(* Title: HOL/Library/Multiset.thy ID: $Id: Multiset.thy,v 1.68 2008/05/07 08:59:33 berghofe Exp $ Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker *) header {* Multisets *} theory Multiset imports List begin subsection {* The type of multisets *} typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}" proof show "(λx. 0::nat) ∈ ?multiset" by simp qed lemmas multiset_typedef [simp] = Abs_multiset_inverse Rep_multiset_inverse Rep_multiset and [simp] = Rep_multiset_inject [symmetric] definition Mempty :: "'a multiset" ("{#}") where "{#} = Abs_multiset (λa. 0)" definition single :: "'a => 'a multiset" where "single a = Abs_multiset (λb. if b = a then 1 else 0)" declare Mempty_def[code func del] single_def[code func del] definition count :: "'a multiset => 'a => nat" where "count = Rep_multiset" definition MCollect :: "'a multiset => ('a => bool) => 'a multiset" where "MCollect M P = Abs_multiset (λx. if P x then Rep_multiset M x else 0)" abbreviation Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where "a :# M == 0 < count M a" notation (xsymbols) Melem (infix "∈#" 50) syntax "_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ :# _./ _#})") translations "{#x :# M. P#}" == "CONST MCollect M (λx. P)" definition set_of :: "'a multiset => 'a set" where "set_of M = {x. x :# M}" instantiation multiset :: (type) "{plus, minus, zero, size}" begin definition union_def[code func del]: "M + N = Abs_multiset (λa. Rep_multiset M a + Rep_multiset N a)" definition diff_def: "M - N = Abs_multiset (λa. Rep_multiset M a - Rep_multiset N a)" definition Zero_multiset_def [simp]: "0 = {#}" definition size_def[code func del]: "size M = setsum (count M) (set_of M)" instance .. end definition multiset_inter :: "'a multiset => 'a multiset => 'a multiset" (infixl "#∩" 70) where "multiset_inter A B = A - (A - B)" text {* Multiset Enumeration *} syntax "_multiset" :: "args => 'a multiset" ("{#(_)#}") translations "{#x, xs#}" == "{#x#} + {#xs#}" "{#x#}" == "CONST single x" text {* \medskip Preservation of the representing set @{term multiset}. *} lemma const0_in_multiset: "(λa. 0) ∈ multiset" by (simp add: multiset_def) lemma only1_in_multiset: "(λb. if b = a then 1 else 0) ∈ multiset" by (simp add: multiset_def) lemma union_preserves_multiset: "M ∈ multiset ==> N ∈ multiset ==> (λa. M a + N a) ∈ multiset" apply (simp add: multiset_def) apply (drule (1) finite_UnI) apply (simp del: finite_Un add: Un_def) done lemma diff_preserves_multiset: "M ∈ multiset ==> (λa. M a - N a) ∈ multiset" apply (simp add: multiset_def) apply (rule finite_subset) apply auto done lemma MCollect_preserves_multiset: "M ∈ multiset ==> (λx. if P x then M x else 0) ∈ multiset" apply (simp add: multiset_def) apply (rule finite_subset, auto) done lemmas in_multiset = const0_in_multiset only1_in_multiset union_preserves_multiset diff_preserves_multiset MCollect_preserves_multiset subsection {* Algebraic properties *} subsubsection {* Union *} lemma union_empty [simp]: "M + {#} = M ∧ {#} + M = M" by (simp add: union_def Mempty_def in_multiset) lemma union_commute: "M + N = N + (M::'a multiset)" by (simp add: union_def add_ac in_multiset) lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))" by (simp add: union_def add_ac in_multiset) lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))" proof - have "M + (N + K) = (N + K) + M" by (rule union_commute) also have "… = N + (K + M)" by (rule union_assoc) also have "K + M = M + K" by (rule union_commute) finally show ?thesis . qed lemmas union_ac = union_assoc union_commute union_lcomm instance multiset :: (type) comm_monoid_add proof fix a b c :: "'a multiset" show "(a + b) + c = a + (b + c)" by (rule union_assoc) show "a + b = b + a" by (rule union_commute) show "0 + a = a" by simp qed subsubsection {* Difference *} lemma diff_empty [simp]: "M - {#} = M ∧ {#} - M = {#}" by (simp add: Mempty_def diff_def in_multiset) lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M" by (simp add: union_def diff_def in_multiset) lemma diff_cancel: "A - A = {#}" by (simp add: diff_def Mempty_def) subsubsection {* Count of elements *} lemma count_empty [simp]: "count {#} a = 0" by (simp add: count_def Mempty_def in_multiset) lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)" by (simp add: count_def single_def in_multiset) lemma count_union [simp]: "count (M + N) a = count M a + count N a" by (simp add: count_def union_def in_multiset) lemma count_diff [simp]: "count (M - N) a = count M a - count N a" by (simp add: count_def diff_def in_multiset) lemma count_MCollect [simp]: "count {# x:#M. P x #} a = (if P a then count M a else 0)" by (simp add: count_def MCollect_def in_multiset) subsubsection {* Set of elements *} lemma set_of_empty [simp]: "set_of {#} = {}" by (simp add: set_of_def) lemma set_of_single [simp]: "set_of {#b#} = {b}" by (simp add: set_of_def) lemma set_of_union [simp]: "set_of (M + N) = set_of M ∪ set_of N" by (auto simp add: set_of_def) lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})" by (auto simp: set_of_def Mempty_def in_multiset count_def expand_fun_eq [where f="Rep_multiset M"]) lemma mem_set_of_iff [simp]: "(x ∈ 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 ∩ {x. P x}" by (auto simp add: set_of_def) subsubsection {* Size *} lemma size_empty [simp,code func]: "size {#} = 0" by (simp add: size_def) lemma size_single [simp,code func]: "size {#b#} = 1" by (simp add: size_def) lemma finite_set_of [iff]: "finite (set_of M)" using Rep_multiset [of M] by (simp add: multiset_def set_of_def count_def) lemma setsum_count_Int: "finite A ==> setsum (count N) (A ∩ set_of N) = setsum (count N) A" apply (induct rule: finite_induct) apply simp apply (simp add: Int_insert_left set_of_def) done lemma size_union[simp,code func]: "size (M + N::'a multiset) = size M + size N" apply (unfold size_def) apply (subgoal_tac "count (M + N) = (λa. count M a + count N a)") prefer 2 apply (rule ext, simp) apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int) apply (subst Int_commute) apply (simp (no_asm_simp) add: setsum_count_Int) done lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})" apply (unfold size_def Mempty_def count_def, auto simp: in_multiset) apply (simp add: set_of_def count_def in_multiset expand_fun_eq) done lemma nonempty_has_size: "(S ≠ {#}) = (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 ==> ∃a. a :# M" apply (unfold size_def) apply (drule setsum_SucD) apply auto done subsubsection {* Equality of multisets *} lemma multiset_eq_conv_count_eq: "(M = N) = (∀a. count M a = count N a)" by (simp add: count_def expand_fun_eq) lemma single_not_empty [simp]: "{#a#} ≠ {#} ∧ {#} ≠ {#a#}" by (simp add: single_def Mempty_def in_multiset expand_fun_eq) lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)" by (auto simp add: single_def in_multiset expand_fun_eq) lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} ∧ N = {#})" by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq) lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} ∧ N = {#})" by (auto simp add: union_def Mempty_def in_multiset expand_fun_eq) lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))" by (simp add: union_def in_multiset expand_fun_eq) lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))" by (simp add: union_def in_multiset expand_fun_eq) lemma union_is_single: "(M + N = {#a#}) = (M = {#a#} ∧ N={#} ∨ M = {#} ∧ N = {#a#})" apply (simp add: Mempty_def single_def union_def in_multiset add_is_1 expand_fun_eq) apply blast done lemma single_is_union: "({#a#} = M + N) <-> ({#a#} = M ∧ N = {#} ∨ M = {#} ∧ {#a#} = N)" apply (unfold Mempty_def single_def union_def) apply (simp add: add_is_1 one_is_add in_multiset expand_fun_eq) apply (blast dest: sym) done lemma add_eq_conv_diff: "(M + {#a#} = N + {#b#}) = (M = N ∧ a = b ∨ M = N - {#a#} + {#b#} ∧ N = M - {#b#} + {#a#})" using [[simproc del: neq]] apply (unfold single_def union_def diff_def) apply (simp (no_asm) add: in_multiset expand_fun_eq) apply (rule conjI, force, safe, simp_all) apply (simp add: eq_sym_conv) done declare Rep_multiset_inject [symmetric, simp del] instance multiset :: (type) cancel_ab_semigroup_add proof fix a b c :: "'a multiset" show "a + b = a + c ==> b = c" by simp qed lemma insert_DiffM: "x ∈# M ==> {#x#} + (M - {#x#}) = M" by (clarsimp simp: multiset_eq_conv_count_eq) lemma insert_DiffM2[simp]: "x ∈# M ==> M - {#x#} + {#x#} = M" by (clarsimp simp: multiset_eq_conv_count_eq) lemma multi_union_self_other_eq: "(A::'a multiset) + X = A + Y ==> X = Y" by (induct A arbitrary: X Y) auto lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False" by (metis single_not_empty union_empty union_left_cancel) lemma insert_noteq_member: assumes BC: "B + {#b#} = C + {#c#}" and bnotc: "b ≠ c" shows "c ∈# B" proof - have "c ∈# C + {#c#}" by simp have nc: "¬ c ∈# {#b#}" using bnotc by simp then have "c ∈# B + {#b#}" using BC by simp then show "c ∈# B" using nc by simp qed lemma add_eq_conv_ex: "(M + {#a#} = N + {#b#}) = (M = N ∧ a = b ∨ (∃K. M = K + {#b#} ∧ N = K + {#a#}))" by (auto simp add: add_eq_conv_diff) lemma empty_multiset_count: "(∀x. count A x = 0) = (A = {#})" by (metis count_empty multiset_eq_conv_count_eq) subsubsection {* Intersection *} lemma multiset_inter_count: "count (A #∩ B) x = min (count A x) (count B x)" by (simp add: multiset_inter_def min_def) lemma multiset_inter_commute: "A #∩ B = B #∩ A" by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_max.inf_commute) lemma multiset_inter_assoc: "A #∩ (B #∩ C) = A #∩ B #∩ C" by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_max.inf_assoc) lemma multiset_inter_left_commute: "A #∩ (B #∩ C) = B #∩ (A #∩ C)" by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def) lemmas multiset_inter_ac = multiset_inter_commute multiset_inter_assoc multiset_inter_left_commute lemma multiset_inter_single: "a ≠ b ==> {#a#} #∩ {#b#} = {#}" by (simp add: multiset_eq_conv_count_eq multiset_inter_count) lemma multiset_union_diff_commute: "B #∩ C = {#} ==> A + B - C = A - C + B" apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def split: split_if_asm) apply clarsimp apply (erule_tac x = a in allE) apply auto done subsubsection {* Comprehension (filter) *} lemma MCollect_empty[simp, code func]: "MCollect {#} P = {#}" by (simp add: MCollect_def Mempty_def Abs_multiset_inject in_multiset expand_fun_eq) lemma MCollect_single[simp, code func]: "MCollect {#x#} P = (if P x then {#x#} else {#})" by (simp add: MCollect_def Mempty_def single_def Abs_multiset_inject in_multiset expand_fun_eq) lemma MCollect_union[simp, code func]: "MCollect (M+N) f = MCollect M f + MCollect N f" by (simp add: MCollect_def union_def Abs_multiset_inject in_multiset expand_fun_eq) subsection {* Induction and case splits *} lemma setsum_decr: "finite F ==> (0::nat) < f a ==> setsum (f (a := f a - 1)) F = (if a∈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 (λa. (0::nat))" and 2: "!!f b. f ∈ multiset ==> P f ==> P (f (b := f b + 1))" shows "∀f. f ∈ multiset --> setsum f {x. f x ≠ 0} = n --> P f" apply (unfold multiset_def) apply (induct_tac n, simp, clarify) apply (subgoal_tac "f = (λa.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 ≠ a --> f x ≠ 0} = {x. f x ≠ 0}") prefer 2 apply blast apply (subgoal_tac "{x. x ≠ a ∧ f x ≠ 0} = {x. f x ≠ 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 ∈ multiset ==> P (λa. 0) ==> (!!f b. f ∈ multiset ==> P f ==> P (f (b := f b + 1))) ==> P f" using rep_multiset_induct_aux by blast theorem multiset_induct [case_names empty add, induct type: multiset]: assumes empty: "P {#}" and add: "!!M x. P M ==> P (M + {#x#})" shows "P M" proof - note defns = union_def single_def Mempty_def show ?thesis apply (rule Rep_multiset_inverse [THEN subst]) apply (rule Rep_multiset [THEN rep_multiset_induct]) apply (rule empty [unfolded defns]) apply (subgoal_tac "f(b := f b + 1) = (λa. f a + (if a=b then 1 else 0))") prefer 2 apply (simp add: expand_fun_eq) apply (erule ssubst) apply (erule Abs_multiset_inverse [THEN subst]) apply (drule add [unfolded defns, simplified]) apply(simp add:in_multiset) done qed lemma multi_nonempty_split: "M ≠ {#} ==> ∃A a. M = A + {#a#}" by (induct M) auto lemma multiset_cases [cases type, case_names empty add]: assumes em: "M = {#} ==> P" assumes add: "!!N x. M = N + {#x#} ==> P" shows "P" proof (cases "M = {#}") assume "M = {#}" then show ?thesis using em by simp next assume "M ≠ {#}" 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 ∈# M ==> ∃A. M = A + {#x#}" apply (cases M) apply simp apply (rule_tac x="M - {#x#}" in exI, simp) done lemma multiset_partition: "M = {# x:#M. P x #} + {# x:#M. ¬ P x #}" apply (subst multiset_eq_conv_count_eq) apply auto done declare multiset_typedef [simp del] lemma multi_drop_mem_not_eq: "c ∈# B ==> B - {#c#} ≠ B" by (cases "B = {#}") (auto dest: multi_member_split) subsection {* Orderings *} subsubsection {* Well-foundedness *} definition mult1 :: "('a × 'a) set => ('a multiset × 'a multiset) set" where "mult1 r = {(N, M). ∃a M0 K. M = M0 + {#a#} ∧ N = M0 + K ∧ (∀b. b :# K --> (b, a) ∈ r)}" definition mult :: "('a × 'a) set => ('a multiset × 'a multiset) set" where "mult r = (mult1 r)+" lemma not_less_empty [iff]: "(M, {#}) ∉ mult1 r" by (simp add: mult1_def) lemma less_add: "(N, M0 + {#a#}) ∈ mult1 r ==> (∃M. (M, M0) ∈ mult1 r ∧ N = M + {#a#}) ∨ (∃K. (∀b. b :# K --> (b, a) ∈ r) ∧ N = M0 + K)" (is "_ ==> ?case1 (mult1 r) ∨ ?case2") proof (unfold mult1_def) let ?r = "λK a. ∀b. b :# K --> (b, a) ∈ r" let ?R = "λN M. ∃a M0 K. M = M0 + {#a#} ∧ N = M0 + K ∧ ?r K a" let ?case1 = "?case1 {(N, M). ?R N M}" assume "(N, M0 + {#a#}) ∈ {(N, M). ?R N M}" then have "∃a' M0' K. M0 + {#a#} = M0' + {#a'#} ∧ N = M0' + K ∧ ?r K a'" by simp then show "?case1 ∨ ?case2" proof (elim exE conjE) fix a' M0' K assume N: "N = M0' + K" and r: "?r K a'" assume "M0 + {#a#} = M0' + {#a'#}" then have "M0 = M0' ∧ a = a' ∨ (∃K'. M0 = K' + {#a'#} ∧ M0' = K' + {#a#})" by (simp only: add_eq_conv_ex) then show ?thesis proof (elim disjE conjE exE) assume "M0 = M0'" "a = a'" with N r have "?r K a ∧ N = M0 + K" by simp then have ?case2 .. then show ?thesis .. next fix K' assume "M0' = K' + {#a#}" with N have n: "N = K' + K + {#a#}" by (simp add: union_ac) assume "M0 = K' + {#a'#}" with r have "?R (K' + K) M0" by blast with n have ?case1 by simp then show ?thesis .. qed qed qed lemma all_accessible: "wf r ==> ∀M. M ∈ acc (mult1 r)" proof let ?R = "mult1 r" let ?W = "acc ?R" { fix M M0 a assume M0: "M0 ∈ ?W" and wf_hyp: "!!b. (b, a) ∈ r ==> (∀M ∈ ?W. M + {#b#} ∈ ?W)" and acc_hyp: "∀M. (M, M0) ∈ ?R --> M + {#a#} ∈ ?W" have "M0 + {#a#} ∈ ?W" proof (rule accI [of "M0 + {#a#}"]) fix N assume "(N, M0 + {#a#}) ∈ ?R" then have "((∃M. (M, M0) ∈ ?R ∧ N = M + {#a#}) ∨ (∃K. (∀b. b :# K --> (b, a) ∈ r) ∧ N = M0 + K))" by (rule less_add) then show "N ∈ ?W" proof (elim exE disjE conjE) fix M assume "(M, M0) ∈ ?R" and N: "N = M + {#a#}" from acc_hyp have "(M, M0) ∈ ?R --> M + {#a#} ∈ ?W" .. from this and `(M, M0) ∈ ?R` have "M + {#a#} ∈ ?W" .. then show "N ∈ ?W" by (simp only: N) next fix K assume N: "N = M0 + K" assume "∀b. b :# K --> (b, a) ∈ r" then have "M0 + K ∈ ?W" proof (induct K) case empty from M0 show "M0 + {#} ∈ ?W" by simp next case (add K x) from add.prems have "(x, a) ∈ r" by simp with wf_hyp have "∀M ∈ ?W. M + {#x#} ∈ ?W" by blast moreover from add have "M0 + K ∈ ?W" by simp ultimately have "(M0 + K) + {#x#} ∈ ?W" .. then show "M0 + (K + {#x#}) ∈ ?W" by (simp only: union_assoc) qed then show "N ∈ ?W" by (simp only: N) qed qed } note tedious_reasoning = this assume wf: "wf r" fix M show "M ∈ ?W" proof (induct M) show "{#} ∈ ?W" proof (rule accI) fix b assume "(b, {#}) ∈ ?R" with not_less_empty show "b ∈ ?W" by contradiction qed fix M a assume "M ∈ ?W" from wf have "∀M ∈ ?W. M + {#a#} ∈ ?W" proof induct fix a assume r: "!!b. (b, a) ∈ r ==> (∀M ∈ ?W. M + {#b#} ∈ ?W)" show "∀M ∈ ?W. M + {#a#} ∈ ?W" proof fix M assume "M ∈ ?W" then show "M + {#a#} ∈ ?W" by (rule acc_induct) (rule tedious_reasoning [OF _ r]) qed qed from this and `M ∈ ?W` show "M + {#a#} ∈ ?W" .. qed qed theorem wf_mult1: "wf r ==> wf (mult1 r)" by (rule acc_wfI) (rule all_accessible) theorem wf_mult: "wf r ==> wf (mult r)" unfolding mult_def by (rule wf_trancl) (rule wf_mult1) subsubsection {* Closure-free presentation *} (*Badly needed: a linear arithmetic procedure for multisets*) lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})" by (simp add: multiset_eq_conv_count_eq) text {* One direction. *} lemma mult_implies_one_step: "trans r ==> (M, N) ∈ mult r ==> ∃I J K. N = I + J ∧ M = I + K ∧ J ≠ {#} ∧ (∀k ∈ set_of K. ∃j ∈ set_of J. (k, j) ∈ r)" apply (unfold mult_def mult1_def set_of_def) apply (erule converse_trancl_induct, clarify) apply (rule_tac x = M0 in exI, simp, clarify) apply (case_tac "a :# K") apply (rule_tac x = I in exI) apply (simp (no_asm)) apply (rule_tac x = "(K - {#a#}) + Ka" in exI) apply (simp (no_asm_simp) add: union_assoc [symmetric]) apply (drule_tac f = "λM. M - {#a#}" in arg_cong) apply (simp add: diff_union_single_conv) apply (simp (no_asm_use) add: trans_def) apply blast apply (subgoal_tac "a :# I") apply (rule_tac x = "I - {#a#}" in exI) apply (rule_tac x = "J + {#a#}" in exI) apply (rule_tac x = "K + Ka" in exI) apply (rule conjI) apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) apply (rule conjI) apply (drule_tac f = "λM. M - {#a#}" in arg_cong, simp) apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) apply (simp (no_asm_use) add: trans_def) apply blast apply (subgoal_tac "a :# (M0 + {#a#})") apply simp apply (simp (no_asm)) done lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}" by (simp add: multiset_eq_conv_count_eq) lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> ∃a N. M = N + {#a#}" apply (erule size_eq_Suc_imp_elem [THEN exE]) apply (drule elem_imp_eq_diff_union, auto) done lemma one_step_implies_mult_aux: "trans r ==> ∀I J K. (size J = n ∧ J ≠ {#} ∧ (∀k ∈ set_of K. ∃j ∈ set_of J. (k, j) ∈ r)) --> (I + K, I + J) ∈ mult r" apply (induct_tac n, auto) apply (frule size_eq_Suc_imp_eq_union, clarify) apply (rename_tac "J'", simp) apply (erule notE, auto) apply (case_tac "J' = {#}") apply (simp add: mult_def) apply (rule r_into_trancl) apply (simp add: mult1_def set_of_def, blast) txt {* Now we know @{term "J' ≠ {#}"}. *} apply (cut_tac M = K and P = "λx. (x, a) ∈ r" in multiset_partition) apply (erule_tac P = "∀k ∈ set_of K. ?P k" in rev_mp) apply (erule ssubst) apply (simp add: Ball_def, auto) apply (subgoal_tac "((I + {# x :# K. (x, a) ∈ r #}) + {# x :# K. (x, a) ∉ r #}, (I + {# x :# K. (x, a) ∈ r #}) + J') ∈ mult r") prefer 2 apply force apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def) apply (erule trancl_trans) apply (rule r_into_trancl) apply (simp add: mult1_def set_of_def) apply (rule_tac x = a in exI) apply (rule_tac x = "I + J'" in exI) apply (simp add: union_ac) done lemma one_step_implies_mult: "trans r ==> J ≠ {#} ==> ∀k ∈ set_of K. ∃j ∈ set_of J. (k, j) ∈ r ==> (I + K, I + J) ∈ mult r" using one_step_implies_mult_aux by blast subsubsection {* Partial-order properties *} instantiation multiset :: (order) order begin definition less_multiset_def: "M' < M <-> (M', M) ∈ mult {(x', x). x' < x}" definition le_multiset_def: "M' <= M <-> M' = M ∨ M' < (M::'a multiset)" lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}" unfolding trans_def by (blast intro: order_less_trans) text {* \medskip Irreflexivity. *} lemma mult_irrefl_aux: "finite A ==> (∀x ∈ A. ∃y ∈ A. x < (y::'a::order)) ==> A = {}" by (induct rule: finite_induct) (auto intro: order_less_trans) lemma mult_less_not_refl: "¬ M < (M::'a::order multiset)" apply (unfold less_multiset_def, auto) apply (drule trans_base_order [THEN mult_implies_one_step], auto) apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]]) apply (simp add: set_of_eq_empty_iff) done lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R" using insert mult_less_not_refl by fast text {* Transitivity. *} theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)" unfolding less_multiset_def mult_def by (blast intro: trancl_trans) text {* Asymmetry. *} theorem mult_less_not_sym: "M < N ==> ¬ N < (M::'a::order multiset)" apply auto apply (rule mult_less_not_refl [THEN notE]) apply (erule mult_less_trans, assumption) done theorem mult_less_asym: "M < N ==> (¬ P ==> N < (M::'a::order multiset)) ==> P" using mult_less_not_sym by blast theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)" unfolding le_multiset_def by auto text {* Anti-symmetry. *} theorem mult_le_antisym: "M <= N ==> N <= M ==> M = (N::'a::order multiset)" unfolding le_multiset_def by (blast dest: mult_less_not_sym) text {* Transitivity. *} theorem mult_le_trans: "K <= M ==> M <= N ==> K <= (N::'a::order multiset)" unfolding le_multiset_def by (blast intro: mult_less_trans) theorem mult_less_le: "(M < N) = (M <= N ∧ M ≠ (N::'a::order multiset))" unfolding le_multiset_def by auto instance apply intro_classes apply (rule mult_less_le) apply (rule mult_le_refl) apply (erule mult_le_trans, assumption) apply (erule mult_le_antisym, assumption) done end subsubsection {* Monotonicity of multiset union *} lemma mult1_union: "(B, D) ∈ mult1 r ==> trans r ==> (C + B, C + D) ∈ mult1 r" apply (unfold mult1_def) apply auto apply (rule_tac x = a in exI) apply (rule_tac x = "C + M0" in exI) apply (simp add: union_assoc) done lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)" apply (unfold less_multiset_def mult_def) apply (erule trancl_induct) apply (blast intro: mult1_union transI order_less_trans r_into_trancl) apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans) done lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)" apply (subst union_commute [of B C]) apply (subst union_commute [of D C]) apply (erule union_less_mono2) done lemma union_less_mono: "A < C ==> B < D ==> A + B < C + (D::'a::order multiset)" by (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans) lemma union_le_mono: "A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)" unfolding le_multiset_def by (blast intro: union_less_mono union_less_mono1 union_less_mono2) lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)" apply (unfold le_multiset_def less_multiset_def) apply (case_tac "M = {#}") prefer 2 apply (subgoal_tac "({#} + {#}, {#} + M) ∈ mult (Collect (split op <))") prefer 2 apply (rule one_step_implies_mult) apply (simp only: trans_def) apply auto done lemma union_upper1: "A <= A + (B::'a::order multiset)" proof - have "A + {#} <= A + B" by (blast intro: union_le_mono) then show ?thesis by simp qed lemma union_upper2: "B <= A + (B::'a::order multiset)" by (subst union_commute) (rule union_upper1) instance multiset :: (order) pordered_ab_semigroup_add apply intro_classes apply (erule union_le_mono[OF mult_le_refl]) done subsection {* Link with lists *} primrec multiset_of :: "'a list => 'a multiset" where "multiset_of [] = {#}" | "multiset_of (a # x) = multiset_of x + {# a #}" lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])" by (induct x) auto lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])" by (induct x) auto lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x" by (induct x) auto lemma mem_set_multiset_eq: "x ∈ set xs = (x :# multiset_of xs)" by (induct xs) auto lemma multiset_of_append [simp]: "multiset_of (xs @ ys) = multiset_of xs + multiset_of ys" by (induct xs arbitrary: ys) (auto simp: union_ac) lemma surj_multiset_of: "surj multiset_of" apply (unfold surj_def) apply (rule allI) apply (rule_tac M = y in multiset_induct) apply auto apply (rule_tac x = "x # xa" in exI) apply auto done lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}" by (induct x) auto lemma distinct_count_atmost_1: "distinct x = (! a. count (multiset_of x) a = (if a ∈ 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 ==> set xs = set ys" by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0) lemma set_eq_iff_multiset_of_eq_distinct: "distinct x ==> distinct y ==> (set x = set y) = (multiset_of x = multiset_of y)" by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) lemma set_eq_iff_multiset_of_remdups_eq: "(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))" apply (rule iffI) apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) apply (drule distinct_remdups [THEN distinct_remdups [THEN set_eq_iff_multiset_of_eq_distinct [THEN iffD2]]]) apply simp done lemma multiset_of_compl_union [simp]: "multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. ¬P x] = multiset_of xs" by (induct xs) (auto simp: union_ac) lemma count_filter: "count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]" by (induct xs) auto lemma nth_mem_multiset_of: "i < length ls ==> (ls ! i) :# multiset_of ls" apply (induct ls arbitrary: i) apply simp apply (case_tac i) apply auto done lemma multiset_of_remove1: "multiset_of (remove1 a xs) = multiset_of xs - {#a#}" by (induct xs) (auto simp add: multiset_eq_conv_count_eq) lemma multiset_of_eq_length: assumes "multiset_of xs = multiset_of ys" shows "length xs = length ys" using assms proof (induct arbitrary: ys rule: length_induct) case (1 xs ys) show ?case proof (cases xs) case Nil with "1.prems" show ?thesis by simp next case (Cons x xs') note xCons = Cons show ?thesis proof (cases ys) case Nil with "1.prems" Cons show ?thesis by simp next case (Cons y ys') have x_in_ys: "x = y ∨ x ∈ set ys'" proof (cases "x = y") case True then show ?thesis .. next case False from "1.prems" [symmetric] xCons Cons have "x :# multiset_of ys' + {#y#}" by simp with False show ?thesis by (simp add: mem_set_multiset_eq) qed from "1.hyps" have IH: "length xs' < length xs --> (∀x. multiset_of xs' = multiset_of x --> length xs' = length x)" by blast from "1.prems" x_in_ys Cons xCons have "multiset_of xs' = multiset_of (remove1 x (y#ys'))" apply - apply (simp add: multiset_of_remove1, simp only: add_eq_conv_diff) apply fastsimp done with IH xCons have IH': "length xs' = length (remove1 x (y#ys'))" by fastsimp from x_in_ys have "x ≠ y ==> length ys' > 0" by auto with Cons xCons x_in_ys IH' show ?thesis by (auto simp add: length_remove1) qed qed qed text {* This lemma shows which properties suffice to show that a function @{text "f"} with @{text "f xs = ys"} behaves like sort. *} lemma properties_for_sort: "multiset_of ys = multiset_of xs ==> sorted ys ==> sort xs = ys" proof (induct xs arbitrary: ys) case Nil then show ?case by simp next case (Cons x xs) then have "x ∈ set ys" by (auto simp add: mem_set_multiset_eq intro!: ccontr) with Cons.prems Cons.hyps [of "remove1 x ys"] show ?case by (simp add: sorted_remove1 multiset_of_remove1 insort_remove1) qed subsection {* Pointwise ordering induced by count *} definition mset_le :: "'a multiset => 'a multiset => bool" (infix "≤#" 50) where "(A ≤# B) = (∀a. count A a ≤ count B a)" definition mset_less :: "'a multiset => 'a multiset => bool" (infix "<#" 50) where "(A <# B) = (A ≤# B ∧ A ≠ B)" notation mset_le (infix "⊆#" 50) notation mset_less (infix "⊂#" 50) lemma mset_le_refl[simp]: "A ≤# A" unfolding mset_le_def by auto lemma mset_le_trans: "A ≤# B ==> B ≤# C ==> A ≤# C" unfolding mset_le_def by (fast intro: order_trans) lemma mset_le_antisym: "A ≤# B ==> B ≤# A ==> A = B" apply (unfold mset_le_def) apply (rule multiset_eq_conv_count_eq [THEN iffD2]) apply (blast intro: order_antisym) done lemma mset_le_exists_conv: "(A ≤# B) = (∃C. B = A + C)" apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI) apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2]) done lemma mset_le_mono_add_right_cancel[simp]: "(A + C ≤# B + C) = (A ≤# B)" unfolding mset_le_def by auto lemma mset_le_mono_add_left_cancel[simp]: "(C + A ≤# C + B) = (A ≤# B)" unfolding mset_le_def by auto lemma mset_le_mono_add: "[| A ≤# B; C ≤# D |] ==> A + C ≤# B + D" apply (unfold mset_le_def) apply auto apply (erule_tac x = a in allE)+ apply auto done lemma mset_le_add_left[simp]: "A ≤# A + B" unfolding mset_le_def by auto lemma mset_le_add_right[simp]: "B ≤# A + B" unfolding mset_le_def by auto lemma mset_le_single: "a :# B ==> {#a#} ≤# B" by (simp add: mset_le_def) lemma multiset_diff_union_assoc: "C ≤# B ==> A + B - C = A + (B - C)" by (simp add: multiset_eq_conv_count_eq mset_le_def) lemma mset_le_multiset_union_diff_commute: assumes "B ≤# A" shows "A - B + C = A + C - B" proof - from mset_le_exists_conv [of "B" "A"] assms have "∃D. A = B + D" .. from this obtain D where "A = B + D" .. then show ?thesis apply simp apply (subst union_commute) apply (subst multiset_diff_union_assoc) apply simp apply (simp add: diff_cancel) apply (subst union_assoc) apply (subst union_commute[of "B" _]) apply (subst multiset_diff_union_assoc) apply simp apply (simp add: diff_cancel) done qed lemma multiset_of_remdups_le: "multiset_of (remdups xs) ≤# multiset_of xs" apply (induct xs) apply auto apply (rule mset_le_trans) apply auto done lemma multiset_of_update: "i < length ls ==> 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 union_assoc) apply (subst union_commute [where M = "{#v#}" and N = "{#x#}"]) apply (subst union_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 ==> j < length ls ==> multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls" apply (case_tac "i = j") apply simp apply (simp add: multiset_of_update) apply (subst elem_imp_eq_diff_union[symmetric]) apply (simp add: nth_mem_multiset_of) apply simp done interpretation mset_order: order ["op ≤#" "op <#"] by (auto intro: order.intro mset_le_refl mset_le_antisym mset_le_trans simp: mset_less_def) interpretation mset_order_cancel_semigroup: pordered_cancel_ab_semigroup_add ["op +" "op ≤#" "op <#"] by unfold_locales (erule mset_le_mono_add [OF mset_le_refl]) interpretation mset_order_semigroup_cancel: pordered_ab_semigroup_add_imp_le ["op +" "op ≤#" "op <#"] by (unfold_locales) simp lemma mset_lessD: "A ⊂# B ==> x ∈# A ==> x ∈# B" apply (clarsimp simp: mset_le_def mset_less_def) apply (erule_tac x=x in allE) apply auto done lemma mset_leD: "A ⊆# B ==> x ∈# A ==> x ∈# 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) ==> (x ∈# B ∧ 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#} ⊆# B) ==> (x ∈# B ∧ A ⊆# 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 ⊂# {#} = False" by (induct A) (auto simp: mset_le_def mset_less_def) 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 = False" by (auto simp: mset_le_def mset_less_def) lemma mset_less_add_bothsides: "T + {#x#} ⊂# S + {#x#} ==> T ⊂# S" by (auto simp: mset_le_def mset_less_def) lemma mset_less_empty_nonempty: "({#} ⊂# S) = (S ≠ {#})" by (auto simp: mset_le_def mset_less_def) lemma mset_less_size: "A ⊂# B ==> size A < size B" proof (induct A arbitrary: B) case (empty M) then have "M ≠ {#}" 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: "!!B. S ⊂# B ==> size S < size B" by fact have SxsubT: "S + {#x#} ⊂# T" by fact then have "x ∈# 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 lemmas mset_less_trans = mset_order.less_eq_less.less_trans lemma mset_less_diff_self: "c ∈# B ==> B - {#c#} ⊂# B" by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq) subsection {* 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 ∈# B" and "b ≠ c" shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}" proof - from `c ∈# 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: union_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: "!!B. ∀A. A ⊂# B --> P A ==> 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 ⊆# A" and empty: "P {#}" and insert: "!!a F. a ∈# A ==> P F ==> P (F + {#a#})" shows "P F" proof - from `F ⊆# A` show ?thesis proof (induct F) show "P {#}" by fact next fix x F assume P: "F ⊆# A ==> P F" and i: "F + {#x#} ⊆# A" show "P (F + {#x#})" proof (rule insert) from i show "x ∈# A" by (auto dest: mset_le_insertD) from i have "F ⊆# A" by (auto dest: mset_le_insertD) with P show "P F" . qed qed qed text{* A consequence: Extensionality. *} lemma multi_count_eq: "(∀x. count A x = count B x) = (A = B)" apply (rule iffI) prefer 2 apply clarsimp apply (induct A arbitrary: B rule: full_multiset_induct) apply (rename_tac C) apply (case_tac B rule: multiset_cases) apply (simp add: empty_multiset_count) apply simp apply (case_tac "x ∈# C") apply (force dest: multi_member_split) apply (erule_tac x = x in allE) apply simp done lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format] subsection {* The fold combinator *} text {* The intended behaviour is @{text "fold_mset f z {#x1, ..., xn#} = f x1 (… (f xn z)…)"} if @{text f} is associative-commutative. *} text {* The graph of @{text "fold_mset"}, @{text "z"}: the start element, @{text "f"}: folding function, @{text "A"}: the multiset, @{text "y"}: the result. *} inductive fold_msetG :: "('a => 'b => 'b) => 'b => 'a multiset => 'b => bool" for f :: "'a => 'b => 'b" and z :: 'b where emptyI [intro]: "fold_msetG f z {#} z" | insertI [intro]: "fold_msetG f z A y ==> fold_msetG f z (A + {#x#}) (f x y)" inductive_cases empty_fold_msetGE [elim!]: "fold_msetG f z {#} x" inductive_cases insert_fold_msetGE: "fold_msetG f z (A + {#}) y" definition fold_mset :: "('a => 'b => 'b) => 'b => 'a multiset => 'b" where "fold_mset f z A = (THE x. fold_msetG f z A x)" lemma Diff1_fold_msetG: "fold_msetG f z (A - {#x#}) y ==> x ∈# A ==> fold_msetG f z A (f x y)" apply (frule_tac x = x in fold_msetG.insertI) apply auto done lemma fold_msetG_nonempty: "∃x. fold_msetG f z A x" apply (induct A) apply blast apply clarsimp apply (drule_tac x = x in fold_msetG.insertI) apply auto done lemma fold_mset_empty[simp]: "fold_mset f z {#} = z" unfolding fold_mset_def by blast locale left_commutative = fixes f :: "'a => 'b => 'b" assumes left_commute: "f x (f y z) = f y (f x z)" begin lemma fold_msetG_determ: "fold_msetG f z A x ==> fold_msetG f z A y ==> y = x" proof (induct arbitrary: x y z rule: full_multiset_induct) case (less M x1 x2 Z) have IH: "∀A. A ⊂# M --> (∀x x' x''. fold_msetG f x'' A x --> fold_msetG f x'' A x' --> x' = x)" by fact have Mfoldx1: "fold_msetG f Z M x1" and Mfoldx2: "fold_msetG f Z M x2" by fact+ show ?case proof (rule fold_msetG.cases [OF Mfoldx1]) assume "M = {#}" and "x1 = Z" then show ?case using Mfoldx2 by auto next fix B b u assume "M = B + {#b#}" and "x1 = f b u" and Bu: "fold_msetG f Z B u" then have MBb: "M = B + {#b#}" and x1: "x1 = f b u" by auto show ?case proof (rule fold_msetG.cases [OF Mfoldx2]) assume "M = {#}" "x2 = Z" then show ?case using Mfoldx1 by auto next fix C c v assume "M = C + {#c#}" and "x2 = f c v" and Cv: "fold_msetG f Z C v" then have MCc: "M = C + {#c#}" and x2: "x2 = f c v" by auto then have CsubM: "C ⊂# M" by simp from MBb have BsubM: "B ⊂# M" by simp show ?case proof cases assume "b=c" then moreover have "B = C" using MBb MCc by auto ultimately show ?thesis using Bu Cv x1 x2 CsubM IH by auto next assume diff: "b ≠ c" let ?D = "B - {#c#}" have cinB: "c ∈# B" and binC: "b ∈# C" using MBb MCc diff by (auto intro: insert_noteq_member dest: sym) have "B - {#c#} ⊂# B" using cinB by (rule mset_less_diff_self) then have DsubM: "?D ⊂# M" using BsubM by (blast intro: mset_less_trans) from MBb MCc have "B + {#b#} = C + {#c#}" by blast then have [simp]: "B + {#b#} - {#c#} = C" using MBb MCc binC cinB by auto have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}" using MBb MCc diff binC cinB by (auto simp: multiset_add_sub_el_shuffle) then obtain d where Dfoldd: "fold_msetG f Z ?D d" using fold_msetG_nonempty by iprover then have "fold_msetG f Z B (f c d)" using cinB by (rule Diff1_fold_msetG) then have "f c d = u" using IH BsubM Bu by blast moreover have "fold_msetG f Z C (f b d)" using binC cinB diff Dfoldd by (auto simp: multiset_add_sub_el_shuffle dest: fold_msetG.insertI [where x=b]) then have "f b d = v" using IH CsubM Cv by blast ultimately show ?thesis using x1 x2 by (auto simp: left_commute) qed qed qed qed lemma fold_mset_insert_aux: "(fold_msetG f z (A + {#x#}) v) = (∃y. fold_msetG f z A y ∧ v = f x y)" apply (rule iffI) prefer 2 apply blast apply (rule_tac A=A and f=f in fold_msetG_nonempty [THEN exE, standard]) apply (blast intro: fold_msetG_determ) done lemma fold_mset_equality: "fold_msetG f z A y ==> fold_mset f z A = y" unfolding fold_mset_def by (blast intro: fold_msetG_determ) lemma fold_mset_insert: "fold_mset f z (A + {#x#}) = f x (fold_mset f z A)" apply (simp add: fold_mset_def fold_mset_insert_aux union_commute) apply (rule the_equality) apply (auto cong add: conj_cong simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty) done lemma fold_mset_insert_idem: "fold_mset f z (A + {#a#}) = f a (fold_mset f z A)" apply (simp add: fold_mset_def fold_mset_insert_aux) apply (rule the_equality) apply (auto cong add: conj_cong simp add: fold_mset_def [symmetric] fold_mset_equality fold_msetG_nonempty) done lemma fold_mset_commute: "f x (fold_mset f z A) = fold_mset f (f x z) A" by (induct A) (auto simp: fold_mset_insert left_commute [of x]) lemma fold_mset_single [simp]: "fold_mset f z {#x#} = f x z" using fold_mset_insert [of z "{#}"] by simp lemma fold_mset_union [simp]: "fold_mset f z (A+B) = fold_mset f (fold_mset f z A) B" proof (induct A) case empty then show ?case by simp next case (add A x) have "A + {#x#} + B = (A+B) + {#x#}" by(simp add:union_ac) then have "fold_mset f z (A + {#x#} + B) = f x (fold_mset f z (A + B))" by (simp add: fold_mset_insert) also have "… = fold_mset f (fold_mset f z (A + {#x#})) B" by (simp add: fold_mset_commute[of x,symmetric] add fold_mset_insert) finally show ?case . qed lemma fold_mset_fusion: includes left_commutative g shows "(!!x y. h (g x y) = f x (h y)) ==> h (fold_mset g w A) = fold_mset f (h w) A" by (induct A) auto lemma fold_mset_rec: assumes "a ∈# A" shows "fold_mset f z A = f a (fold_mset f z (A - {#a#}))" proof - from assms obtain A' where "A = A' + {#a#}" by (blast dest: multi_member_split) then show ?thesis by simp qed end text {* A note on code generation: When defining some function containing a subterm @{term"fold_mset F"}, code generation is not automatic. When interpreting locale @{text left_commutative} with @{text F}, the would be code thms for @{const fold_mset} become thms like @{term"fold_mset F z {#} = z"} where @{text F} is not a pattern but contains defined symbols, i.e.\ is not a code thm. Hence a separate constant with its own code thms needs to be introduced for @{text F}. See the image operator below. *} subsection {* Image *} definition [code func del]: "image_mset f == fold_mset (op + o single o f) {#}" interpretation image_left_comm: left_commutative ["op + o single o f"] by (unfold_locales) (simp add:union_ac) lemma image_mset_empty [simp, code func]: "image_mset f {#} = {#}" by (simp add: image_mset_def) lemma image_mset_single [simp, code func]: "image_mset f {#x#} = {#f x#}" by (simp add: image_mset_def) lemma image_mset_insert: "image_mset f (M + {#a#}) = image_mset f M + {#f a#}" by (simp add: image_mset_def add_ac) lemma image_mset_union[simp, code func]: "image_mset f (M+N) = image_mset f M + image_mset f N" apply (induct N) apply simp apply (simp add: union_assoc [symmetric] image_mset_insert) done lemma size_image_mset [simp]: "size (image_mset f M) = size M" by (induct M) simp_all lemma image_mset_is_empty_iff [simp]: "image_mset f M = {#} <-> M = {#}" by (cases M) auto syntax comprehension1_mset :: "'a => 'b => 'b multiset => 'a multiset" ("({#_/. _ :# _#})") translations "{#e. x:#M#}" == "CONST image_mset (%x. e) M" syntax comprehension2_mset :: "'a => 'b => 'b multiset => bool => 'a multiset" ("({#_/ | _ :# _./ _#})") translations "{#e | x:#M. P#}" => "{#e. x :# {# x:#M. P#}#}" text {* This allows to write not just filters like @{term "{#x:#M. x<c#}"} but also images like @{term "{#x+x. x:#M #}"} and @{term [source] "{#x+x|x:#M. x<c#}"}, where the latter is currently displayed as @{term "{#x+x|x:#M. x<c#}"}. *} end
lemma multiset_typedef:
y ∈ multiset ==> Rep_multiset (Abs_multiset y) = y
Abs_multiset (Rep_multiset x) = x
Rep_multiset x ∈ multiset
and
(x = y) = (Rep_multiset x = Rep_multiset y)
lemma const0_in_multiset:
(λa. 0) ∈ multiset
lemma only1_in_multiset:
(λb. if b = a then 1 else 0) ∈ multiset
lemma union_preserves_multiset:
[| M ∈ multiset; N ∈ multiset |] ==> (λa. M a + N a) ∈ multiset
lemma diff_preserves_multiset:
M ∈ multiset ==> (λa. M a - N a) ∈ multiset
lemma MCollect_preserves_multiset:
M ∈ multiset ==> (λx. if P x then M x else 0) ∈ multiset
lemma in_multiset:
(λa. 0) ∈ multiset
(λb. if b = a then 1 else 0) ∈ multiset
[| M ∈ multiset; N ∈ multiset |] ==> (λa. M a + N a) ∈ multiset
M ∈ multiset ==> (λa. M a - N a) ∈ multiset
M ∈ multiset ==> (λx. if P x then M x else 0) ∈ multiset
lemma union_empty:
M + {#} = M ∧ {#} + M = M
lemma union_commute:
M + N = N + M
lemma union_assoc:
M + N + K = M + (N + K)
lemma union_lcomm:
M + (N + K) = N + (M + K)
lemma union_ac:
M + N + K = M + (N + K)
M + N = N + M
M + (N + K) = N + (M + K)
lemma diff_empty:
M - {#} = M ∧ {#} - M = {#}
lemma diff_union_inverse2:
M + {#a#} - {#a#} = M
lemma diff_cancel:
A - A = {#}
lemma count_empty:
count {#} a = 0
lemma count_single:
count {#b#} a = (if b = a then 1 else 0)
lemma count_union:
count (M + N) a = count M a + count N a
lemma count_diff:
count (M - N) a = count M a - count N a
lemma count_MCollect:
count (MCollect M P) a = (if P a then count M a else 0)
lemma set_of_empty:
set_of {#} = {}
lemma set_of_single:
set_of {#b#} = {b}
lemma set_of_union:
set_of (M + N) = set_of M ∪ set_of N
lemma set_of_eq_empty_iff:
(set_of M = {}) = (M = {#})
lemma mem_set_of_iff:
(x ∈ set_of M) = (x :# M)
lemma set_of_MCollect:
set_of (MCollect M P) = set_of M ∩ {x. P x}
lemma size_empty:
size {#} = 0
lemma size_single:
size {#b#} = 1
lemma finite_set_of:
finite (set_of M)
lemma setsum_count_Int:
finite A ==> setsum (count N) (A ∩ set_of N) = setsum (count N) A
lemma size_union:
size (M + N) = size M + size N
lemma size_eq_0_iff_empty:
(size M = 0) = (M = {#})
lemma nonempty_has_size:
(S ≠ {#}) = (0 < size S)
lemma size_eq_Suc_imp_elem:
size M = Suc n ==> ∃a. a :# M
lemma multiset_eq_conv_count_eq:
(M = N) = (∀a. count M a = count N a)
lemma single_not_empty:
{#a#} ≠ {#} ∧ {#} ≠ {#a#}
lemma single_eq_single:
({#a#} = {#b#}) = (a = b)
lemma union_eq_empty:
(M + N = {#}) = (M = {#} ∧ N = {#})
lemma empty_eq_union:
({#} = M + N) = (M = {#} ∧ N = {#})
lemma union_right_cancel:
(M + K = N + K) = (M = N)
lemma union_left_cancel:
(K + M = K + N) = (M = N)
lemma union_is_single:
(M + N = {#a#}) = (M = {#a#} ∧ N = {#} ∨ M = {#} ∧ N = {#a#})
lemma single_is_union:
({#a#} = M + N) = ({#a#} = M ∧ N = {#} ∨ M = {#} ∧ {#a#} = N)
lemma add_eq_conv_diff:
(M + {#a#} = N + {#b#}) =
(M = N ∧ a = b ∨ M = N - {#a#} + {#b#} ∧ N = M - {#b#} + {#a#})
lemma insert_DiffM:
x :# M ==> {#x#} + (M - {#x#}) = M
lemma insert_DiffM2:
x :# M ==> M - {#x#} + {#x#} = M
lemma multi_union_self_other_eq:
A + X = A + Y ==> X = Y
lemma multi_self_add_other_not_self:
(A = A + {#x#}) = False
lemma insert_noteq_member:
[| B + {#b#} = C + {#c#}; b ≠ c |] ==> c :# B
lemma add_eq_conv_ex:
(M + {#a#} = N + {#b#}) = (M = N ∧ a = b ∨ (∃K. M = K + {#b#} ∧ N = K + {#a#}))
lemma empty_multiset_count:
(∀x. count A x = 0) = (A = {#})
lemma multiset_inter_count:
count (A #∩ B) x = min (count A x) (count B x)
lemma multiset_inter_commute:
A #∩ B = B #∩ A
lemma multiset_inter_assoc:
A #∩ (B #∩ C) = A #∩ B #∩ C
lemma multiset_inter_left_commute:
A #∩ (B #∩ C) = B #∩ (A #∩ C)
lemma multiset_inter_ac:
A #∩ B = B #∩ A
A #∩ (B #∩ C) = A #∩ B #∩ C
A #∩ (B #∩ C) = B #∩ (A #∩ C)
lemma multiset_inter_single:
a ≠ b ==> {#a#} #∩ {#b#} = {#}
lemma multiset_union_diff_commute:
B #∩ C = {#} ==> A + B - C = A - C + B
lemma MCollect_empty:
MCollect {#} P = {#}
lemma MCollect_single:
MCollect {#x#} P = (if P x then {#x#} else {#})
lemma MCollect_union:
MCollect (M + N) f = MCollect M f + MCollect N f
lemma setsum_decr:
[| finite F; 0 < f a |]
==> setsum (f(a := f a - 1)) F = (if a ∈ F then setsum f F - 1 else setsum f F)
lemma rep_multiset_induct_aux:
[| P (λa. 0); !!f b. [| f ∈ multiset; P f |] ==> P (f(b := f b + 1)) |]
==> ∀f. f ∈ multiset --> setsum f {x. f x ≠ 0} = n --> P f
theorem rep_multiset_induct:
[| f ∈ multiset; P (λa. 0);
!!f b. [| f ∈ multiset; P f |] ==> P (f(b := f b + 1)) |]
==> P f
theorem multiset_induct:
[| P {#}; !!M x. P M ==> P (M + {#x#}) |] ==> P M
lemma multi_nonempty_split:
M ≠ {#} ==> ∃A a. M = A + {#a#}
lemma multiset_cases:
[| M = {#} ==> P; !!N x. M = N + {#x#} ==> P |] ==> P
lemma multi_member_split:
x :# M ==> ∃A. M = A + {#x#}
lemma multiset_partition:
M = MCollect M P + {# x :# M. ¬ P x#}
lemma multi_drop_mem_not_eq:
c :# B ==> B - {#c#} ≠ B
lemma not_less_empty:
(M, {#}) ∉ mult1 r
lemma less_add:
(N, M0.0 + {#a#}) ∈ mult1 r
==> (∃M. (M, M0.0) ∈ mult1 r ∧ N = M + {#a#}) ∨
(∃K. (∀b. b :# K --> (b, a) ∈ r) ∧ N = M0.0 + K)
lemma all_accessible:
wf r ==> ∀M. M ∈ acc (mult1 r)
theorem wf_mult1:
wf r ==> wf (mult1 r)
theorem wf_mult:
wf r ==> wf (mult r)
lemma diff_union_single_conv:
a :# J ==> I + J - {#a#} = I + (J - {#a#})
lemma mult_implies_one_step:
[| trans r; (M, N) ∈ mult r |]
==> ∃I J K.
N = I + J ∧ M = I + K ∧ J ≠ {#} ∧ (∀k∈set_of K. ∃j∈set_of J. (k, j) ∈ r)
lemma elem_imp_eq_diff_union:
a :# M ==> M = M - {#a#} + {#a#}
lemma size_eq_Suc_imp_eq_union:
size M = Suc n ==> ∃a N. M = N + {#a#}
lemma one_step_implies_mult_aux:
trans r
==> ∀I J K.
size J = n ∧ J ≠ {#} ∧ (∀k∈set_of K. ∃j∈set_of J. (k, j) ∈ r) -->
(I + K, I + J) ∈ mult r
lemma one_step_implies_mult:
[| trans r; J ≠ {#}; ∀k∈set_of K. ∃j∈set_of J. (k, j) ∈ r |]
==> (I + K, I + J) ∈ mult r
lemma trans_base_order:
transP op <
lemma mult_irrefl_aux:
[| finite A; ∀x∈A. ∃y∈A. x < y |] ==> A = {}
lemma mult_less_not_refl:
¬ M < M
lemma mult_less_irrefl:
M < M ==> R
theorem mult_less_trans:
[| K < M; M < N |] ==> K < N
theorem mult_less_not_sym:
M < N ==> ¬ N < M
theorem mult_less_asym:
[| M < N; ¬ P ==> N < M |] ==> P
theorem mult_le_refl:
M ≤ M
theorem mult_le_antisym:
[| M ≤ N; N ≤ M |] ==> M = N
theorem mult_le_trans:
[| K ≤ M; M ≤ N |] ==> K ≤ N
theorem mult_less_le:
(M < N) = (M ≤ N ∧ M ≠ N)
lemma mult1_union:
[| (B, D) ∈ mult1 r; trans r |] ==> (C + B, C + D) ∈ mult1 r
lemma union_less_mono2:
B < D ==> C + B < C + D
lemma union_less_mono1:
B < D ==> B + C < D + C
lemma union_less_mono:
[| A < C; B < D |] ==> A + B < C + D
lemma union_le_mono:
[| A ≤ C; B ≤ D |] ==> A + B ≤ C + D
lemma empty_leI:
{#} ≤ M
lemma union_upper1:
A ≤ A + B
lemma union_upper2:
B ≤ A + B
lemma multiset_of_zero_iff:
(multiset_of x = {#}) = (x = [])
lemma multiset_of_zero_iff_right:
({#} = multiset_of x) = (x = [])
lemma set_of_multiset_of:
set_of (multiset_of x) = set x
lemma mem_set_multiset_eq:
(x ∈ set xs) = (x :# multiset_of xs)
lemma multiset_of_append:
multiset_of (xs @ ys) = multiset_of xs + multiset_of ys
lemma surj_multiset_of:
surj multiset_of
lemma set_count_greater_0:
set x = {a. a :# multiset_of x}
lemma distinct_count_atmost_1:
distinct x = (∀a. count (multiset_of x) a = (if a ∈ set x then 1 else 0))
lemma multiset_of_eq_setD:
multiset_of xs = multiset_of ys ==> set xs = set ys
lemma set_eq_iff_multiset_of_eq_distinct:
[| distinct x; distinct y |]
==> (set x = set y) = (multiset_of x = multiset_of y)
lemma set_eq_iff_multiset_of_remdups_eq:
(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))
lemma multiset_of_compl_union:
multiset_of (filter P xs) + multiset_of [x\<leftarrow>xs . ¬ P x] =
multiset_of xs
lemma count_filter:
count (multiset_of xs) x = length [y\<leftarrow>xs . y = x]
lemma nth_mem_multiset_of:
i < length ls ==> ls ! i :# multiset_of ls
lemma multiset_of_remove1:
multiset_of (remove1 a xs) = multiset_of xs - {#a#}
lemma multiset_of_eq_length:
multiset_of xs = multiset_of ys ==> length xs = length ys
lemma properties_for_sort:
[| multiset_of ys = multiset_of xs; sorted ys |] ==> sort xs = ys
lemma mset_le_refl:
A ⊆# A
lemma mset_le_trans:
[| A ⊆# B; B ⊆# C |] ==> A ⊆# C
lemma mset_le_antisym:
[| A ⊆# B; B ⊆# A |] ==> A = B
lemma mset_le_exists_conv:
(A ⊆# B) = (∃C. B = A + C)
lemma mset_le_mono_add_right_cancel:
(A + C ⊆# B + C) = (A ⊆# B)
lemma mset_le_mono_add_left_cancel:
(C + A ⊆# C + B) = (A ⊆# B)
lemma mset_le_mono_add:
[| A ⊆# B; C ⊆# D |] ==> A + C ⊆# B + D
lemma mset_le_add_left:
A ⊆# A + B
lemma mset_le_add_right:
B ⊆# A + B
lemma mset_le_single:
a :# B ==> {#a#} ⊆# B
lemma multiset_diff_union_assoc:
C ⊆# B ==> A + B - C = A + (B - C)
lemma mset_le_multiset_union_diff_commute:
B ⊆# A ==> A - B + C = A + C - B
lemma multiset_of_remdups_le:
multiset_of (remdups xs) ⊆# multiset_of xs
lemma multiset_of_update:
i < length ls ==> multiset_of (ls[i := v]) = multiset_of ls - {#ls ! i#} + {#v#}
lemma multiset_of_swap:
[| i < length ls; j < length ls |]
==> multiset_of (ls[j := ls ! i, i := ls ! j]) = multiset_of ls
lemma mset_lessD:
[| A ⊂# B; x :# A |] ==> x :# B
lemma mset_leD:
[| A ⊆# B; x :# A |] ==> x :# B
lemma mset_less_insertD:
A + {#x#} ⊂# B ==> x :# B ∧ A ⊂# B
lemma mset_le_insertD:
A + {#x#} ⊆# B ==> x :# B ∧ A ⊆# B
lemma mset_less_of_empty:
(A ⊂# {#}) = False
lemma multi_psub_of_add_self:
A ⊂# A + {#x#}
lemma multi_psub_self:
(A ⊂# A) = False
lemma mset_less_add_bothsides:
T + {#x#} ⊂# S + {#x#} ==> T ⊂# S
lemma mset_less_empty_nonempty:
({#} ⊂# S) = (S ≠ {#})
lemma mset_less_size:
A ⊂# B ==> size A < size B
lemma mset_less_trans:
[| x ⊂# y; y ⊂# z |] ==> x ⊂# z
lemma mset_less_diff_self:
c :# B ==> B - {#c#} ⊂# B
lemma multiset_add_sub_el_shuffle:
[| c :# B; b ≠ c |] ==> B - {#c#} + {#b#} = B + {#b#} - {#c#}
lemma wf_mset_less_rel:
wf mset_less_rel
lemma full_multiset_induct:
(!!B. ∀A. A ⊂# B --> P A ==> P B) ==> P B
lemma multi_subset_induct:
[| F ⊆# A; P {#}; !!a F. [| a :# A; P F |] ==> P (F + {#a#}) |] ==> P F
lemma multi_count_eq:
(∀x. count A x = count B x) = (A = B)
lemma multi_count_ext:
(!!x. count A x = count B x) ==> A = B
lemma Diff1_fold_msetG:
[| fold_msetG f z (A - {#x#}) y; x :# A |] ==> fold_msetG f z A (f x y)
lemma fold_msetG_nonempty:
∃x. fold_msetG f z A x
lemma fold_mset_empty:
fold_mset f z {#} = z
lemma fold_msetG_determ:
[| fold_msetG f z A x; fold_msetG f z A y |] ==> y = x
lemma fold_mset_insert_aux:
fold_msetG f z (A + {#x#}) v = (∃y. fold_msetG f z A y ∧ v = f x y)
lemma fold_mset_equality:
fold_msetG f z A y ==> fold_mset f z A = y
lemma fold_mset_insert:
fold_mset f z (A + {#x#}) = f x (fold_mset f z A)
lemma fold_mset_insert_idem:
fold_mset f z (A + {#a#}) = f a (fold_mset f z A)
lemma fold_mset_commute:
f x (fold_mset f z A) = fold_mset f (f x z) A
lemma fold_mset_single:
fold_mset f z {#x#} = f x z
lemma fold_mset_union:
fold_mset f z (A + B) = fold_mset f (fold_mset f z A) B
lemma fold_mset_fusion:
[| left_commutative g; !!x y. h (g x y) = f x (h y) |]
==> h (fold_mset g w A) = fold_mset f (h w) A
lemma fold_mset_rec:
a :# A ==> fold_mset f z A = f a (fold_mset f z (A - {#a#}))
lemma image_mset_empty:
image_mset f {#} = {#}
lemma image_mset_single:
image_mset f {#x#} = {#f x#}
lemma image_mset_insert:
image_mset f (M + {#a#}) = image_mset f M + {#f a#}
lemma image_mset_union:
image_mset f (M + N) = image_mset f M + image_mset f N
lemma size_image_mset:
size (image_mset f M) = size M
lemma image_mset_is_empty_iff:
(image_mset f M = {#}) = (M = {#})