author | kleing |
Thu, 13 Dec 2007 06:51:22 +0100 | |
changeset 25610 | 72e1563aee09 |
parent 25595 | 6c48275f9c76 |
child 25622 | 6067d838041a |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Multiset.thy |
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ID: $Id$ |
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Author: Tobias Nipkow, Markus Wenzel, Lawrence C Paulson, Norbert Voelker |
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*) |
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header {* Multisets *} |
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theory Multiset |
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imports List |
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begin |
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subsection {* The type of multisets *} |
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typedef 'a multiset = "{f::'a => nat. finite {x . f x > 0}}" |
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proof |
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show "(\<lambda>x. 0::nat) \<in> ?multiset" by simp |
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qed |
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lemmas multiset_typedef [simp] = |
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Abs_multiset_inverse Rep_multiset_inverse Rep_multiset |
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and [simp] = Rep_multiset_inject [symmetric] |
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definition |
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Mempty :: "'a multiset" ("{#}") where |
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"{#} = Abs_multiset (\<lambda>a. 0)" |
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definition |
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single :: "'a => 'a multiset" where |
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"single a = Abs_multiset (\<lambda>b. if b = a then 1 else 0)" |
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definition |
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count :: "'a multiset => 'a => nat" where |
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"count = Rep_multiset" |
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definition |
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MCollect :: "'a multiset => ('a => bool) => 'a multiset" where |
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"MCollect M P = Abs_multiset (\<lambda>x. if P x then Rep_multiset M x else 0)" |
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abbreviation |
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Melem :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) where |
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"a :# M == 0 < count M a" |
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notation (xsymbols) Melem (infix "\<in>#" 50) |
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syntax |
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"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ : _./ _#})") |
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translations |
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"{#x:M. P#}" == "CONST MCollect M (\<lambda>x. P)" |
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definition |
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set_of :: "'a multiset => 'a set" where |
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"set_of M = {x. x :# M}" |
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instantiation multiset :: (type) "{plus, minus, zero, size}" |
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begin |
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definition |
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union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)" |
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definition |
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diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)" |
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definition |
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Zero_multiset_def [simp]: "0 == {#}" |
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definition |
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size_def: "size M == setsum (count M) (set_of M)" |
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instance .. |
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end |
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definition |
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multiset_inter :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> 'a multiset" (infixl "#\<inter>" 70) where |
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"multiset_inter A B = A - (A - B)" |
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syntax -- "Multiset Enumeration" |
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"@multiset" :: "args => 'a multiset" ("{#(_)#}") |
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translations |
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"{#x, xs#}" == "{#x#} + {#xs#}" |
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"{#x#}" == "CONST single x" |
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text {* |
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\medskip Preservation of the representing set @{term multiset}. |
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*} |
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lemma const0_in_multiset [simp]: "(\<lambda>a. 0) \<in> multiset" |
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by (simp add: multiset_def) |
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lemma only1_in_multiset [simp]: "(\<lambda>b. if b = a then 1 else 0) \<in> multiset" |
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by (simp add: multiset_def) |
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lemma union_preserves_multiset [simp]: |
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"M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset" |
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apply (simp add: multiset_def) |
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apply (drule (1) finite_UnI) |
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apply (simp del: finite_Un add: Un_def) |
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done |
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lemma diff_preserves_multiset [simp]: |
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"M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset" |
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apply (simp add: multiset_def) |
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apply (rule finite_subset) |
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apply auto |
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done |
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subsection {* Algebraic properties of multisets *} |
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subsubsection {* Union *} |
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lemma union_empty [simp]: "M + {#} = M \<and> {#} + M = M" |
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by (simp add: union_def Mempty_def) |
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lemma union_commute: "M + N = N + (M::'a multiset)" |
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by (simp add: union_def add_ac) |
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lemma union_assoc: "(M + N) + K = M + (N + (K::'a multiset))" |
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by (simp add: union_def add_ac) |
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lemma union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))" |
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proof - |
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have "M + (N + K) = (N + K) + M" |
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by (rule union_commute) |
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also have "\<dots> = N + (K + M)" |
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by (rule union_assoc) |
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also have "K + M = M + K" |
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by (rule union_commute) |
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finally show ?thesis . |
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qed |
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lemmas union_ac = union_assoc union_commute union_lcomm |
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instance multiset :: (type) comm_monoid_add |
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proof |
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fix a b c :: "'a multiset" |
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show "(a + b) + c = a + (b + c)" by (rule union_assoc) |
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show "a + b = b + a" by (rule union_commute) |
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show "0 + a = a" by simp |
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qed |
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subsubsection {* Difference *} |
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lemma diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}" |
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by (simp add: Mempty_def diff_def) |
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lemma diff_union_inverse2 [simp]: "M + {#a#} - {#a#} = M" |
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by (simp add: union_def diff_def) |
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subsubsection {* Count of elements *} |
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lemma count_empty [simp]: "count {#} a = 0" |
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by (simp add: count_def Mempty_def) |
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lemma count_single [simp]: "count {#b#} a = (if b = a then 1 else 0)" |
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by (simp add: count_def single_def) |
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lemma count_union [simp]: "count (M + N) a = count M a + count N a" |
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by (simp add: count_def union_def) |
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lemma count_diff [simp]: "count (M - N) a = count M a - count N a" |
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by (simp add: count_def diff_def) |
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subsubsection {* Set of elements *} |
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lemma set_of_empty [simp]: "set_of {#} = {}" |
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by (simp add: set_of_def) |
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lemma set_of_single [simp]: "set_of {#b#} = {b}" |
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by (simp add: set_of_def) |
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lemma set_of_union [simp]: "set_of (M + N) = set_of M \<union> set_of N" |
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by (auto simp add: set_of_def) |
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lemma set_of_eq_empty_iff [simp]: "(set_of M = {}) = (M = {#})" |
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by (auto simp add: set_of_def Mempty_def count_def expand_fun_eq) |
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lemma mem_set_of_iff [simp]: "(x \<in> set_of M) = (x :# M)" |
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by (auto simp add: set_of_def) |
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subsubsection {* Size *} |
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lemma size_empty [simp]: "size {#} = 0" |
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by (simp add: size_def) |
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lemma size_single [simp]: "size {#b#} = 1" |
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by (simp add: size_def) |
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lemma finite_set_of [iff]: "finite (set_of M)" |
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using Rep_multiset [of M] |
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by (simp add: multiset_def set_of_def count_def) |
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lemma setsum_count_Int: |
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"finite A ==> setsum (count N) (A \<inter> set_of N) = setsum (count N) A" |
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apply (induct rule: finite_induct) |
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apply simp |
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apply (simp add: Int_insert_left set_of_def) |
204 |
done |
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||
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lemma size_union [simp]: "size (M + N::'a multiset) = size M + size N" |
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apply (unfold size_def) |
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apply (subgoal_tac "count (M + N) = (\<lambda>a. count M a + count N a)") |
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prefer 2 |
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apply (rule ext, simp) |
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apply (simp (no_asm_simp) add: setsum_Un_nat setsum_addf setsum_count_Int) |
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apply (subst Int_commute) |
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apply (simp (no_asm_simp) add: setsum_count_Int) |
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done |
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lemma size_eq_0_iff_empty [iff]: "(size M = 0) = (M = {#})" |
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apply (unfold size_def Mempty_def count_def, auto) |
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apply (simp add: set_of_def count_def expand_fun_eq) |
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done |
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lemma size_eq_Suc_imp_elem: "size M = Suc n ==> \<exists>a. a :# M" |
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apply (unfold size_def) |
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apply (drule setsum_SucD, auto) |
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done |
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subsubsection {* Equality of multisets *} |
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lemma multiset_eq_conv_count_eq: "(M = N) = (\<forall>a. count M a = count N a)" |
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by (simp add: count_def expand_fun_eq) |
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lemma single_not_empty [simp]: "{#a#} \<noteq> {#} \<and> {#} \<noteq> {#a#}" |
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by (simp add: single_def Mempty_def expand_fun_eq) |
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lemma single_eq_single [simp]: "({#a#} = {#b#}) = (a = b)" |
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by (auto simp add: single_def expand_fun_eq) |
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lemma union_eq_empty [iff]: "(M + N = {#}) = (M = {#} \<and> N = {#})" |
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by (auto simp add: union_def Mempty_def expand_fun_eq) |
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lemma empty_eq_union [iff]: "({#} = M + N) = (M = {#} \<and> N = {#})" |
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by (auto simp add: union_def Mempty_def expand_fun_eq) |
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lemma union_right_cancel [simp]: "(M + K = N + K) = (M = (N::'a multiset))" |
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by (simp add: union_def expand_fun_eq) |
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lemma union_left_cancel [simp]: "(K + M = K + N) = (M = (N::'a multiset))" |
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by (simp add: union_def expand_fun_eq) |
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lemma union_is_single: |
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"(M + N = {#a#}) = (M = {#a#} \<and> N={#} \<or> M = {#} \<and> N = {#a#})" |
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apply (simp add: Mempty_def single_def union_def add_is_1 expand_fun_eq) |
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apply blast |
253 |
done |
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lemma single_is_union: |
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"({#a#} = M + N) = ({#a#} = M \<and> N = {#} \<or> M = {#} \<and> {#a#} = N)" |
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apply (unfold Mempty_def single_def union_def) |
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apply (simp add: add_is_1 one_is_add expand_fun_eq) |
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apply (blast dest: sym) |
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done |
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lemma add_eq_conv_diff: |
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"(M + {#a#} = N + {#b#}) = |
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(M = N \<and> a = b \<or> M = N - {#a#} + {#b#} \<and> N = M - {#b#} + {#a#})" |
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using [[simproc del: neq]] |
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apply (unfold single_def union_def diff_def) |
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apply (simp (no_asm) add: expand_fun_eq) |
|
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apply (rule conjI, force, safe, simp_all) |
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apply (simp add: eq_sym_conv) |
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done |
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declare Rep_multiset_inject [symmetric, simp del] |
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instance multiset :: (type) cancel_ab_semigroup_add |
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proof |
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fix a b c :: "'a multiset" |
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show "a + b = a + c \<Longrightarrow> b = c" by simp |
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qed |
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lemma insert_DiffM: |
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"x \<in># M \<Longrightarrow> {#x#} + (M - {#x#}) = M" |
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by (clarsimp simp: multiset_eq_conv_count_eq) |
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lemma insert_DiffM2[simp]: |
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"x \<in># M \<Longrightarrow> M - {#x#} + {#x#} = M" |
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by (clarsimp simp: multiset_eq_conv_count_eq) |
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lemma multi_union_self_other_eq: |
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"(A::'a multiset) + X = A + Y \<Longrightarrow> X = Y" |
|
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by (induct A arbitrary: X Y, auto) |
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lemma multi_self_add_other_not_self[simp]: "(A = A + {#x#}) = False" |
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proof - |
|
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{ |
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assume a: "A = A + {#x#}" |
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have "A = A + {#}" by simp |
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hence "A + {#} = A + {#x#}" using a by auto |
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hence "{#} = {#x#}" |
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by - (drule multi_union_self_other_eq) |
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hence False by auto |
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} |
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thus ?thesis by blast |
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qed |
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lemma insert_noteq_member: |
|
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assumes BC: "B + {#b#} = C + {#c#}" |
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and bnotc: "b \<noteq> c" |
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shows "c \<in># B" |
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proof - |
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have "c \<in># C + {#c#}" by simp |
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have nc: "\<not> c \<in># {#b#}" using bnotc by simp |
|
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hence "c \<in># B + {#b#}" using BC by simp |
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thus "c \<in># B" using nc by simp |
|
314 |
qed |
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316 |
||
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subsubsection {* Intersection *} |
318 |
||
319 |
lemma multiset_inter_count: |
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"count (A #\<inter> B) x = min (count A x) (count B x)" |
321 |
by (simp add: multiset_inter_def min_def) |
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15869 | 322 |
|
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lemma multiset_inter_commute: "A #\<inter> B = B #\<inter> A" |
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by (simp add: multiset_eq_conv_count_eq multiset_inter_count |
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325 |
min_max.inf_commute) |
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|
327 |
lemma multiset_inter_assoc: "A #\<inter> (B #\<inter> C) = A #\<inter> B #\<inter> C" |
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by (simp add: multiset_eq_conv_count_eq multiset_inter_count |
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haftmann
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20770
diff
changeset
|
329 |
min_max.inf_assoc) |
15869 | 330 |
|
331 |
lemma multiset_inter_left_commute: "A #\<inter> (B #\<inter> C) = B #\<inter> (A #\<inter> C)" |
|
332 |
by (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def) |
|
333 |
||
17161 | 334 |
lemmas multiset_inter_ac = |
335 |
multiset_inter_commute |
|
336 |
multiset_inter_assoc |
|
337 |
multiset_inter_left_commute |
|
15869 | 338 |
|
339 |
lemma multiset_union_diff_commute: "B #\<inter> C = {#} \<Longrightarrow> A + B - C = A - C + B" |
|
17200 | 340 |
apply (simp add: multiset_eq_conv_count_eq multiset_inter_count min_def |
17161 | 341 |
split: split_if_asm) |
15869 | 342 |
apply clarsimp |
17161 | 343 |
apply (erule_tac x = a in allE) |
15869 | 344 |
apply auto |
345 |
done |
|
346 |
||
10249 | 347 |
|
348 |
subsection {* Induction over multisets *} |
|
349 |
||
350 |
lemma setsum_decr: |
|
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11655
diff
changeset
|
351 |
"finite F ==> (0::nat) < f a ==> |
15072 | 352 |
setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)" |
18258 | 353 |
apply (induct rule: finite_induct) |
354 |
apply auto |
|
15072 | 355 |
apply (drule_tac a = a in mk_disjoint_insert, auto) |
10249 | 356 |
done |
357 |
||
10313 | 358 |
lemma rep_multiset_induct_aux: |
18730 | 359 |
assumes 1: "P (\<lambda>a. (0::nat))" |
360 |
and 2: "!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))" |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
24035
diff
changeset
|
361 |
shows "\<forall>f. f \<in> multiset --> setsum f {x. f x \<noteq> 0} = n --> P f" |
18730 | 362 |
apply (unfold multiset_def) |
363 |
apply (induct_tac n, simp, clarify) |
|
364 |
apply (subgoal_tac "f = (\<lambda>a.0)") |
|
365 |
apply simp |
|
366 |
apply (rule 1) |
|
367 |
apply (rule ext, force, clarify) |
|
368 |
apply (frule setsum_SucD, clarify) |
|
369 |
apply (rename_tac a) |
|
25162 | 370 |
apply (subgoal_tac "finite {x. (f (a := f a - 1)) x > 0}") |
18730 | 371 |
prefer 2 |
372 |
apply (rule finite_subset) |
|
373 |
prefer 2 |
|
374 |
apply assumption |
|
375 |
apply simp |
|
376 |
apply blast |
|
377 |
apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)") |
|
378 |
prefer 2 |
|
379 |
apply (rule ext) |
|
380 |
apply (simp (no_asm_simp)) |
|
381 |
apply (erule ssubst, rule 2 [unfolded multiset_def], blast) |
|
382 |
apply (erule allE, erule impE, erule_tac [2] mp, blast) |
|
383 |
apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def) |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
24035
diff
changeset
|
384 |
apply (subgoal_tac "{x. x \<noteq> a --> f x \<noteq> 0} = {x. f x \<noteq> 0}") |
18730 | 385 |
prefer 2 |
386 |
apply blast |
|
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
24035
diff
changeset
|
387 |
apply (subgoal_tac "{x. x \<noteq> a \<and> f x \<noteq> 0} = {x. f x \<noteq> 0} - {a}") |
18730 | 388 |
prefer 2 |
389 |
apply blast |
|
390 |
apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong) |
|
391 |
done |
|
10249 | 392 |
|
10313 | 393 |
theorem rep_multiset_induct: |
11464 | 394 |
"f \<in> multiset ==> P (\<lambda>a. 0) ==> |
11701
3d51fbf81c17
sane numerals (stage 1): added generic 1, removed 1' and 2 on nat,
wenzelm
parents:
11655
diff
changeset
|
395 |
(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f" |
17161 | 396 |
using rep_multiset_induct_aux by blast |
10249 | 397 |
|
18258 | 398 |
theorem multiset_induct [case_names empty add, induct type: multiset]: |
399 |
assumes empty: "P {#}" |
|
400 |
and add: "!!M x. P M ==> P (M + {#x#})" |
|
17161 | 401 |
shows "P M" |
10249 | 402 |
proof - |
403 |
note defns = union_def single_def Mempty_def |
|
404 |
show ?thesis |
|
405 |
apply (rule Rep_multiset_inverse [THEN subst]) |
|
10313 | 406 |
apply (rule Rep_multiset [THEN rep_multiset_induct]) |
18258 | 407 |
apply (rule empty [unfolded defns]) |
15072 | 408 |
apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))") |
10249 | 409 |
prefer 2 |
410 |
apply (simp add: expand_fun_eq) |
|
411 |
apply (erule ssubst) |
|
17200 | 412 |
apply (erule Abs_multiset_inverse [THEN subst]) |
18258 | 413 |
apply (erule add [unfolded defns, simplified]) |
10249 | 414 |
done |
415 |
qed |
|
416 |
||
25610 | 417 |
lemma empty_multiset_count: |
418 |
"(\<forall>x. count A x = 0) = (A = {#})" |
|
419 |
apply (rule iffI) |
|
420 |
apply (induct A, simp) |
|
421 |
apply (erule_tac x=x in allE) |
|
422 |
apply auto |
|
423 |
done |
|
424 |
||
425 |
subsection {* Case splits *} |
|
426 |
||
427 |
lemma multi_nonempty_split: "M \<noteq> {#} \<Longrightarrow> \<exists>A a. M = A + {#a#}" |
|
428 |
by (induct M, auto) |
|
429 |
||
430 |
lemma multiset_cases [cases type, case_names empty add]: |
|
431 |
assumes em: "M = {#} \<Longrightarrow> P" |
|
432 |
assumes add: "\<And>N x. M = N + {#x#} \<Longrightarrow> P" |
|
433 |
shows "P" |
|
434 |
proof (cases "M = {#}") |
|
435 |
assume "M = {#}" thus ?thesis using em by simp |
|
436 |
next |
|
437 |
assume "M \<noteq> {#}" |
|
438 |
then obtain M' m where "M = M' + {#m#}" |
|
439 |
by (blast dest: multi_nonempty_split) |
|
440 |
thus ?thesis using add by simp |
|
441 |
qed |
|
442 |
||
443 |
lemma multi_member_split: "x \<in># M \<Longrightarrow> \<exists>A. M = A + {#x#}" |
|
444 |
apply (cases M, simp) |
|
445 |
apply (rule_tac x="M - {#x#}" in exI, simp) |
|
446 |
done |
|
447 |
||
10249 | 448 |
lemma MCollect_preserves_multiset: |
11464 | 449 |
"M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset" |
10249 | 450 |
apply (simp add: multiset_def) |
15072 | 451 |
apply (rule finite_subset, auto) |
10249 | 452 |
done |
453 |
||
17161 | 454 |
lemma count_MCollect [simp]: |
10249 | 455 |
"count {# x:M. P x #} a = (if P a then count M a else 0)" |
15072 | 456 |
by (simp add: count_def MCollect_def MCollect_preserves_multiset) |
10249 | 457 |
|
17161 | 458 |
lemma set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}" |
459 |
by (auto simp add: set_of_def) |
|
10249 | 460 |
|
17161 | 461 |
lemma multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}" |
462 |
by (subst multiset_eq_conv_count_eq, auto) |
|
10249 | 463 |
|
17161 | 464 |
lemma add_eq_conv_ex: |
465 |
"(M + {#a#} = N + {#b#}) = |
|
466 |
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))" |
|
15072 | 467 |
by (auto simp add: add_eq_conv_diff) |
10249 | 468 |
|
15869 | 469 |
declare multiset_typedef [simp del] |
10249 | 470 |
|
25610 | 471 |
lemma nonempty_has_size: "(S \<noteq> {#}) = (0 < size S)" |
472 |
apply (rule iffI) |
|
473 |
apply (case_tac "size S = 0") |
|
474 |
apply clarsimp |
|
475 |
apply (subst (asm) neq0_conv) |
|
476 |
apply auto |
|
477 |
done |
|
478 |
||
479 |
lemma multi_drop_mem_not_eq: "c \<in># B \<Longrightarrow> B - {#c#} \<noteq> B" |
|
480 |
by (cases "B={#}", auto dest: multi_member_split) |
|
17161 | 481 |
|
10249 | 482 |
subsection {* Multiset orderings *} |
483 |
||
484 |
subsubsection {* Well-foundedness *} |
|
485 |
||
19086 | 486 |
definition |
23751 | 487 |
mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where |
19086 | 488 |
"mult1 r = |
23751 | 489 |
{(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> |
490 |
(\<forall>b. b :# K --> (b, a) \<in> r)}" |
|
10249 | 491 |
|
21404
eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
21214
diff
changeset
|
492 |
definition |
23751 | 493 |
mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" where |
494 |
"mult r = (mult1 r)\<^sup>+" |
|
10249 | 495 |
|
23751 | 496 |
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r" |
10277 | 497 |
by (simp add: mult1_def) |
10249 | 498 |
|
23751 | 499 |
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==> |
500 |
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or> |
|
501 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)" |
|
19582 | 502 |
(is "_ \<Longrightarrow> ?case1 (mult1 r) \<or> ?case2") |
10249 | 503 |
proof (unfold mult1_def) |
23751 | 504 |
let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r" |
11464 | 505 |
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a" |
23751 | 506 |
let ?case1 = "?case1 {(N, M). ?R N M}" |
10249 | 507 |
|
23751 | 508 |
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}" |
18258 | 509 |
then have "\<exists>a' M0' K. |
11464 | 510 |
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp |
18258 | 511 |
then show "?case1 \<or> ?case2" |
10249 | 512 |
proof (elim exE conjE) |
513 |
fix a' M0' K |
|
514 |
assume N: "N = M0' + K" and r: "?r K a'" |
|
515 |
assume "M0 + {#a#} = M0' + {#a'#}" |
|
18258 | 516 |
then have "M0 = M0' \<and> a = a' \<or> |
11464 | 517 |
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})" |
10249 | 518 |
by (simp only: add_eq_conv_ex) |
18258 | 519 |
then show ?thesis |
10249 | 520 |
proof (elim disjE conjE exE) |
521 |
assume "M0 = M0'" "a = a'" |
|
11464 | 522 |
with N r have "?r K a \<and> N = M0 + K" by simp |
18258 | 523 |
then have ?case2 .. then show ?thesis .. |
10249 | 524 |
next |
525 |
fix K' |
|
526 |
assume "M0' = K' + {#a#}" |
|
527 |
with N have n: "N = K' + K + {#a#}" by (simp add: union_ac) |
|
528 |
||
529 |
assume "M0 = K' + {#a'#}" |
|
530 |
with r have "?R (K' + K) M0" by blast |
|
18258 | 531 |
with n have ?case1 by simp then show ?thesis .. |
10249 | 532 |
qed |
533 |
qed |
|
534 |
qed |
|
535 |
||
23751 | 536 |
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)" |
10249 | 537 |
proof |
538 |
let ?R = "mult1 r" |
|
539 |
let ?W = "acc ?R" |
|
540 |
{ |
|
541 |
fix M M0 a |
|
23751 | 542 |
assume M0: "M0 \<in> ?W" |
543 |
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
|
544 |
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W" |
|
545 |
have "M0 + {#a#} \<in> ?W" |
|
546 |
proof (rule accI [of "M0 + {#a#}"]) |
|
10249 | 547 |
fix N |
23751 | 548 |
assume "(N, M0 + {#a#}) \<in> ?R" |
549 |
then have "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or> |
|
550 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))" |
|
10249 | 551 |
by (rule less_add) |
23751 | 552 |
then show "N \<in> ?W" |
10249 | 553 |
proof (elim exE disjE conjE) |
23751 | 554 |
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}" |
555 |
from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" .. |
|
556 |
from this and `(M, M0) \<in> ?R` have "M + {#a#} \<in> ?W" .. |
|
557 |
then show "N \<in> ?W" by (simp only: N) |
|
10249 | 558 |
next |
559 |
fix K |
|
560 |
assume N: "N = M0 + K" |
|
23751 | 561 |
assume "\<forall>b. b :# K --> (b, a) \<in> r" |
562 |
then have "M0 + K \<in> ?W" |
|
10249 | 563 |
proof (induct K) |
18730 | 564 |
case empty |
23751 | 565 |
from M0 show "M0 + {#} \<in> ?W" by simp |
18730 | 566 |
next |
567 |
case (add K x) |
|
23751 | 568 |
from add.prems have "(x, a) \<in> r" by simp |
569 |
with wf_hyp have "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast |
|
570 |
moreover from add have "M0 + K \<in> ?W" by simp |
|
571 |
ultimately have "(M0 + K) + {#x#} \<in> ?W" .. |
|
572 |
then show "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc) |
|
10249 | 573 |
qed |
23751 | 574 |
then show "N \<in> ?W" by (simp only: N) |
10249 | 575 |
qed |
576 |
qed |
|
577 |
} note tedious_reasoning = this |
|
578 |
||
23751 | 579 |
assume wf: "wf r" |
10249 | 580 |
fix M |
23751 | 581 |
show "M \<in> ?W" |
10249 | 582 |
proof (induct M) |
23751 | 583 |
show "{#} \<in> ?W" |
10249 | 584 |
proof (rule accI) |
23751 | 585 |
fix b assume "(b, {#}) \<in> ?R" |
586 |
with not_less_empty show "b \<in> ?W" by contradiction |
|
10249 | 587 |
qed |
588 |
||
23751 | 589 |
fix M a assume "M \<in> ?W" |
590 |
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 591 |
proof induct |
592 |
fix a |
|
23751 | 593 |
assume r: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
594 |
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 595 |
proof |
23751 | 596 |
fix M assume "M \<in> ?W" |
597 |
then show "M + {#a#} \<in> ?W" |
|
23373 | 598 |
by (rule acc_induct) (rule tedious_reasoning [OF _ r]) |
10249 | 599 |
qed |
600 |
qed |
|
23751 | 601 |
from this and `M \<in> ?W` show "M + {#a#} \<in> ?W" .. |
10249 | 602 |
qed |
603 |
qed |
|
604 |
||
23751 | 605 |
theorem wf_mult1: "wf r ==> wf (mult1 r)" |
23373 | 606 |
by (rule acc_wfI) (rule all_accessible) |
10249 | 607 |
|
23751 | 608 |
theorem wf_mult: "wf r ==> wf (mult r)" |
609 |
unfolding mult_def by (rule wf_trancl) (rule wf_mult1) |
|
10249 | 610 |
|
611 |
||
612 |
subsubsection {* Closure-free presentation *} |
|
613 |
||
614 |
(*Badly needed: a linear arithmetic procedure for multisets*) |
|
615 |
||
616 |
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})" |
|
23373 | 617 |
by (simp add: multiset_eq_conv_count_eq) |
10249 | 618 |
|
619 |
text {* One direction. *} |
|
620 |
||
621 |
lemma mult_implies_one_step: |
|
23751 | 622 |
"trans r ==> (M, N) \<in> mult r ==> |
11464 | 623 |
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> |
23751 | 624 |
(\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)" |
10249 | 625 |
apply (unfold mult_def mult1_def set_of_def) |
23751 | 626 |
apply (erule converse_trancl_induct, clarify) |
15072 | 627 |
apply (rule_tac x = M0 in exI, simp, clarify) |
23751 | 628 |
apply (case_tac "a :# K") |
10249 | 629 |
apply (rule_tac x = I in exI) |
630 |
apply (simp (no_asm)) |
|
23751 | 631 |
apply (rule_tac x = "(K - {#a#}) + Ka" in exI) |
10249 | 632 |
apply (simp (no_asm_simp) add: union_assoc [symmetric]) |
11464 | 633 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong) |
10249 | 634 |
apply (simp add: diff_union_single_conv) |
635 |
apply (simp (no_asm_use) add: trans_def) |
|
636 |
apply blast |
|
637 |
apply (subgoal_tac "a :# I") |
|
638 |
apply (rule_tac x = "I - {#a#}" in exI) |
|
639 |
apply (rule_tac x = "J + {#a#}" in exI) |
|
640 |
apply (rule_tac x = "K + Ka" in exI) |
|
641 |
apply (rule conjI) |
|
642 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) |
|
643 |
apply (rule conjI) |
|
15072 | 644 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp) |
10249 | 645 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) |
646 |
apply (simp (no_asm_use) add: trans_def) |
|
647 |
apply blast |
|
10277 | 648 |
apply (subgoal_tac "a :# (M0 + {#a#})") |
10249 | 649 |
apply simp |
650 |
apply (simp (no_asm)) |
|
651 |
done |
|
652 |
||
653 |
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}" |
|
23373 | 654 |
by (simp add: multiset_eq_conv_count_eq) |
10249 | 655 |
|
11464 | 656 |
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}" |
10249 | 657 |
apply (erule size_eq_Suc_imp_elem [THEN exE]) |
15072 | 658 |
apply (drule elem_imp_eq_diff_union, auto) |
10249 | 659 |
done |
660 |
||
661 |
lemma one_step_implies_mult_aux: |
|
23751 | 662 |
"trans r ==> |
663 |
\<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)) |
|
664 |
--> (I + K, I + J) \<in> mult r" |
|
15072 | 665 |
apply (induct_tac n, auto) |
666 |
apply (frule size_eq_Suc_imp_eq_union, clarify) |
|
667 |
apply (rename_tac "J'", simp) |
|
668 |
apply (erule notE, auto) |
|
10249 | 669 |
apply (case_tac "J' = {#}") |
670 |
apply (simp add: mult_def) |
|
23751 | 671 |
apply (rule r_into_trancl) |
15072 | 672 |
apply (simp add: mult1_def set_of_def, blast) |
11464 | 673 |
txt {* Now we know @{term "J' \<noteq> {#}"}. *} |
23751 | 674 |
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition) |
11464 | 675 |
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp) |
10249 | 676 |
apply (erule ssubst) |
15072 | 677 |
apply (simp add: Ball_def, auto) |
10249 | 678 |
apply (subgoal_tac |
23751 | 679 |
"((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #}, |
680 |
(I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r") |
|
10249 | 681 |
prefer 2 |
682 |
apply force |
|
683 |
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def) |
|
23751 | 684 |
apply (erule trancl_trans) |
685 |
apply (rule r_into_trancl) |
|
10249 | 686 |
apply (simp add: mult1_def set_of_def) |
687 |
apply (rule_tac x = a in exI) |
|
688 |
apply (rule_tac x = "I + J'" in exI) |
|
689 |
apply (simp add: union_ac) |
|
690 |
done |
|
691 |
||
17161 | 692 |
lemma one_step_implies_mult: |
23751 | 693 |
"trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r |
694 |
==> (I + K, I + J) \<in> mult r" |
|
23373 | 695 |
using one_step_implies_mult_aux by blast |
10249 | 696 |
|
697 |
||
698 |
subsubsection {* Partial-order properties *} |
|
699 |
||
12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents:
11868
diff
changeset
|
700 |
instance multiset :: (type) ord .. |
10249 | 701 |
|
702 |
defs (overloaded) |
|
23751 | 703 |
less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}" |
11464 | 704 |
le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)" |
10249 | 705 |
|
23751 | 706 |
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}" |
18730 | 707 |
unfolding trans_def by (blast intro: order_less_trans) |
10249 | 708 |
|
709 |
text {* |
|
710 |
\medskip Irreflexivity. |
|
711 |
*} |
|
712 |
||
713 |
lemma mult_irrefl_aux: |
|
18258 | 714 |
"finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) \<Longrightarrow> A = {}" |
23373 | 715 |
by (induct rule: finite_induct) (auto intro: order_less_trans) |
10249 | 716 |
|
17161 | 717 |
lemma mult_less_not_refl: "\<not> M < (M::'a::order multiset)" |
15072 | 718 |
apply (unfold less_multiset_def, auto) |
719 |
apply (drule trans_base_order [THEN mult_implies_one_step], auto) |
|
10249 | 720 |
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]]) |
721 |
apply (simp add: set_of_eq_empty_iff) |
|
722 |
done |
|
723 |
||
724 |
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R" |
|
23373 | 725 |
using insert mult_less_not_refl by fast |
10249 | 726 |
|
727 |
||
728 |
text {* Transitivity. *} |
|
729 |
||
730 |
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)" |
|
23751 | 731 |
unfolding less_multiset_def mult_def by (blast intro: trancl_trans) |
10249 | 732 |
|
733 |
text {* Asymmetry. *} |
|
734 |
||
11464 | 735 |
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)" |
10249 | 736 |
apply auto |
737 |
apply (rule mult_less_not_refl [THEN notE]) |
|
15072 | 738 |
apply (erule mult_less_trans, assumption) |
10249 | 739 |
done |
740 |
||
741 |
theorem mult_less_asym: |
|
11464 | 742 |
"M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P" |
15072 | 743 |
by (insert mult_less_not_sym, blast) |
10249 | 744 |
|
745 |
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)" |
|
18730 | 746 |
unfolding le_multiset_def by auto |
10249 | 747 |
|
748 |
text {* Anti-symmetry. *} |
|
749 |
||
750 |
theorem mult_le_antisym: |
|
751 |
"M <= N ==> N <= M ==> M = (N::'a::order multiset)" |
|
18730 | 752 |
unfolding le_multiset_def by (blast dest: mult_less_not_sym) |
10249 | 753 |
|
754 |
text {* Transitivity. *} |
|
755 |
||
756 |
theorem mult_le_trans: |
|
757 |
"K <= M ==> M <= N ==> K <= (N::'a::order multiset)" |
|
18730 | 758 |
unfolding le_multiset_def by (blast intro: mult_less_trans) |
10249 | 759 |
|
11655 | 760 |
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))" |
18730 | 761 |
unfolding le_multiset_def by auto |
10249 | 762 |
|
10277 | 763 |
text {* Partial order. *} |
764 |
||
765 |
instance multiset :: (order) order |
|
766 |
apply intro_classes |
|
23751 | 767 |
apply (rule mult_less_le) |
768 |
apply (rule mult_le_refl) |
|
769 |
apply (erule mult_le_trans, assumption) |
|
770 |
apply (erule mult_le_antisym, assumption) |
|
10277 | 771 |
done |
772 |
||
10249 | 773 |
|
774 |
subsubsection {* Monotonicity of multiset union *} |
|
775 |
||
17161 | 776 |
lemma mult1_union: |
23751 | 777 |
"(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r" |
15072 | 778 |
apply (unfold mult1_def, auto) |
10249 | 779 |
apply (rule_tac x = a in exI) |
780 |
apply (rule_tac x = "C + M0" in exI) |
|
781 |
apply (simp add: union_assoc) |
|
782 |
done |
|
783 |
||
784 |
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)" |
|
785 |
apply (unfold less_multiset_def mult_def) |
|
23751 | 786 |
apply (erule trancl_induct) |
787 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl) |
|
788 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans) |
|
10249 | 789 |
done |
790 |
||
791 |
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)" |
|
792 |
apply (subst union_commute [of B C]) |
|
793 |
apply (subst union_commute [of D C]) |
|
794 |
apply (erule union_less_mono2) |
|
795 |
done |
|
796 |
||
17161 | 797 |
lemma union_less_mono: |
10249 | 798 |
"A < C ==> B < D ==> A + B < C + (D::'a::order multiset)" |
799 |
apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans) |
|
800 |
done |
|
801 |
||
17161 | 802 |
lemma union_le_mono: |
10249 | 803 |
"A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)" |
18730 | 804 |
unfolding le_multiset_def |
805 |
by (blast intro: union_less_mono union_less_mono1 union_less_mono2) |
|
10249 | 806 |
|
17161 | 807 |
lemma empty_leI [iff]: "{#} <= (M::'a::order multiset)" |
10249 | 808 |
apply (unfold le_multiset_def less_multiset_def) |
809 |
apply (case_tac "M = {#}") |
|
810 |
prefer 2 |
|
23751 | 811 |
apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))") |
10249 | 812 |
prefer 2 |
813 |
apply (rule one_step_implies_mult) |
|
23751 | 814 |
apply (simp only: trans_def, auto) |
10249 | 815 |
done |
816 |
||
17161 | 817 |
lemma union_upper1: "A <= A + (B::'a::order multiset)" |
15072 | 818 |
proof - |
17200 | 819 |
have "A + {#} <= A + B" by (blast intro: union_le_mono) |
18258 | 820 |
then show ?thesis by simp |
15072 | 821 |
qed |
822 |
||
17161 | 823 |
lemma union_upper2: "B <= A + (B::'a::order multiset)" |
18258 | 824 |
by (subst union_commute) (rule union_upper1) |
15072 | 825 |
|
23611 | 826 |
instance multiset :: (order) pordered_ab_semigroup_add |
827 |
apply intro_classes |
|
828 |
apply(erule union_le_mono[OF mult_le_refl]) |
|
829 |
done |
|
15072 | 830 |
|
17200 | 831 |
subsection {* Link with lists *} |
15072 | 832 |
|
17200 | 833 |
consts |
15072 | 834 |
multiset_of :: "'a list \<Rightarrow> 'a multiset" |
835 |
primrec |
|
836 |
"multiset_of [] = {#}" |
|
837 |
"multiset_of (a # x) = multiset_of x + {# a #}" |
|
838 |
||
839 |
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])" |
|
18258 | 840 |
by (induct x) auto |
15072 | 841 |
|
842 |
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])" |
|
18258 | 843 |
by (induct x) auto |
15072 | 844 |
|
845 |
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x" |
|
18258 | 846 |
by (induct x) auto |
15867 | 847 |
|
848 |
lemma mem_set_multiset_eq: "x \<in> set xs = (x :# multiset_of xs)" |
|
849 |
by (induct xs) auto |
|
15072 | 850 |
|
18258 | 851 |
lemma multiset_of_append [simp]: |
852 |
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys" |
|
20503 | 853 |
by (induct xs arbitrary: ys) (auto simp: union_ac) |
18730 | 854 |
|
15072 | 855 |
lemma surj_multiset_of: "surj multiset_of" |
17200 | 856 |
apply (unfold surj_def, rule allI) |
857 |
apply (rule_tac M=y in multiset_induct, auto) |
|
858 |
apply (rule_tac x = "x # xa" in exI, auto) |
|
10249 | 859 |
done |
860 |
||
25162 | 861 |
lemma set_count_greater_0: "set x = {a. count (multiset_of x) a > 0}" |
18258 | 862 |
by (induct x) auto |
15072 | 863 |
|
17200 | 864 |
lemma distinct_count_atmost_1: |
15072 | 865 |
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))" |
18258 | 866 |
apply (induct x, simp, rule iffI, simp_all) |
17200 | 867 |
apply (rule conjI) |
868 |
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) |
|
15072 | 869 |
apply (erule_tac x=a in allE, simp, clarify) |
17200 | 870 |
apply (erule_tac x=aa in allE, simp) |
15072 | 871 |
done |
872 |
||
17200 | 873 |
lemma multiset_of_eq_setD: |
15867 | 874 |
"multiset_of xs = multiset_of ys \<Longrightarrow> set xs = set ys" |
875 |
by (rule) (auto simp add:multiset_eq_conv_count_eq set_count_greater_0) |
|
876 |
||
17200 | 877 |
lemma set_eq_iff_multiset_of_eq_distinct: |
878 |
"\<lbrakk>distinct x; distinct y\<rbrakk> |
|
15072 | 879 |
\<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)" |
17200 | 880 |
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) |
15072 | 881 |
|
17200 | 882 |
lemma set_eq_iff_multiset_of_remdups_eq: |
15072 | 883 |
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))" |
17200 | 884 |
apply (rule iffI) |
885 |
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) |
|
886 |
apply (drule distinct_remdups[THEN distinct_remdups |
|
887 |
[THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]]) |
|
15072 | 888 |
apply simp |
10249 | 889 |
done |
890 |
||
18258 | 891 |
lemma multiset_of_compl_union [simp]: |
23281 | 892 |
"multiset_of [x\<leftarrow>xs. P x] + multiset_of [x\<leftarrow>xs. \<not>P x] = multiset_of xs" |
15630 | 893 |
by (induct xs) (auto simp: union_ac) |
15072 | 894 |
|
17200 | 895 |
lemma count_filter: |
23281 | 896 |
"count (multiset_of xs) x = length [y \<leftarrow> xs. y = x]" |
18258 | 897 |
by (induct xs) auto |
15867 | 898 |
|
899 |
||
15072 | 900 |
subsection {* Pointwise ordering induced by count *} |
901 |
||
19086 | 902 |
definition |
25610 | 903 |
mset_le :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "\<le>#" 50) where |
904 |
"(A \<le># B) = (\<forall>a. count A a \<le> count B a)" |
|
23611 | 905 |
definition |
25610 | 906 |
mset_less :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" (infix "<#" 50) where |
907 |
"(A <# B) = (A \<le># B \<and> A \<noteq> B)" |
|
908 |
||
909 |
notation mset_le (infix "\<subseteq>#" 50) |
|
910 |
notation mset_less (infix "\<subset>#" 50) |
|
15072 | 911 |
|
23611 | 912 |
lemma mset_le_refl[simp]: "A \<le># A" |
18730 | 913 |
unfolding mset_le_def by auto |
15072 | 914 |
|
23611 | 915 |
lemma mset_le_trans: "\<lbrakk> A \<le># B; B \<le># C \<rbrakk> \<Longrightarrow> A \<le># C" |
18730 | 916 |
unfolding mset_le_def by (fast intro: order_trans) |
15072 | 917 |
|
23611 | 918 |
lemma mset_le_antisym: "\<lbrakk> A \<le># B; B \<le># A \<rbrakk> \<Longrightarrow> A = B" |
17200 | 919 |
apply (unfold mset_le_def) |
920 |
apply (rule multiset_eq_conv_count_eq[THEN iffD2]) |
|
15072 | 921 |
apply (blast intro: order_antisym) |
922 |
done |
|
923 |
||
17200 | 924 |
lemma mset_le_exists_conv: |
23611 | 925 |
"(A \<le># B) = (\<exists>C. B = A + C)" |
926 |
apply (unfold mset_le_def, rule iffI, rule_tac x = "B - A" in exI) |
|
15072 | 927 |
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2]) |
928 |
done |
|
929 |
||
23611 | 930 |
lemma mset_le_mono_add_right_cancel[simp]: "(A + C \<le># B + C) = (A \<le># B)" |
18730 | 931 |
unfolding mset_le_def by auto |
15072 | 932 |
|
23611 | 933 |
lemma mset_le_mono_add_left_cancel[simp]: "(C + A \<le># C + B) = (A \<le># B)" |
18730 | 934 |
unfolding mset_le_def by auto |
15072 | 935 |
|
23611 | 936 |
lemma mset_le_mono_add: "\<lbrakk> A \<le># B; C \<le># D \<rbrakk> \<Longrightarrow> A + C \<le># B + D" |
17200 | 937 |
apply (unfold mset_le_def) |
938 |
apply auto |
|
15072 | 939 |
apply (erule_tac x=a in allE)+ |
940 |
apply auto |
|
941 |
done |
|
942 |
||
23611 | 943 |
lemma mset_le_add_left[simp]: "A \<le># A + B" |
18730 | 944 |
unfolding mset_le_def by auto |
15072 | 945 |
|
23611 | 946 |
lemma mset_le_add_right[simp]: "B \<le># A + B" |
18730 | 947 |
unfolding mset_le_def by auto |
15072 | 948 |
|
23611 | 949 |
lemma multiset_of_remdups_le: "multiset_of (remdups xs) \<le># multiset_of xs" |
950 |
apply (induct xs) |
|
951 |
apply auto |
|
952 |
apply (rule mset_le_trans) |
|
953 |
apply auto |
|
954 |
done |
|
955 |
||
25208 | 956 |
interpretation mset_order: |
957 |
order ["op \<le>#" "op <#"] |
|
958 |
by (auto intro: order.intro mset_le_refl mset_le_antisym |
|
959 |
mset_le_trans simp: mset_less_def) |
|
23611 | 960 |
|
961 |
interpretation mset_order_cancel_semigroup: |
|
25208 | 962 |
pordered_cancel_ab_semigroup_add ["op \<le>#" "op <#" "op +"] |
963 |
by unfold_locales (erule mset_le_mono_add [OF mset_le_refl]) |
|
23611 | 964 |
|
965 |
interpretation mset_order_semigroup_cancel: |
|
25208 | 966 |
pordered_ab_semigroup_add_imp_le ["op \<le>#" "op <#" "op +"] |
967 |
by (unfold_locales) simp |
|
15072 | 968 |
|
25610 | 969 |
|
970 |
lemma mset_lessD: |
|
971 |
"\<lbrakk> A \<subset># B ; x \<in># A \<rbrakk> \<Longrightarrow> x \<in># B" |
|
972 |
apply (clarsimp simp: mset_le_def mset_less_def) |
|
973 |
apply (erule_tac x=x in allE) |
|
974 |
apply auto |
|
975 |
done |
|
976 |
||
977 |
lemma mset_leD: |
|
978 |
"\<lbrakk> A \<subseteq># B ; x \<in># A \<rbrakk> \<Longrightarrow> x \<in># B" |
|
979 |
apply (clarsimp simp: mset_le_def mset_less_def) |
|
980 |
apply (erule_tac x=x in allE) |
|
981 |
apply auto |
|
982 |
done |
|
983 |
||
984 |
lemma mset_less_insertD: |
|
985 |
"(A + {#x#} \<subset># B) \<Longrightarrow> (x \<in># B \<and> A \<subset># B)" |
|
986 |
apply (rule conjI) |
|
987 |
apply (simp add: mset_lessD) |
|
988 |
apply (clarsimp simp: mset_le_def mset_less_def) |
|
989 |
apply safe |
|
990 |
apply (erule_tac x=a in allE) |
|
991 |
apply (auto split: split_if_asm) |
|
992 |
done |
|
993 |
||
994 |
lemma mset_le_insertD: |
|
995 |
"(A + {#x#} \<subseteq># B) \<Longrightarrow> (x \<in># B \<and> A \<subseteq># B)" |
|
996 |
apply (rule conjI) |
|
997 |
apply (simp add: mset_leD) |
|
998 |
apply (force simp: mset_le_def mset_less_def split: split_if_asm) |
|
999 |
done |
|
1000 |
||
1001 |
lemma mset_less_of_empty[simp]: "A \<subset># {#} = False" |
|
1002 |
by (induct A, auto simp: mset_le_def mset_less_def) |
|
1003 |
||
1004 |
lemma multi_psub_of_add_self[simp]: "A \<subset># A + {#x#}" |
|
1005 |
by (clarsimp simp: mset_le_def mset_less_def) |
|
1006 |
||
1007 |
lemma multi_psub_self[simp]: "A \<subset># A = False" |
|
1008 |
by (clarsimp simp: mset_le_def mset_less_def) |
|
1009 |
||
1010 |
lemma mset_less_add_bothsides: |
|
1011 |
"T + {#x#} \<subset># S + {#x#} \<Longrightarrow> T \<subset># S" |
|
1012 |
by (clarsimp simp: mset_le_def mset_less_def) |
|
1013 |
||
1014 |
lemma mset_less_empty_nonempty: "({#} \<subset># S) = (S \<noteq> {#})" |
|
1015 |
by (auto simp: mset_le_def mset_less_def) |
|
1016 |
||
1017 |
lemma mset_less_size: "A \<subset># B \<Longrightarrow> size A < size B" |
|
1018 |
proof (induct A arbitrary: B) |
|
1019 |
case (empty M) |
|
1020 |
hence "M \<noteq> {#}" by (simp add: mset_less_empty_nonempty) |
|
1021 |
then obtain M' x where "M = M' + {#x#}" |
|
1022 |
by (blast dest: multi_nonempty_split) |
|
1023 |
thus ?case by simp |
|
1024 |
next |
|
1025 |
case (add S x T) |
|
1026 |
have IH: "\<And>B. S \<subset># B \<Longrightarrow> size S < size B" by fact |
|
1027 |
have SxsubT: "S + {#x#} \<subset># T" by fact |
|
1028 |
hence "x \<in># T" and "S \<subset># T" by (auto dest: mset_less_insertD) |
|
1029 |
then obtain T' where T: "T = T' + {#x#}" |
|
1030 |
by (blast dest: multi_member_split) |
|
1031 |
hence "S \<subset># T'" using SxsubT |
|
1032 |
by (blast intro: mset_less_add_bothsides) |
|
1033 |
hence "size S < size T'" using IH by simp |
|
1034 |
thus ?case using T by simp |
|
1035 |
qed |
|
1036 |
||
1037 |
lemmas mset_less_trans = mset_order.less_eq_less.less_trans |
|
1038 |
||
1039 |
lemma mset_less_diff_self: "c \<in># B \<Longrightarrow> B - {#c#} \<subset># B" |
|
1040 |
by (auto simp: mset_le_def mset_less_def multi_drop_mem_not_eq) |
|
1041 |
||
1042 |
subsection {* Strong induction and subset induction for multisets *} |
|
1043 |
||
1044 |
subsubsection {* Well-foundedness of proper subset operator *} |
|
1045 |
||
1046 |
definition |
|
1047 |
mset_less_rel :: "('a multiset * 'a multiset) set" |
|
1048 |
where |
|
1049 |
--{* proper multiset subset *} |
|
1050 |
"mset_less_rel \<equiv> {(A,B). A \<subset># B}" |
|
1051 |
||
1052 |
lemma multiset_add_sub_el_shuffle: |
|
1053 |
assumes cinB: "c \<in># B" and bnotc: "b \<noteq> c" |
|
1054 |
shows "B - {#c#} + {#b#} = B + {#b#} - {#c#}" |
|
1055 |
proof - |
|
1056 |
from cinB obtain A where B: "B = A + {#c#}" |
|
1057 |
by (blast dest: multi_member_split) |
|
1058 |
have "A + {#b#} = A + {#b#} + {#c#} - {#c#}" by simp |
|
1059 |
hence "A + {#b#} = A + {#c#} + {#b#} - {#c#}" |
|
1060 |
by (simp add: union_ac) |
|
1061 |
thus ?thesis using B by simp |
|
1062 |
qed |
|
1063 |
||
1064 |
lemma wf_mset_less_rel: "wf mset_less_rel" |
|
1065 |
apply (unfold mset_less_rel_def) |
|
1066 |
apply (rule wf_measure [THEN wf_subset, where f1=size]) |
|
1067 |
apply (clarsimp simp: measure_def inv_image_def mset_less_size) |
|
1068 |
done |
|
1069 |
||
1070 |
subsubsection {* The induction rules *} |
|
1071 |
||
1072 |
lemma full_multiset_induct [case_names less]: |
|
1073 |
assumes ih: "\<And>B. \<forall>A. A \<subset># B \<longrightarrow> P A \<Longrightarrow> P B" |
|
1074 |
shows "P B" |
|
1075 |
apply (rule wf_mset_less_rel [THEN wf_induct]) |
|
1076 |
apply (rule ih, auto simp: mset_less_rel_def) |
|
1077 |
done |
|
1078 |
||
1079 |
lemma multi_subset_induct [consumes 2, case_names empty add]: |
|
1080 |
assumes "F \<subseteq># A" |
|
1081 |
and empty: "P {#}" |
|
1082 |
and insert: "\<And>a F. a \<in># A \<Longrightarrow> P F \<Longrightarrow> P (F + {#a#})" |
|
1083 |
shows "P F" |
|
1084 |
proof - |
|
1085 |
from `F \<subseteq># A` |
|
1086 |
show ?thesis |
|
1087 |
proof (induct F) |
|
1088 |
show "P {#}" by fact |
|
1089 |
next |
|
1090 |
fix x F |
|
1091 |
assume P: "F \<subseteq># A \<Longrightarrow> P F" and i: "F + {#x#} \<subseteq># A" |
|
1092 |
show "P (F + {#x#})" |
|
1093 |
proof (rule insert) |
|
1094 |
from i show "x \<in># A" by (auto dest: mset_le_insertD) |
|
1095 |
from i have "F \<subseteq># A" by (auto simp: mset_le_insertD) |
|
1096 |
with P show "P F" . |
|
1097 |
qed |
|
1098 |
qed |
|
1099 |
qed |
|
1100 |
||
1101 |
subsection {* Multiset extensionality *} |
|
1102 |
||
1103 |
lemma multi_count_eq: |
|
1104 |
"(\<forall>x. count A x = count B x) = (A = B)" |
|
1105 |
apply (rule iffI) |
|
1106 |
prefer 2 |
|
1107 |
apply clarsimp |
|
1108 |
apply (induct A arbitrary: B rule: full_multiset_induct) |
|
1109 |
apply (rename_tac C) |
|
1110 |
apply (case_tac B rule: multiset_cases) |
|
1111 |
apply (simp add: empty_multiset_count) |
|
1112 |
apply simp |
|
1113 |
apply (case_tac "x \<in># C") |
|
1114 |
apply (force dest: multi_member_split) |
|
1115 |
apply (erule_tac x=x in allE) |
|
1116 |
apply simp |
|
1117 |
done |
|
1118 |
||
1119 |
lemmas multi_count_ext = multi_count_eq [THEN iffD1, rule_format] |
|
1120 |
||
1121 |
subsection {* The fold combinator *} |
|
1122 |
||
1123 |
text {* The intended behaviour is |
|
1124 |
@{text "foldM f z {#x\<^isub>1, ..., x\<^isub>n#} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"} |
|
1125 |
if @{text f} is associative-commutative. |
|
1126 |
*} |
|
1127 |
||
1128 |
(* the graph of foldM, z = the start element, f = folding function, |
|
1129 |
A the multiset, y the result *) |
|
1130 |
inductive |
|
1131 |
foldMG :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b \<Rightarrow> bool" |
|
1132 |
for f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" |
|
1133 |
and z :: 'b |
|
1134 |
where |
|
1135 |
emptyI [intro]: "foldMG f z {#} z" |
|
1136 |
| insertI [intro]: "foldMG f z A y \<Longrightarrow> foldMG f z (A + {#x#}) (f x y)" |
|
1137 |
||
1138 |
inductive_cases empty_foldMGE [elim!]: "foldMG f z {#} x" |
|
1139 |
inductive_cases insert_foldMGE: "foldMG f z (A + {#}) y" |
|
1140 |
||
1141 |
definition |
|
1142 |
foldM :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> 'b \<Rightarrow> 'a multiset \<Rightarrow> 'b" |
|
1143 |
where |
|
1144 |
"foldM f z A \<equiv> THE x. foldMG f z A x" |
|
1145 |
||
1146 |
lemma Diff1_foldMG: |
|
1147 |
"\<lbrakk> foldMG f z (A - {#x#}) y; x \<in># A \<rbrakk> \<Longrightarrow> foldMG f z A (f x y)" |
|
1148 |
by (frule_tac x=x in foldMG.insertI, auto) |
|
1149 |
||
1150 |
lemma foldMG_nonempty: "\<exists>x. foldMG f z A x" |
|
1151 |
apply (induct A) |
|
1152 |
apply blast |
|
1153 |
apply clarsimp |
|
1154 |
apply (drule_tac x=x in foldMG.insertI) |
|
1155 |
apply auto |
|
1156 |
done |
|
1157 |
||
1158 |
lemma foldM_empty[simp]: "foldM f z {#} = z" |
|
1159 |
by (unfold foldM_def, blast) |
|
1160 |
||
1161 |
locale left_commutative = |
|
1162 |
fixes f :: "'a => 'b => 'b" (infixl "\<cdot>" 70) |
|
1163 |
assumes left_commute: "x \<cdot> (y \<cdot> z) = y \<cdot> (x \<cdot> z)" |
|
1164 |
||
1165 |
lemma (in left_commutative) foldMG_determ: |
|
1166 |
"\<lbrakk>foldMG f z A x; foldMG f z A y\<rbrakk> \<Longrightarrow> y = x" |
|
1167 |
proof (induct arbitrary: x y z rule: full_multiset_induct) |
|
1168 |
case (less M x\<^isub>1 x\<^isub>2 Z) |
|
1169 |
have IH: "\<forall>A. A \<subset># M \<longrightarrow> |
|
1170 |
(\<forall>x x' x''. foldMG op \<cdot> x'' A x \<longrightarrow> foldMG op \<cdot> x'' A x' |
|
1171 |
\<longrightarrow> x' = x)" by fact |
|
1172 |
have Mfoldx\<^isub>1: "foldMG f Z M x\<^isub>1" and Mfoldx\<^isub>2: "foldMG f Z M x\<^isub>2" by fact+ |
|
1173 |
show ?case |
|
1174 |
proof (rule foldMG.cases [OF Mfoldx\<^isub>1]) |
|
1175 |
assume "M = {#}" and "x\<^isub>1 = Z" |
|
1176 |
thus ?case using Mfoldx\<^isub>2 by auto |
|
1177 |
next |
|
1178 |
fix B b u |
|
1179 |
assume "M = B + {#b#}" and "x\<^isub>1 = b \<cdot> u" and Bu: "foldMG op \<cdot> Z B u" |
|
1180 |
hence MBb: "M = B + {#b#}" and x\<^isub>1: "x\<^isub>1 = b \<cdot> u" by auto |
|
1181 |
show ?case |
|
1182 |
proof (rule foldMG.cases [OF Mfoldx\<^isub>2]) |
|
1183 |
assume "M = {#}" "x\<^isub>2 = Z" |
|
1184 |
thus ?case using Mfoldx\<^isub>1 by auto |
|
1185 |
next |
|
1186 |
fix C c v |
|
1187 |
assume "M = C + {#c#}" and "x\<^isub>2 = c \<cdot> v" and Cv: "foldMG op \<cdot> Z C v" |
|
1188 |
hence MCc: "M = C + {#c#}" and x\<^isub>2: "x\<^isub>2 = c \<cdot> v" by auto |
|
1189 |
hence CsubM: "C \<subset># M" by simp |
|
1190 |
from MBb have BsubM: "B \<subset># M" by simp |
|
1191 |
show ?case |
|
1192 |
proof cases |
|
1193 |
assume "b=c" |
|
1194 |
then moreover have "B = C" using MBb MCc by auto |
|
1195 |
ultimately show ?thesis using Bu Cv x\<^isub>1 x\<^isub>2 CsubM IH by auto |
|
1196 |
next |
|
1197 |
assume diff: "b \<noteq> c" |
|
1198 |
let ?D = "B - {#c#}" |
|
1199 |
have cinB: "c \<in># B" and binC: "b \<in># C" using MBb MCc diff |
|
1200 |
by (auto intro: insert_noteq_member dest: sym) |
|
1201 |
have "B - {#c#} \<subset># B" using cinB by (rule mset_less_diff_self) |
|
1202 |
hence DsubM: "?D \<subset># M" using BsubM by (blast intro: mset_less_trans) |
|
1203 |
from MBb MCc have "B + {#b#} = C + {#c#}" by blast |
|
1204 |
hence [simp]: "B + {#b#} - {#c#} = C" |
|
1205 |
using MBb MCc binC cinB by auto |
|
1206 |
have B: "B = ?D + {#c#}" and C: "C = ?D + {#b#}" |
|
1207 |
using MBb MCc diff binC cinB |
|
1208 |
by (auto simp: multiset_add_sub_el_shuffle) |
|
1209 |
then obtain d where Dfoldd: "foldMG f Z ?D d" |
|
1210 |
using foldMG_nonempty by iprover |
|
1211 |
hence "foldMG f Z B (c \<cdot> d)" using cinB |
|
1212 |
by (rule Diff1_foldMG) |
|
1213 |
hence "c \<cdot> d = u" using IH BsubM Bu by blast |
|
1214 |
moreover |
|
1215 |
have "foldMG f Z C (b \<cdot> d)" using binC cinB diff Dfoldd |
|
1216 |
by (auto simp: multiset_add_sub_el_shuffle |
|
1217 |
dest: foldMG.insertI [where x=b]) |
|
1218 |
hence "b \<cdot> d = v" using IH CsubM Cv by blast |
|
1219 |
ultimately show ?thesis using x\<^isub>1 x\<^isub>2 |
|
1220 |
by (auto simp: left_commute) |
|
1221 |
qed |
|
1222 |
qed |
|
1223 |
qed |
|
1224 |
qed |
|
1225 |
||
1226 |
lemma (in left_commutative) foldM_insert_aux: " |
|
1227 |
(foldMG f z (A + {#x#}) v) = |
|
1228 |
(\<exists>y. foldMG f z A y \<and> v = f x y)" |
|
1229 |
apply (rule iffI) |
|
1230 |
prefer 2 |
|
1231 |
apply blast |
|
1232 |
apply (rule_tac A=A and f=f in foldMG_nonempty [THEN exE, standard]) |
|
1233 |
apply (blast intro: foldMG_determ) |
|
1234 |
done |
|
1235 |
||
1236 |
lemma (in left_commutative) foldM_equality: "foldMG f z A y \<Longrightarrow> foldM f z A = y" |
|
1237 |
by (unfold foldM_def) (blast intro: foldMG_determ) |
|
1238 |
||
1239 |
lemma (in left_commutative) foldM_insert[simp]: |
|
1240 |
"foldM f z (A + {#x#}) = f x (foldM f z A)" |
|
1241 |
apply (simp add: foldM_def foldM_insert_aux union_commute) |
|
1242 |
apply (rule the_equality) |
|
1243 |
apply (auto cong add: conj_cong |
|
1244 |
simp add: foldM_def [symmetric] foldM_equality foldMG_nonempty) |
|
1245 |
done |
|
1246 |
||
1247 |
lemma (in left_commutative) foldM_insert_idem: |
|
1248 |
shows "foldM f z (A + {#a#}) = a \<cdot> foldM f z A" |
|
1249 |
apply (simp add: foldM_def foldM_insert_aux) |
|
1250 |
apply (rule the_equality) |
|
1251 |
apply (auto cong add: conj_cong |
|
1252 |
simp add: foldM_def [symmetric] foldM_equality foldMG_nonempty) |
|
1253 |
done |
|
1254 |
||
1255 |
lemma (in left_commutative) foldM_fusion: |
|
1256 |
includes left_commutative g |
|
1257 |
shows "\<lbrakk>\<And>x y. h (g x y) = f x (h y) \<rbrakk> \<Longrightarrow> h (foldM g w A) = foldM f (h w) A" |
|
1258 |
by (induct A, auto) |
|
1259 |
||
1260 |
lemma (in left_commutative) foldM_commute: |
|
1261 |
"f x (foldM f z A) = foldM f (f x z) A" |
|
1262 |
by (induct A, auto simp: left_commute [of x]) |
|
1263 |
||
1264 |
lemma (in left_commutative) foldM_rec: |
|
1265 |
assumes a: "a \<in># A" |
|
1266 |
shows "foldM f z A = f a (foldM f z (A - {#a#}))" |
|
1267 |
proof - |
|
1268 |
from a obtain A' where "A = A' + {#a#}" by (blast dest: multi_member_split) |
|
1269 |
thus ?thesis by simp |
|
1270 |
qed |
|
1271 |
||
1272 |
||
10249 | 1273 |
end |