise ordering induced by count *}
paulson@15072: 
haftmann@28708: definition mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "\<le>#" 50) where
haftmann@28708:   [code del]: "A \<le># B \<longleftrightarrow> (\<forall>a. count A a \<le> count B a)"
wenzelm@26145: 
haftmann@28708: definition mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool"  (infix "<#" 50) where
haftmann@28708:   [code del]: "A <# B \<longleftrightarrow> A \<le># B \<and> A \<noteq> B"
kleing@25610: 
wenzelm@26145: notation mset_le  (infix "\<subseteq>#" 50)
wenzelm@26145: notation mset_less  (infix "\<subset>#" 50)
paulson@15072: 
nipkow@23611: lemma mset_le_refl[simp]: "A \<le># A"
nipkow@26178: unfolding mset_le_def by auto
paulson@15072: 
wenzelm@26145: lemma mset_le_trans: "A \<le># B \<Longrightarrow> B \<le># C \<Longrightarrow> A \<le># C"
nipkow@26178: unfolding mset_le_def by (fast intro: order_trans)
paulson@15072: 
wenzelm@26145: lemma mset_le_antisym: "A \<le># B \<Longrightarrow> B \<le># A \<Longrightarrow> A = B"
nipkow@26178: apply (unfold mset_le_def)
nipkow@26178: apply (rule multiset_eq_conv_count_eq [THEN iffD2])
nipkow@26178: apply (blast intro: order_antisym)
nipkow@26178: done
paulson@15072: 
wenzelm@26145: lemma mset_le_exists_conv: "(A \<le># B) = (\<exists>C. B = A + C)"
nipkow@26178: apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI)
nipkow@26178: apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2])
nipkow@26178: done
paulson@15072: 
nipkow@23611: lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)"
nipkow@26178: unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)"
nipkow@26178: unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D"
nipkow@26178: apply (unfold mset_le_def)
nipkow@26178: apply auto
nipkow@26178: apply (erule_tac x = a in allE)+
nipkow@26178: apply auto
nipkow@26178: done
paulson@15072: 
nipkow@23611: lemma mset_le_add_left[simp]: "A \<le># A + B"
nipkow@26178: unfolding mset_le_def by auto
paulson@15072: 
nipkow@23611: lemma mset_le_add_right[simp]: "B \<le># A + B"
nipkow@26178: unfolding mset_le_def by auto
paulson@15072: 
bulwahn@26143: lemma mset_le_single: "a :# B \<Longrightarrow> {#a#} \<le># B"
nipkow@26178: by (simp add: mset_le_def)
bulwahn@26143: 
bulwahn@26143: lemma multiset_diff_union_assoc: "C \<le># B \<Longrightarrow> A + B - C = A + (B - C)"
nipkow@26178: by (simp add: multiset_eq_conv_count_eq mset_le_def)
bulwahn@26143: 
bulwahn@26143: lemma mset_le_multiset_union_diff_commute:
nipkow@26178: assumes "B \<le># A"
nipkow@26178: shows "A - B + C = A + C - B"
bulwahn@26143: proof -
wenzelm@26145:   from mset_le_exists_conv [of "B" "A"] assms have "\<exists>D. A = B + D" ..
wenzelm@26145:   from this obtain D where "A = B + D" ..
wenzelm@26145:   then show ?thesis
wenzelm@26145:     apply simp
wenzelm@26145:     apply (subst union_commute)
wenzelm@26145:     apply (subst multiset_diff_union_assoc)
wenzelm@26145:     apply simp
wenzelm@26145:     apply (simp add: diff_cancel)
wenzelm@26145:     apply (subst union_assoc)
wenzelm@26145:     apply (subst union_commute[of "B" _])
wenzelm@26145:     apply (subst multiset_diff_union_assoc)
wenzelm@26145:     apply simp
wenzelm@26145:     apply (simp add: diff_cancel)
wenzelm@26145:     done
bulwahn@26143: qed
bulwahn@26143: 
nipkow@23611: lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs"
nipkow@26178: apply (induct xs)
nipkow@26178:  apply auto
nipkow@26178: apply (rule mset_le_trans)
nipkow@26178:  apply auto
nipkow@26178: done
nipkow@23611: 
wenzelm@26145: lemma multiset_of_update:
wenzelm@26145:   "i < length ls \<Longrightarrow> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}"
bulwahn@26143: proof (induct ls arbitrary: i)
wenzelm@26145:   case Nil then show ?case by simp
bulwahn@26143: next
bulwahn@26143:   case (Cons x xs)
bulwahn@26143:   show ?case
wenzelm@26145:   proof (cases i)
wenzelm@26145:     case 0 then show ?thesis by simp
wenzelm@26145:   next
wenzelm@26145:     case (Suc i')
wenzelm@26145:     with Cons show ?thesis
wenzelm@26145:       apply simp
wenzelm@26145:       apply (subst union_assoc)
wenzelm@26145:       apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"])
wenzelm@26145:       apply (subst union_assoc [symmetric])
wenzelm@26145:       apply simp
wenzelm@26145:       apply (rule mset_le_multiset_union_diff_commute)
wenzelm@26145:       apply (simp add: mset_le_single nth_mem_multiset_of)
wenzelm@26145:       done
bulwahn@26143:   qed
bulwahn@26143: qed
bulwahn@26143: 
wenzelm@26145: lemma multiset_of_swap:
wenzelm@26145:   "i < length ls \<Longrightarrow> j < length ls \<Longrightarrow>
wenzelm@26145:     multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls"
nipkow@26178: apply (case_tac "i = j")
nipkow@26178:  apply simp
nipkow@26178: apply (simp add: multiset_of_update)
nipkow@26178: apply (subst elem_imp_eq_diff_union[symmetric])
nipkow@26178:  apply (simp add: nth_mem_multiset_of)
nipkow@26178: apply simp
nipkow@26178: done
bulwahn@26143: 
wenzelm@26145: interpretation mset_order: order ["op \<le>#" "op <#"]
haftmann@27682: proof qed (auto intro: order.intro mset_le_refl mset_le_antisym
haftmann@27682:   mset_le_trans simp: mset_less_def)
nipkow@23611: 
nipkow@23611: interpretation mset_order_cancel_semigroup:
haftmann@28823:   pordered_cancel_ab_semigroup_add ["op +" "op \<le>#" "op <#"]
haftmann@27682: proof qed (erule mset_le_mono_add [OF mset_le_refl])
nipkow@23611: 
nipkow@23611: interpretation mset_order_semigroup_cancel:
haftmann@28823:   pordered_ab_semigroup_add_imp_le ["op +" "op \<le>#" "op <#"]
haftmann@27682: proof qed simp
paulson@15072: 
kleing@25610: 
wenzelm@26145: lemma mset_lessD: "A \<subset># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
nipkow@26178: apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178: apply (erule_tac x=x in allE)
nipkow@26178: apply auto
nipkow@26178: done
kleing@25610: 
wenzelm@26145: lemma mset_leD: "A \<subseteq># B \<Longrightarrow> x \<in># A \<Longrightarrow> x \<in># B"
nipkow@26178: apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178: apply (erule_tac x = x in allE)
nipkow@26178: apply auto
nipkow@26178: done
kleing@25610:   
wenzelm@26145: lemma mset_less_insertD: "(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)"
nipkow@26178: apply (rule conjI)
nipkow@26178:  apply (simp add: mset_lessD)
nipkow@26178: apply (clarsimp simp: mset_le_def mset_less_def)
nipkow@26178: apply safe
nipkow@26178:  apply (erule_tac x = a in allE)
nipkow@26178:  apply (auto split: split_if_asm)
nipkow@26178: done
kleing@25610: 
wenzelm@26145: lemma mset_le_insertD: "(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)"
nipkow@26178: apply (rule conjI)
nipkow@26178:  apply (simp add: mset_leD)
nipkow@26178: apply (force simp: mset_le_def mset_less_def split: split_if_asm)
nipkow@26178: done
kleing@25610: 
kleing@25610: lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" 
nipkow@26178: by (induct A) (auto simp: mset_le_def mset_less_def)
kleing@25610: 
kleing@25610: lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}"
nipkow@26178: by (auto simp: mset_le_def mset_less_def)
kleing@25610: 
kleing@25610: lemma multi_psub_self[simp]: "A \<subset># A = False"
nipkow@26178: by (auto simp: mset_le_def mset_less_def)
kleing@25610: 
kleing@25610: lemma mset_less_add_bothsides:
kleing@25610:   "T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S"
nipkow@26178: by (auto simp: mset_le_def mset_less_def)
kleing@25610: 
kleing@25610: lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})"
nipkow@26178: by (auto simp: mset_le_def mset_less_def)
kleing@25610: 
kleing@25610: lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B"
kleing@25610: proof (induct A arbitrary: B)
kleing@25610:   case (empty M)
wenzelm@26145:   then have "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty)
kleing@25610:   then obtain M' x where "M = M' + {#x#}" 
kleing@25610:     by (blast dest: multi_nonempty_split)
wenzelm@26145:   then show ?case by simp
kleing@25610: next
kleing@25610:   case (add S x T)
kleing@25610:   have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact
kleing@25610:   have SxsubT: "S + {#x#} \<subset># T" by fact
wenzelm@26145:   then have "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD)
kleing@25610:   then obtain T' where T: "T = T' + {#x#}" 
kleing@25610:     by (blast dest: multi_member_split)
wenzelm@26145:   then have "S \<subset># T'" using SxsubT 
kleing@25610:     by (blast intro: mset_less_add_bothsides)
wenzelm@26145:   then have "size S < size T'" using IH by simp
wenzelm@26145:   then show ?case using T by simp
kleing@25610: qed
kleing@25610: 
kleing@25610: lemmas mset_less_trans = mset_order.less_eq_less.less_trans
kleing@25610: 
kleing@25610: lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B"
nipkow@26178: by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq)
kleing@25610: 
wenzelm@26145: 
kleing@25610: subsection {* Strong induction and subset induction for multisets *}
kleing@25610: 
nipkow@26016: text {* Well-foundedness of proper subset operator: *}
kleing@25610: 
wenzelm@26145: text {* proper multiset subset *}
kleing@25610: definition
wenzelm@26145:   mset_less_rel :: "('a multiset * 'a multiset) set" where
wenzelm@26145:   "mset_less_rel = {(A,B). A \<subset># B}"
kleing@25610: 
kleing@25610: lemma multiset_add_sub_el_shuffle: 
wenzelm@26145:   assumes "c \<in># B" and "b \<noteq> c" 
kleing@25610:   shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}"
kleing@25610: proof -
wenzelm@26145:   from `c \<in># B` obtain A where B: "B = A + {#c#}" 
kleing@25610:     by (blast dest: multi_member_split)
kleing@25610:   have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp
wenzelm@26145:   then have "A + {#b#} = A + {#c#} + {#b#} - {#c#}" 
kleing@25610:     by (simp add: union_ac)
wenzelm@26145:   then show ?thesis using B by simp
kleing@25610: qed
kleing@25610: 
kleing@25610: lemma wf_mset_less_rel: "wf mset_less_rel"
nipkow@26178: apply (unfold mset_less_rel_def)
nipkow@26178: apply (rule wf_measure [THEN wf_subset, where f1=size])
nipkow@26178: apply (clarsimp simp: measure_def inv_image_def mset_less_size)
nipkow@26178: done
kleing@25610: 
nipkow@26016: text {* The induction rules: *}
kleing@25610: 
kleing@25610: lemma full_multiset_induct [case_names less]:
nipkow@26178: assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B"
nipkow@26178: shows "P B"
nipkow@26178: apply (rule wf_mset_less_rel [THEN wf_induct])
nipkow@26178: apply (rule ih, auto simp: mset_less_rel_def)
nipkow@26178: done
kleing@25610: 
kleing@25610: lemma multi_subset_induct [consumes 2, case_names empty add]:
nipkow@26178: assumes "F \<subseteq># A"
nipkow@26178:   and empty: "P {#}"
nipkow@26178:   and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})"
nipkow@26178: shows "P F"
kleing@25610: proof -
kleing@25610:   from `F \<subseteq># A`
kleing@25610:   show ?thesis
kleing@25610:   proof (induct F)
kleing@25610:     show "P {#}" by fact
kleing@25610:   next
kleing@25610:     fix x F
kleing@25610:     assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A"
kleing@25610:     show "P (F + {#x#})"
kleing@25610:     proof (rule insert)
kleing@25610:       from i show "x \<in># A" by (auto dest: mset_le_insertD)
wenzelm@26145:       from i have "F \<subseteq># A" by (auto dest: mset_le_insertD)
kleing@25610:       with P show "P F" .
kleing@25610:     qed
kleing@25610:   qed
kleing@25610: qed 
kleing@25610: 
nipkow@26016: text{* A consequence: Extensionality. *}
kleing@25610: 
wenzelm@26145: lemma multi_count_eq: "(\<forall>x. count A x = count B x) = (A = B)"
nipkow@26178: apply (rule iffI)
nipkow@26178:  prefer 2
nipkow@26178:  apply clarsimp 
nipkow@26178: apply (induct A arbitrary: B rule: full_multiset_induct)
nipkow@26178: apply (rename_tac C)
nipkow@26178: apply (case_tac B rule: multiset_cases)
nipkow@26178:  apply (simp add: empty_multiset_count)
nipkow@26178: apply simp
nipkow@26178: apply (case_tac "x \<in># C")
nipkow@26178:  apply (force dest: multi_member_split)
nipkow@26178: apply (erule_tac x = x in allE)
nipkow@26178: apply simp
nipkow@26178: done
kleing@25610: 
kleing@25610: lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format]
kleing@25610: 
wenzelm@26145: 
kleing@25610: subsection {* The fold combinator *}
kleing@25610: 
wenzelm@26145: text {*
wenzelm@26145:   The intended behaviour is
wenzelm@26145:   @{text "fold_mset f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
wenzelm@26145:   if @{text f} is associative-commutative. 
kleing@25610: *}
kleing@25610: 
wenzelm@26145: text {*
wenzelm@26145:   The graph of @{text "fold_mset"}, @{text "z"}: the start element,
wenzelm@26145:   @{text "f"}: folding function, @{text "A"}: the multiset, @{text
wenzelm@26145:   "y"}: the result.
wenzelm@26145: *}
kleing@25610: inductive 
kleing@25759:   fold_msetG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" 
kleing@25610:   for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" 
kleing@25610:   and z :: 'b
kleing@25610: where
kleing@25759:   emptyI [intro]:  "fold_msetG f z {#} z"
kleing@25759: | insertI [intro]: "fold_msetG f z A y \<Longrightarrow> fold_msetG f z (A + {#x#}) (f x y)"
kleing@25610: 
kleing@25759: inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x"
kleing@25759: inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" 
kleing@25610: 
kleing@25610: definition
wenzelm@26145:   fold_mset :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" where
wenzelm@26145:   "fold_mset f z A = (THE x. fold_msetG f z A x)"
kleing@25610: 
kleing@25759: lemma Diff1_fold_msetG:
wenzelm@26145:   "fold_msetG f z (A - {#x#}) y \<Longrightarrow> x \<in># A \<Longrightarrow> fold_msetG f z A (f x y)"
nipkow@26178: apply (frule_tac x = x in fold_msetG.insertI)
nipkow@26178: apply auto
nipkow@26178: done
kleing@25610: 
kleing@25759: lemma fold_msetG_nonempty: "\<exists>x. fold_msetG f z A x"
nipkow@26178: apply (induct A)
nipkow@26178:  apply blast
nipkow@26178: apply clarsimp
nipkow@26178: apply (drule_tac x = x in fold_msetG.insertI)
nipkow@26178: apply auto
nipkow@26178: done
kleing@25610: 
kleing@25759: lemma fold_mset_empty[simp]: "fold_mset f z {#} = z"
nipkow@26178: unfolding fold_mset_def by blast
kleing@25610: 
kleing@25610: locale left_commutative = 
nipkow@26178: fixes f :: "'a => 'b => 'b"
nipkow@26178: assumes left_commute: "f x (f y z) = f y (f x z)"
wenzelm@26145: begin
kleing@25610: 
wenzelm@26145: lemma fold_msetG_determ:
wenzelm@26145:   "fold_msetG f z A x \<Longrightarrow> fold_msetG f z A y \<Longrightarrow> y = x"
kleing@25610: proof (induct arbitrary: x y z rule: full_multiset_induct)
kleing@25610:   case (less M x\<^isub>1 x\<^isub>2 Z)
kleing@25610:   have IH: "\<forall>A. A \<subset># M \<longrightarrow> 
kleing@25759:     (\<forall>x x' x''. fold_msetG f x'' A x \<longrightarrow> fold_msetG f x'' A x'
kleing@25610:                \<longrightarrow> x' = x)" by fact
kleing@25759:   have Mfoldx\<^isub>1: "fold_msetG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "fold_msetG f Z M x\<^isub>2" by fact+
kleing@25610:   show ?case
kleing@25759:   proof (rule fold_msetG.cases [OF Mfoldx\<^isub>1])
kleing@25610:     assume "M = {#}" and "x\<^isub>1 = Z"
wenzelm@26145:     then show ?case using Mfoldx\<^isub>2 by auto 
kleing@25610:   next
kleing@25610:     fix B b u
kleing@25759:     assume "M = B + {#b#}" and "x\<^isub>1 = f b u" and Bu: "fold_msetG f Z B u"
wenzelm@26145:     then have MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = f b u" by auto
kleing@25610:     show ?case
kleing@25759:     proof (rule fold_msetG.cases [OF Mfoldx\<^isub>2])
kleing@25610:       assume "M = {#}" "x\<^isub>2 = Z"
wenzelm@26145:       then show ?case using Mfoldx\<^isub>1 by auto
kleing@25610:     next
kleing@25610:       fix C c v
kleing@25759:       assume "M = C + {#c#}" and "x\<^isub>2 = f c v" and Cv: "fold_msetG f Z C v"
wenzelm@26145:       then have MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = f c v" by auto
wenzelm@26145:       then have CsubM: "C \<subset># M" by simp
kleing@25610:       from MBb have BsubM: "B \<subset># M" by simp
kleing@25610:       show ?case
kleing@25610:       proof cases
kleing@25610:         assume "b=c"
kleing@25610:         then moreover have "B = C" using MBb MCc by auto
kleing@25610:         ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto
kleing@25610:       next
kleing@25610:         assume diff: "b \<noteq> c"
kleing@25610:         let ?D = "B - {#c#}"
kleing@25610:         have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff
kleing@25610:           by (auto intro: insert_noteq_member dest: sym)
kleing@25610:         have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self)
wenzelm@26145:         then have DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans)
kleing@25610:         from MBb MCc have "B + {#b#} = C + {#c#}" by blast
wenzelm@26145:         then have [simp]: "B + {#b#} - {#c#} = C"
kleing@25610:           using MBb MCc binC cinB by auto
kleing@25610:         have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}"
kleing@25610:           using MBb MCc diff binC cinB
kleing@25610:           by (auto simp: multiset_add_sub_el_shuffle)
kleing@25759:         then obtain d where Dfoldd: "fold_msetG f Z ?D d"
kleing@25759:           using fold_msetG_nonempty by iprover
wenzelm@26145:         then have "fold_msetG f Z B (f c d)" using cinB
kleing@25759:           by (rule Diff1_fold_msetG)
wenzelm@26145:         then have "f c d = u" using IH BsubM Bu by blast
kleing@25610:         moreover 
kleing@25759:         have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd
kleing@25610:           by (auto simp: multiset_add_sub_el_shuffle 
kleing@25759:             dest: fold_msetG.insertI [where x=b])
wenzelm@26145:         then have "f b d = v" using IH CsubM Cv by blast
kleing@25610:         ultimately show ?thesis using x\<^isub>1 x\<^isub>2
kleing@25610:           by (auto simp: left_commute)
kleing@25610:       qed
kleing@25610:     qed
kleing@25610:   qed
kleing@25610: qed
kleing@25610:         
wenzelm@26145: lemma fold_mset_insert_aux:
wenzelm@26145:   "(fold_msetG f z (A + {#x#}) v) =
kleing@25759:     (\<exists>y. fold_msetG f z A y \<and> v = f x y)"
nipkow@26178: apply (rule iffI)
nipkow@26178:  prefer 2
nipkow@26178:  apply blast
nipkow@26178: apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard])
nipkow@26178: apply (blast intro: fold_msetG_determ)
nipkow@26178: done
kleing@25610: 
wenzelm@26145: lemma fold_mset_equality: "fold_msetG f z A y \<Longrightarrow> fold_mset f z A = y"
nipkow@26178: unfolding fold_mset_def by (blast intro: fold_msetG_determ)
kleing@25610: 
wenzelm@26145: lemma fold_mset_insert:
nipkow@26178:   "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)"
nipkow@26178: apply (simp add: fold_mset_def fold_mset_insert_aux union_commute)  
nipkow@26178: apply (rule the_equality)
nipkow@26178:  apply (auto cong add: conj_cong 
wenzelm@26145:      simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
nipkow@26178: done
kleing@25759: 
wenzelm@26145: lemma fold_mset_insert_idem:
nipkow@26178:   "fold_mset f z (A + {#a#}) = f a (fold_mset f z A)"
nipkow@26178: apply (simp add: fold_mset_def fold_mset_insert_aux)
nipkow@26178: apply (rule the_equality)
nipkow@26178:  apply (auto cong add: conj_cong 
wenzelm@26145:      simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty)
nipkow@26178: done
kleing@25610: 
wenzelm@26145: lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A"
nipkow@26178: by (induct A) (auto simp: fold_mset_insert left_commute [of x])
nipkow@26178: 
wenzelm@26145: lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z"
nipkow@26178: using fold_mset_insert [of z "{#}"] by simp
kleing@25610: 
wenzelm@26145: lemma fold_mset_union [simp]:
wenzelm@26145:   "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B"
kleing@25759: proof (induct A)
wenzelm@26145:   case empty then show ?case by simp
kleing@25759: next
wenzelm@26145:   case (add A x)
wenzelm@26145:   have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac)
wenzelm@26145:   then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" 
wenzelm@26145:     by (simp add: fold_mset_insert)
wenzelm@26145:   also have "\<dots> = fold_mset f (fold_mset f z (A + {#x#})) B"
wenzelm@26145:     by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert)
wenzelm@26145:   finally show ?case .
kleing@25759: qed
kleing@25759: 
wenzelm@26145: lemma fold_mset_fusion:
ballarin@27611:   assumes "left_commutative g"
ballarin@27611:   shows "(\<And>x y. h (g x y) = f x (h y)) \<Longrightarrow> h (fold_mset g w A) = fold_mset f (h w) A" (is "PROP ?P")
ballarin@27611: proof -
ballarin@27611:   interpret left_commutative [g] by fact
ballarin@27611:   show "PROP ?P" by (induct A) auto
ballarin@27611: qed
kleing@25610: 
wenzelm@26145: lemma fold_mset_rec:
wenzelm@26145:   assumes "a \<in># A" 
kleing@25759:   shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))"
kleing@25610: proof -
wenzelm@26145:   from assms obtain A' where "A = A' + {#a#}"
wenzelm@26145:     by (blast dest: multi_member_split)
wenzelm@26145:   then show ?thesis by simp
kleing@25610: qed
kleing@25610: 
wenzelm@26145: end
wenzelm@26145: 
wenzelm@26145: text {*
wenzelm@26145:   A note on code generation: When defining some function containing a
wenzelm@26145:   subterm @{term"fold_mset F"}, code generation is not automatic. When
wenzelm@26145:   interpreting locale @{text left_commutative} with @{text F}, the
wenzelm@26145:   would be code thms for @{const fold_mset} become thms like
wenzelm@26145:   @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but
wenzelm@26145:   contains defined symbols, i.e.\ is not a code thm. Hence a separate
wenzelm@26145:   constant with its own code thms needs to be introduced for @{text
wenzelm@26145:   F}. See the image operator below.
wenzelm@26145: *}
wenzelm@26145: 
nipkow@26016: 
nipkow@26016: subsection {* Image *}
nipkow@26016: 
haftmann@28708: definition [code del]:
haftmann@28708:  "image_mset f = fold_mset (op + o single o f) {#}"
nipkow@26016: 
wenzelm@26145: interpretation image_left_comm: left_commutative ["op + o single o f"]
haftmann@28823:   proof qed (simp add:union_ac)
nipkow@26016: 
haftmann@28708: lemma image_mset_empty [simp]: "image_mset f {#} = {#}"
nipkow@26178: by (simp add: image_mset_def)
nipkow@26016: 
haftmann@28708: lemma image_mset_single [simp]: "image_mset f {#x#} = {#f x#}"
nipkow@26178: by (simp add: image_mset_def)
nipkow@26016: 
nipkow@26016: lemma image_mset_insert:
nipkow@26016:   "image_mset f (M + {#a#}) = image_mset f M + {#f a#}"
nipkow@26178: by (simp add: image_mset_def add_ac)
nipkow@26016: 
haftmann@28708: lemma image_mset_union [simp]:
nipkow@26016:   "image_mset f (M+N) = image_mset f M + image_mset f N"
nipkow@26178: apply (induct N)
nipkow@26178:  apply simp
nipkow@26178: apply (simp add: union_assoc [symmetric] image_mset_insert)
nipkow@26178: done
nipkow@26016: 
wenzelm@26145: lemma size_image_mset [simp]: "size (image_mset f M) = size M"
nipkow@26178: by (induct M) simp_all
nipkow@26016: 
wenzelm@26145: lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} \<longleftrightarrow> M = {#}"
nipkow@26178: by (cases M) auto
nipkow@26016: 
wenzelm@26145: syntax
wenzelm@26145:   comprehension1_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> 'a multiset"
wenzelm@26145:       ("({#_/. _ :# _#})")
wenzelm@26145: translations
wenzelm@26145:   "{#e. x:#M#}" == "CONST image_mset (%x. e) M"
nipkow@26016: 
wenzelm@26145: syntax
wenzelm@26145:   comprehension2_mset :: "'a \<Rightarrow> 'b \<Rightarrow> 'b multiset \<Rightarrow> bool \<Rightarrow> 'a multiset"
wenzelm@26145:       ("({#_/ | _ :# _./ _#})")
nipkow@26016: translations
nipkow@26033:   "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}"
nipkow@26016: 
wenzelm@26145: text {*
wenzelm@26145:   This allows to write not just filters like @{term "{#x:#M. x<c#}"}
wenzelm@26145:   but also images like @{term "{#x+x. x:#M #}"} and @{term [source]
wenzelm@26145:   "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as
wenzelm@26145:   @{term "{#x+x|x:#M. x<c#}"}.
wenzelm@26145: *}
nipkow@26016: 
krauss@29125: 
krauss@29125: subsection {* Termination proofs with multiset orders *}
krauss@29125: 
krauss@29125: lemma multi_member_skip: "x \<in># XS \<Longrightarrow> x \<in># {# y #} + XS"
krauss@29125:   and multi_member_this: "x \<in># {# x #} + XS"
krauss@29125:   and multi_member_last: "x \<in># {# x #}"
krauss@29125:   by auto
krauss@29125: 
krauss@29125: definition "ms_strict = mult pair_less"
krauss@29125: definition "ms_weak = ms_strict \<union> Id"
krauss@29125: 
krauss@29125: lemma ms_reduction_pair: "reduction_pair (ms_strict, ms_weak)"
krauss@29125: unfolding reduction_pair_def ms_strict_def ms_weak_def pair_less_def
krauss@29125: by (auto intro: wf_mult1 wf_trancl simp: mult_def)
krauss@29125: 
krauss@29125: lemma smsI:
krauss@29125:   "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z + B) \<in> ms_strict"
krauss@29125:   unfolding ms_strict_def
krauss@29125: by (rule one_step_implies_mult) (auto simp add: max_strict_def pair_less_def elim!:max_ext.cases)
krauss@29125: 
krauss@29125: lemma wmsI:
krauss@29125:   "(set_of A, set_of B) \<in> max_strict \<or> A = {#} \<and> B = {#}
krauss@29125:   \<Longrightarrow> (Z + A, Z + B) \<in> ms_weak"
krauss@29125: unfolding ms_weak_def ms_strict_def
krauss@29125: by (auto simp add: pair_less_def max_strict_def elim!:max_ext.cases intro: one_step_implies_mult)
krauss@29125: 
krauss@29125: inductive pw_leq
krauss@29125: where
krauss@29125:   pw_leq_empty: "pw_leq {#} {#}"
krauss@29125: | pw_leq_step:  "\<lbrakk>(x,y) \<in> pair_leq; pw_leq X Y \<rbrakk> \<Longrightarrow> pw_leq ({#x#} + X) ({#y#} + Y)"
krauss@29125: 
krauss@29125: lemma pw_leq_lstep:
krauss@29125:   "(x, y) \<in> pair_leq \<Longrightarrow> pw_leq {#x#} {#y#}"
krauss@29125: by (drule pw_leq_step) (rule pw_leq_empty, simp)
krauss@29125: 
krauss@29125: lemma pw_leq_split:
krauss@29125:   assumes "pw_leq X Y"
krauss@29125:   shows "\<exists>A B Z. X = A + Z \<and> Y = B + Z \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125:   using assms
krauss@29125: proof (induct)
krauss@29125:   case pw_leq_empty thus ?case by auto
krauss@29125: next
krauss@29125:   case (pw_leq_step x y X Y)
krauss@29125:   then obtain A B Z where
krauss@29125:     [simp]: "X = A + Z" "Y = B + Z" 
krauss@29125:       and 1[simp]: "(set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#})" 
krauss@29125:     by auto
krauss@29125:   from pw_leq_step have "x = y \<or> (x, y) \<in> pair_less" 
krauss@29125:     unfolding pair_leq_def by auto
krauss@29125:   thus ?case
krauss@29125:   proof
krauss@29125:     assume [simp]: "x = y"
krauss@29125:     have
krauss@29125:       "{#x#} + X = A + ({#y#}+Z) 
krauss@29125:       \<and> {#y#} + Y = B + ({#y#}+Z)
krauss@29125:       \<and> ((set_of A, set_of B) \<in> max_strict \<or> (B = {#} \<and> A = {#}))"
krauss@29125:       by (auto simp: add_ac)
krauss@29125:     thus ?case by (intro exI)
krauss@29125:   next
krauss@29125:     assume A: "(x, y) \<in> pair_less"
krauss@29125:     let ?A' = "{#x#} + A" and ?B' = "{#y#} + B"
krauss@29125:     have "{#x#} + X = ?A' + Z"
krauss@29125:       "{#y#} + Y = ?B' + Z"
krauss@29125:       by (auto simp add: add_ac)
krauss@29125:     moreover have 
krauss@29125:       "(set_of ?A', set_of ?B') \<in> max_strict"
krauss@29125:       using 1 A unfolding max_strict_def 
krauss@29125:       by (auto elim!: max_ext.cases)
krauss@29125:     ultimately show ?thesis by blast
krauss@29125:   qed
krauss@29125: qed
krauss@29125: 
krauss@29125: lemma 
krauss@29125:   assumes pwleq: "pw_leq Z Z'"
krauss@29125:   shows ms_strictI: "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_strict"
krauss@29125:   and   ms_weakI1:  "(set_of A, set_of B) \<in> max_strict \<Longrightarrow> (Z + A, Z' + B) \<in> ms_weak"
krauss@29125:   and   ms_weakI2:  "(Z + {#}, Z' + {#}) \<in> ms_weak"
krauss@29125: proof -
krauss@29125:   from pw_leq_split[OF pwleq] 
krauss@29125:   obtain A' B' Z''
krauss@29125:     where [simp]: "Z = A' + Z''" "Z' = B' + Z''"
krauss@29125:     and mx_or_empty: "(set_of A', set_of B') \<in> max_strict \<or> (A' = {#} \<and> B' = {#})"
krauss@29125:     by blast
krauss@29125:   {
krauss@29125:     assume max: "(set_of A, set_of B) \<in> max_strict"
krauss@29125:     from mx_or_empty
krauss@29125:     have "(Z'' + (A + A'), Z'' + (B + B')) \<in> ms_strict"
krauss@29125:     proof
krauss@29125:       assume max': "(set_of A', set_of B') \<in> max_strict"
krauss@29125:       with max have "(set_of (A + A'), set_of (B + B')) \<in> max_strict"
krauss@29125:         by (auto simp: max_strict_def intro: max_ext_additive)
krauss@29125:       thus ?thesis by (rule smsI) 
krauss@29125:     next
krauss@29125:       assume [simp]: "A' = {#} \<and> B' = {#}"
krauss@29125:       show ?thesis by (rule smsI) (auto intro: max)
krauss@29125:     qed
krauss@29125:     thus "(Z + A, Z' + B) \<in> ms_strict" by (simp add:add_ac)
krauss@29125:     thus "(Z + A, Z' + B) \<in> ms_weak" by (simp add: ms_weak_def)
krauss@29125:   }
krauss@29125:   from mx_or_empty
krauss@29125:   have "(Z'' + A', Z'' + B') \<in> ms_weak" by (rule wmsI)
krauss@29125:   thus "(Z + {#}, Z' + {#}) \<in> ms_weak" by (simp add:add_ac)
krauss@29125: qed
krauss@29125: 
krauss@29125: lemma empty_idemp: "{#} + x = x" "x + {#} = x"
krauss@29125: and nonempty_plus: "{# x #} + rs \<noteq> {#}"
krauss@29125: and nonempty_single: "{# x #} \<noteq> {#}"
krauss@29125: by auto
krauss@29125: 
krauss@29125: setup {*
krauss@29125: let
krauss@29125:   fun msetT T = Type ("Multiset.multiset", [T]);
krauss@29125: 
krauss@29125:   fun mk_mset T [] = Const (@{const_name Mempty}, msetT T)
krauss@29125:     | mk_mset T [x] = Const (@{const_name single}, T --> msetT T) $ x
krauss@29125:     | mk_mset T (x :: xs) =
krauss@29125:           Const (@{const_name plus}, msetT T --> msetT T --> msetT T) $
krauss@29125:                 mk_mset T [x] $ mk_mset T xs
krauss@29125: 
krauss@29125:   fun mset_member_tac m i =
krauss@29125:       (if m <= 0 then
krauss@29125:            rtac @{thm multi_member_this} i ORELSE rtac @{thm multi_member_last} i
krauss@29125:        else
krauss@29125:            rtac @{thm multi_member_skip} i THEN mset_member_tac (m - 1) i)
krauss@29125: 
krauss@29125:   val mset_nonempty_tac =
krauss@29125:       rtac @{thm nonempty_plus} ORELSE' rtac @{thm nonempty_single}
krauss@29125: 
krauss@29125:   val regroup_munion_conv =
krauss@29125:       FundefLib.regroup_conv @{const_name Multiset.Mempty} @{const_name plus}
krauss@29125:         (map (fn t => t RS eq_reflection) (@{thms union_ac} @ @{thms empty_idemp}))
krauss@29125: 
krauss@29125:   fun unfold_pwleq_tac i =
krauss@29125:     (rtac @{thm pw_leq_step} i THEN (fn st => unfold_pwleq_tac (i + 1) st))
krauss@29125:       ORELSE (rtac @{thm pw_leq_lstep} i)
krauss@29125:       ORELSE (rtac @{thm pw_leq_empty} i)
krauss@29125: 
krauss@29125:   val set_of_simps = [@{thm set_of_empty}, @{thm set_of_single}, @{thm set_of_union},
krauss@29125:                       @{thm Un_insert_left}, @{thm Un_empty_left}]
krauss@29125: in
krauss@29125:   ScnpReconstruct.multiset_setup (ScnpReconstruct.Multiset 
krauss@29125:   {
krauss@29125:     msetT=msetT, mk_mset=mk_mset, mset_regroup_conv=regroup_munion_conv,
krauss@29125:     mset_member_tac=mset_member_tac, mset_nonempty_tac=mset_nonempty_tac,
krauss@29125:     mset_pwleq_tac=unfold_pwleq_tac, set_of_simps=set_of_simps,
krauss@29125:     smsI'=@{thm ms_strictI}, wmsI2''=@{thm ms_weakI2}, wmsI1=@{thm ms_weakI1},
krauss@29125:     reduction_pair=@{thm ms_reduction_pair}
krauss@29125:   })
wenzelm@10249: end
krauss@29125: *}
krauss@29125: 
krauss@29125: end