author | nipkow |
Sun, 12 Dec 2004 16:25:47 +0100 | |
changeset 15402 | 97204f3b4705 |
parent 15316 | 2a6ff941a115 |
child 15630 | cc3776f004e2 |
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 Accessible_Part |
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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 . 0 < f x}}" |
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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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constdefs |
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Mempty :: "'a multiset" ("{#}") |
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"{#} == Abs_multiset (\<lambda>a. 0)" |
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single :: "'a => 'a multiset" ("{#_#}") |
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"{#a#} == Abs_multiset (\<lambda>b. if b = a then 1 else 0)" |
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count :: "'a multiset => 'a => nat" |
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"count == Rep_multiset" |
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MCollect :: "'a multiset => ('a => bool) => 'a multiset" |
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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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syntax |
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"_Melem" :: "'a => 'a multiset => bool" ("(_/ :# _)" [50, 51] 50) |
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"_MCollect" :: "pttrn => 'a multiset => bool => 'a multiset" ("(1{# _ : _./ _#})") |
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translations |
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"a :# M" == "0 < count M a" |
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"{#x:M. P#}" == "MCollect M (\<lambda>x. P)" |
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constdefs |
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set_of :: "'a multiset => 'a set" |
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"set_of M == {x. x :# M}" |
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instance multiset :: (type) "{plus, minus, zero}" .. |
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defs (overloaded) |
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union_def: "M + N == Abs_multiset (\<lambda>a. Rep_multiset M a + Rep_multiset N a)" |
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diff_def: "M - N == Abs_multiset (\<lambda>a. Rep_multiset M a - Rep_multiset N a)" |
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Zero_multiset_def [simp]: "0 == {#}" |
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size_def: "size M == setsum (count M) (set_of M)" |
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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 (unfold multiset_def, simp) |
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apply (drule finite_UnI, assumption) |
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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 (unfold multiset_def, simp) |
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apply (rule finite_subset) |
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prefer 2 |
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apply assumption |
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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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theorem union_empty [simp]: "M + {#} = M \<and> {#} + M = M" |
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by (simp add: union_def Mempty_def) |
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theorem union_commute: "M + N = N + (M::'a multiset)" |
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by (simp add: union_def add_ac) |
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theorem 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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theorem union_lcomm: "M + (N + K) = N + (M + (K::'a multiset))" |
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apply (rule union_commute [THEN trans]) |
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apply (rule union_assoc [THEN trans]) |
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apply (rule union_commute [THEN arg_cong]) |
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done |
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theorems 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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theorem diff_empty [simp]: "M - {#} = M \<and> {#} - M = {#}" |
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by (simp add: Mempty_def diff_def) |
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theorem 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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theorem count_empty [simp]: "count {#} a = 0" |
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by (simp add: count_def Mempty_def) |
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theorem 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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theorem 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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theorem 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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theorem set_of_empty [simp]: "set_of {#} = {}" |
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by (simp add: set_of_def) |
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theorem set_of_single [simp]: "set_of {#b#} = {b}" |
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by (simp add: set_of_def) |
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theorem 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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theorem 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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theorem 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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theorem size_empty [simp]: "size {#} = 0" |
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by (simp add: size_def) |
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theorem size_single [simp]: "size {#b#} = 1" |
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by (simp add: size_def) |
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theorem finite_set_of [iff]: "finite (set_of M)" |
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apply (cut_tac x = M in Rep_multiset) |
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apply (simp add: multiset_def set_of_def count_def) |
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done |
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theorem 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 (erule finite_induct, simp) |
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apply (simp add: Int_insert_left set_of_def) |
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done |
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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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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theorem 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 |
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done |
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theorem 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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theorem 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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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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(* |
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val prems = Goal |
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"[| !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> finite F --> P F"; |
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by (res_inst_tac [("a","F"),("f","\<lambda>A. if finite A then card A else 0")] |
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measure_induct 1); |
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by (Clarify_tac 1) |
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by (resolve_tac prems 1) |
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by (assume_tac 1) |
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by (Clarify_tac 1) |
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by (subgoal_tac "finite G" 1) |
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by (fast_tac (claset() addDs [finite_subset,order_less_le RS iffD1]) 2); |
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by (etac allE 1) |
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by (etac impE 1) |
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by (Blast_tac 2) |
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by (asm_simp_tac (simpset() addsimps [psubset_card]) 1); |
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no_qed(); |
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val lemma = result(); |
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val prems = Goal |
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"[| finite F; !!F. [| finite F; !G. G < F --> P G |] ==> P F |] ==> P F"; |
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by (rtac (lemma RS mp) 1); |
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by (REPEAT(ares_tac prems 1)); |
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qed "finite_psubset_induct"; |
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Better: use wf_finite_psubset in WF_Rel |
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*) |
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subsection {* Induction over multisets *} |
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lemma setsum_decr: |
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"finite F ==> (0::nat) < f a ==> |
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setsum (f (a := f a - 1)) F = (if a\<in>F then setsum f F - 1 else setsum f F)" |
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apply (erule finite_induct, auto) |
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apply (drule_tac a = a in mk_disjoint_insert, auto) |
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done |
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lemma rep_multiset_induct_aux: |
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"P (\<lambda>a. (0::nat)) ==> (!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) |
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==> \<forall>f. f \<in> multiset --> setsum f {x. 0 < f x} = n --> P f" |
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proof - |
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case rule_context |
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note premises = this [unfolded multiset_def] |
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show ?thesis |
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apply (unfold multiset_def) |
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apply (induct_tac n, simp, clarify) |
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apply (subgoal_tac "f = (\<lambda>a.0)") |
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apply simp |
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apply (rule premises) |
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apply (rule ext, force, clarify) |
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apply (frule setsum_SucD, clarify) |
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apply (rename_tac a) |
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apply (subgoal_tac "finite {x. 0 < (f (a := f a - 1)) x}") |
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prefer 2 |
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apply (rule finite_subset) |
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prefer 2 |
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apply assumption |
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apply simp |
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apply blast |
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apply (subgoal_tac "f = (f (a := f a - 1))(a := (f (a := f a - 1)) a + 1)") |
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prefer 2 |
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apply (rule ext) |
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apply (simp (no_asm_simp)) |
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apply (erule ssubst, rule premises, blast) |
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apply (erule allE, erule impE, erule_tac [2] mp, blast) |
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apply (simp (no_asm_simp) add: setsum_decr del: fun_upd_apply One_nat_def) |
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apply (subgoal_tac "{x. x \<noteq> a --> 0 < f x} = {x. 0 < f x}") |
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prefer 2 |
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apply blast |
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apply (subgoal_tac "{x. x \<noteq> a \<and> 0 < f x} = {x. 0 < f x} - {a}") |
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prefer 2 |
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apply blast |
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apply (simp add: le_imp_diff_is_add setsum_diff1_nat cong: conj_cong) |
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done |
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qed |
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theorem rep_multiset_induct: |
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"f \<in> multiset ==> P (\<lambda>a. 0) ==> |
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(!!f b. f \<in> multiset ==> P f ==> P (f (b := f b + 1))) ==> P f" |
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by (insert rep_multiset_induct_aux, blast) |
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theorem multiset_induct [induct type: multiset]: |
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"P {#} ==> (!!M x. P M ==> P (M + {#x#})) ==> P M" |
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proof - |
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note defns = union_def single_def Mempty_def |
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assume prem1 [unfolded defns]: "P {#}" |
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assume prem2 [unfolded defns]: "!!M x. P M ==> P (M + {#x#})" |
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show ?thesis |
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apply (rule Rep_multiset_inverse [THEN subst]) |
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apply (rule Rep_multiset [THEN rep_multiset_induct]) |
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apply (rule prem1) |
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apply (subgoal_tac "f(b := f b + 1) = (\<lambda>a. f a + (if a=b then 1 else 0))") |
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prefer 2 |
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apply (simp add: expand_fun_eq) |
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apply (erule ssubst) |
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apply (erule Abs_multiset_inverse [THEN subst]) |
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apply (erule prem2 [simplified]) |
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done |
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qed |
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lemma MCollect_preserves_multiset: |
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"M \<in> multiset ==> (\<lambda>x. if P x then M x else 0) \<in> multiset" |
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apply (simp add: multiset_def) |
15072 | 344 |
apply (rule finite_subset, auto) |
10249 | 345 |
done |
346 |
||
347 |
theorem count_MCollect [simp]: |
|
348 |
"count {# x:M. P x #} a = (if P a then count M a else 0)" |
|
15072 | 349 |
by (simp add: count_def MCollect_def MCollect_preserves_multiset) |
10249 | 350 |
|
11464 | 351 |
theorem set_of_MCollect [simp]: "set_of {# x:M. P x #} = set_of M \<inter> {x. P x}" |
15072 | 352 |
by (auto simp add: set_of_def) |
10249 | 353 |
|
11464 | 354 |
theorem multiset_partition: "M = {# x:M. P x #} + {# x:M. \<not> P x #}" |
15072 | 355 |
by (subst multiset_eq_conv_count_eq, auto) |
10249 | 356 |
|
10277 | 357 |
declare Rep_multiset_inject [symmetric, simp del] |
10249 | 358 |
declare multiset_typedef [simp del] |
359 |
||
360 |
theorem add_eq_conv_ex: |
|
15072 | 361 |
"(M + {#a#} = N + {#b#}) = |
362 |
(M = N \<and> a = b \<or> (\<exists>K. M = K + {#b#} \<and> N = K + {#a#}))" |
|
363 |
by (auto simp add: add_eq_conv_diff) |
|
10249 | 364 |
|
365 |
||
366 |
subsection {* Multiset orderings *} |
|
367 |
||
368 |
subsubsection {* Well-foundedness *} |
|
369 |
||
370 |
constdefs |
|
11464 | 371 |
mult1 :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" |
10249 | 372 |
"mult1 r == |
11464 | 373 |
{(N, M). \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> |
374 |
(\<forall>b. b :# K --> (b, a) \<in> r)}" |
|
10249 | 375 |
|
11464 | 376 |
mult :: "('a \<times> 'a) set => ('a multiset \<times> 'a multiset) set" |
10392 | 377 |
"mult r == (mult1 r)\<^sup>+" |
10249 | 378 |
|
11464 | 379 |
lemma not_less_empty [iff]: "(M, {#}) \<notin> mult1 r" |
10277 | 380 |
by (simp add: mult1_def) |
10249 | 381 |
|
11464 | 382 |
lemma less_add: "(N, M0 + {#a#}) \<in> mult1 r ==> |
383 |
(\<exists>M. (M, M0) \<in> mult1 r \<and> N = M + {#a#}) \<or> |
|
384 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K)" |
|
385 |
(concl is "?case1 (mult1 r) \<or> ?case2") |
|
10249 | 386 |
proof (unfold mult1_def) |
11464 | 387 |
let ?r = "\<lambda>K a. \<forall>b. b :# K --> (b, a) \<in> r" |
388 |
let ?R = "\<lambda>N M. \<exists>a M0 K. M = M0 + {#a#} \<and> N = M0 + K \<and> ?r K a" |
|
10249 | 389 |
let ?case1 = "?case1 {(N, M). ?R N M}" |
390 |
||
11464 | 391 |
assume "(N, M0 + {#a#}) \<in> {(N, M). ?R N M}" |
392 |
hence "\<exists>a' M0' K. |
|
393 |
M0 + {#a#} = M0' + {#a'#} \<and> N = M0' + K \<and> ?r K a'" by simp |
|
394 |
thus "?case1 \<or> ?case2" |
|
10249 | 395 |
proof (elim exE conjE) |
396 |
fix a' M0' K |
|
397 |
assume N: "N = M0' + K" and r: "?r K a'" |
|
398 |
assume "M0 + {#a#} = M0' + {#a'#}" |
|
11464 | 399 |
hence "M0 = M0' \<and> a = a' \<or> |
400 |
(\<exists>K'. M0 = K' + {#a'#} \<and> M0' = K' + {#a#})" |
|
10249 | 401 |
by (simp only: add_eq_conv_ex) |
402 |
thus ?thesis |
|
403 |
proof (elim disjE conjE exE) |
|
404 |
assume "M0 = M0'" "a = a'" |
|
11464 | 405 |
with N r have "?r K a \<and> N = M0 + K" by simp |
10249 | 406 |
hence ?case2 .. thus ?thesis .. |
407 |
next |
|
408 |
fix K' |
|
409 |
assume "M0' = K' + {#a#}" |
|
410 |
with N have n: "N = K' + K + {#a#}" by (simp add: union_ac) |
|
411 |
||
412 |
assume "M0 = K' + {#a'#}" |
|
413 |
with r have "?R (K' + K) M0" by blast |
|
414 |
with n have ?case1 by simp thus ?thesis .. |
|
415 |
qed |
|
416 |
qed |
|
417 |
qed |
|
418 |
||
11464 | 419 |
lemma all_accessible: "wf r ==> \<forall>M. M \<in> acc (mult1 r)" |
10249 | 420 |
proof |
421 |
let ?R = "mult1 r" |
|
422 |
let ?W = "acc ?R" |
|
423 |
{ |
|
424 |
fix M M0 a |
|
11464 | 425 |
assume M0: "M0 \<in> ?W" |
12399 | 426 |
and wf_hyp: "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
11464 | 427 |
and acc_hyp: "\<forall>M. (M, M0) \<in> ?R --> M + {#a#} \<in> ?W" |
428 |
have "M0 + {#a#} \<in> ?W" |
|
10249 | 429 |
proof (rule accI [of "M0 + {#a#}"]) |
430 |
fix N |
|
11464 | 431 |
assume "(N, M0 + {#a#}) \<in> ?R" |
432 |
hence "((\<exists>M. (M, M0) \<in> ?R \<and> N = M + {#a#}) \<or> |
|
433 |
(\<exists>K. (\<forall>b. b :# K --> (b, a) \<in> r) \<and> N = M0 + K))" |
|
10249 | 434 |
by (rule less_add) |
11464 | 435 |
thus "N \<in> ?W" |
10249 | 436 |
proof (elim exE disjE conjE) |
11464 | 437 |
fix M assume "(M, M0) \<in> ?R" and N: "N = M + {#a#}" |
438 |
from acc_hyp have "(M, M0) \<in> ?R --> M + {#a#} \<in> ?W" .. |
|
439 |
hence "M + {#a#} \<in> ?W" .. |
|
440 |
thus "N \<in> ?W" by (simp only: N) |
|
10249 | 441 |
next |
442 |
fix K |
|
443 |
assume N: "N = M0 + K" |
|
11464 | 444 |
assume "\<forall>b. b :# K --> (b, a) \<in> r" |
445 |
have "?this --> M0 + K \<in> ?W" (is "?P K") |
|
10249 | 446 |
proof (induct K) |
11464 | 447 |
from M0 have "M0 + {#} \<in> ?W" by simp |
10249 | 448 |
thus "?P {#}" .. |
449 |
||
450 |
fix K x assume hyp: "?P K" |
|
451 |
show "?P (K + {#x#})" |
|
452 |
proof |
|
11464 | 453 |
assume a: "\<forall>b. b :# (K + {#x#}) --> (b, a) \<in> r" |
454 |
hence "(x, a) \<in> r" by simp |
|
455 |
with wf_hyp have b: "\<forall>M \<in> ?W. M + {#x#} \<in> ?W" by blast |
|
10249 | 456 |
|
11464 | 457 |
from a hyp have "M0 + K \<in> ?W" by simp |
458 |
with b have "(M0 + K) + {#x#} \<in> ?W" .. |
|
459 |
thus "M0 + (K + {#x#}) \<in> ?W" by (simp only: union_assoc) |
|
10249 | 460 |
qed |
461 |
qed |
|
11464 | 462 |
hence "M0 + K \<in> ?W" .. |
463 |
thus "N \<in> ?W" by (simp only: N) |
|
10249 | 464 |
qed |
465 |
qed |
|
466 |
} note tedious_reasoning = this |
|
467 |
||
468 |
assume wf: "wf r" |
|
469 |
fix M |
|
11464 | 470 |
show "M \<in> ?W" |
10249 | 471 |
proof (induct M) |
11464 | 472 |
show "{#} \<in> ?W" |
10249 | 473 |
proof (rule accI) |
11464 | 474 |
fix b assume "(b, {#}) \<in> ?R" |
475 |
with not_less_empty show "b \<in> ?W" by contradiction |
|
10249 | 476 |
qed |
477 |
||
11464 | 478 |
fix M a assume "M \<in> ?W" |
479 |
from wf have "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
|
10249 | 480 |
proof induct |
481 |
fix a |
|
12399 | 482 |
assume "!!b. (b, a) \<in> r ==> (\<forall>M \<in> ?W. M + {#b#} \<in> ?W)" |
11464 | 483 |
show "\<forall>M \<in> ?W. M + {#a#} \<in> ?W" |
10249 | 484 |
proof |
11464 | 485 |
fix M assume "M \<in> ?W" |
486 |
thus "M + {#a#} \<in> ?W" |
|
10249 | 487 |
by (rule acc_induct) (rule tedious_reasoning) |
488 |
qed |
|
489 |
qed |
|
11464 | 490 |
thus "M + {#a#} \<in> ?W" .. |
10249 | 491 |
qed |
492 |
qed |
|
493 |
||
494 |
theorem wf_mult1: "wf r ==> wf (mult1 r)" |
|
495 |
by (rule acc_wfI, rule all_accessible) |
|
496 |
||
497 |
theorem wf_mult: "wf r ==> wf (mult r)" |
|
498 |
by (unfold mult_def, rule wf_trancl, rule wf_mult1) |
|
499 |
||
500 |
||
501 |
subsubsection {* Closure-free presentation *} |
|
502 |
||
503 |
(*Badly needed: a linear arithmetic procedure for multisets*) |
|
504 |
||
505 |
lemma diff_union_single_conv: "a :# J ==> I + J - {#a#} = I + (J - {#a#})" |
|
15072 | 506 |
by (simp add: multiset_eq_conv_count_eq) |
10249 | 507 |
|
508 |
text {* One direction. *} |
|
509 |
||
510 |
lemma mult_implies_one_step: |
|
11464 | 511 |
"trans r ==> (M, N) \<in> mult r ==> |
512 |
\<exists>I J K. N = I + J \<and> M = I + K \<and> J \<noteq> {#} \<and> |
|
513 |
(\<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r)" |
|
10249 | 514 |
apply (unfold mult_def mult1_def set_of_def) |
15072 | 515 |
apply (erule converse_trancl_induct, clarify) |
516 |
apply (rule_tac x = M0 in exI, simp, clarify) |
|
10249 | 517 |
apply (case_tac "a :# K") |
518 |
apply (rule_tac x = I in exI) |
|
519 |
apply (simp (no_asm)) |
|
520 |
apply (rule_tac x = "(K - {#a#}) + Ka" in exI) |
|
521 |
apply (simp (no_asm_simp) add: union_assoc [symmetric]) |
|
11464 | 522 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong) |
10249 | 523 |
apply (simp add: diff_union_single_conv) |
524 |
apply (simp (no_asm_use) add: trans_def) |
|
525 |
apply blast |
|
526 |
apply (subgoal_tac "a :# I") |
|
527 |
apply (rule_tac x = "I - {#a#}" in exI) |
|
528 |
apply (rule_tac x = "J + {#a#}" in exI) |
|
529 |
apply (rule_tac x = "K + Ka" in exI) |
|
530 |
apply (rule conjI) |
|
531 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) |
|
532 |
apply (rule conjI) |
|
15072 | 533 |
apply (drule_tac f = "\<lambda>M. M - {#a#}" in arg_cong, simp) |
10249 | 534 |
apply (simp add: multiset_eq_conv_count_eq split: nat_diff_split) |
535 |
apply (simp (no_asm_use) add: trans_def) |
|
536 |
apply blast |
|
10277 | 537 |
apply (subgoal_tac "a :# (M0 + {#a#})") |
10249 | 538 |
apply simp |
539 |
apply (simp (no_asm)) |
|
540 |
done |
|
541 |
||
542 |
lemma elem_imp_eq_diff_union: "a :# M ==> M = M - {#a#} + {#a#}" |
|
15072 | 543 |
by (simp add: multiset_eq_conv_count_eq) |
10249 | 544 |
|
11464 | 545 |
lemma size_eq_Suc_imp_eq_union: "size M = Suc n ==> \<exists>a N. M = N + {#a#}" |
10249 | 546 |
apply (erule size_eq_Suc_imp_elem [THEN exE]) |
15072 | 547 |
apply (drule elem_imp_eq_diff_union, auto) |
10249 | 548 |
done |
549 |
||
550 |
lemma one_step_implies_mult_aux: |
|
551 |
"trans r ==> |
|
11464 | 552 |
\<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)) |
553 |
--> (I + K, I + J) \<in> mult r" |
|
15072 | 554 |
apply (induct_tac n, auto) |
555 |
apply (frule size_eq_Suc_imp_eq_union, clarify) |
|
556 |
apply (rename_tac "J'", simp) |
|
557 |
apply (erule notE, auto) |
|
10249 | 558 |
apply (case_tac "J' = {#}") |
559 |
apply (simp add: mult_def) |
|
560 |
apply (rule r_into_trancl) |
|
15072 | 561 |
apply (simp add: mult1_def set_of_def, blast) |
11464 | 562 |
txt {* Now we know @{term "J' \<noteq> {#}"}. *} |
563 |
apply (cut_tac M = K and P = "\<lambda>x. (x, a) \<in> r" in multiset_partition) |
|
564 |
apply (erule_tac P = "\<forall>k \<in> set_of K. ?P k" in rev_mp) |
|
10249 | 565 |
apply (erule ssubst) |
15072 | 566 |
apply (simp add: Ball_def, auto) |
10249 | 567 |
apply (subgoal_tac |
11464 | 568 |
"((I + {# x : K. (x, a) \<in> r #}) + {# x : K. (x, a) \<notin> r #}, |
569 |
(I + {# x : K. (x, a) \<in> r #}) + J') \<in> mult r") |
|
10249 | 570 |
prefer 2 |
571 |
apply force |
|
572 |
apply (simp (no_asm_use) add: union_assoc [symmetric] mult_def) |
|
573 |
apply (erule trancl_trans) |
|
574 |
apply (rule r_into_trancl) |
|
575 |
apply (simp add: mult1_def set_of_def) |
|
576 |
apply (rule_tac x = a in exI) |
|
577 |
apply (rule_tac x = "I + J'" in exI) |
|
578 |
apply (simp add: union_ac) |
|
579 |
done |
|
580 |
||
581 |
theorem one_step_implies_mult: |
|
11464 | 582 |
"trans r ==> J \<noteq> {#} ==> \<forall>k \<in> set_of K. \<exists>j \<in> set_of J. (k, j) \<in> r |
583 |
==> (I + K, I + J) \<in> mult r" |
|
15072 | 584 |
apply (insert one_step_implies_mult_aux, blast) |
10249 | 585 |
done |
586 |
||
587 |
||
588 |
subsubsection {* Partial-order properties *} |
|
589 |
||
12338
de0f4a63baa5
renamed class "term" to "type" (actually "HOL.type");
wenzelm
parents:
11868
diff
changeset
|
590 |
instance multiset :: (type) ord .. |
10249 | 591 |
|
592 |
defs (overloaded) |
|
11464 | 593 |
less_multiset_def: "M' < M == (M', M) \<in> mult {(x', x). x' < x}" |
594 |
le_multiset_def: "M' <= M == M' = M \<or> M' < (M::'a multiset)" |
|
10249 | 595 |
|
596 |
lemma trans_base_order: "trans {(x', x). x' < (x::'a::order)}" |
|
597 |
apply (unfold trans_def) |
|
598 |
apply (blast intro: order_less_trans) |
|
599 |
done |
|
600 |
||
601 |
text {* |
|
602 |
\medskip Irreflexivity. |
|
603 |
*} |
|
604 |
||
605 |
lemma mult_irrefl_aux: |
|
11464 | 606 |
"finite A ==> (\<forall>x \<in> A. \<exists>y \<in> A. x < (y::'a::order)) --> A = {}" |
10249 | 607 |
apply (erule finite_induct) |
608 |
apply (auto intro: order_less_trans) |
|
609 |
done |
|
610 |
||
11464 | 611 |
theorem mult_less_not_refl: "\<not> M < (M::'a::order multiset)" |
15072 | 612 |
apply (unfold less_multiset_def, auto) |
613 |
apply (drule trans_base_order [THEN mult_implies_one_step], auto) |
|
10249 | 614 |
apply (drule finite_set_of [THEN mult_irrefl_aux [rule_format (no_asm)]]) |
615 |
apply (simp add: set_of_eq_empty_iff) |
|
616 |
done |
|
617 |
||
618 |
lemma mult_less_irrefl [elim!]: "M < (M::'a::order multiset) ==> R" |
|
15072 | 619 |
by (insert mult_less_not_refl, fast) |
10249 | 620 |
|
621 |
||
622 |
text {* Transitivity. *} |
|
623 |
||
624 |
theorem mult_less_trans: "K < M ==> M < N ==> K < (N::'a::order multiset)" |
|
625 |
apply (unfold less_multiset_def mult_def) |
|
626 |
apply (blast intro: trancl_trans) |
|
627 |
done |
|
628 |
||
629 |
text {* Asymmetry. *} |
|
630 |
||
11464 | 631 |
theorem mult_less_not_sym: "M < N ==> \<not> N < (M::'a::order multiset)" |
10249 | 632 |
apply auto |
633 |
apply (rule mult_less_not_refl [THEN notE]) |
|
15072 | 634 |
apply (erule mult_less_trans, assumption) |
10249 | 635 |
done |
636 |
||
637 |
theorem mult_less_asym: |
|
11464 | 638 |
"M < N ==> (\<not> P ==> N < (M::'a::order multiset)) ==> P" |
15072 | 639 |
by (insert mult_less_not_sym, blast) |
10249 | 640 |
|
641 |
theorem mult_le_refl [iff]: "M <= (M::'a::order multiset)" |
|
15072 | 642 |
by (unfold le_multiset_def, auto) |
10249 | 643 |
|
644 |
text {* Anti-symmetry. *} |
|
645 |
||
646 |
theorem mult_le_antisym: |
|
647 |
"M <= N ==> N <= M ==> M = (N::'a::order multiset)" |
|
648 |
apply (unfold le_multiset_def) |
|
649 |
apply (blast dest: mult_less_not_sym) |
|
650 |
done |
|
651 |
||
652 |
text {* Transitivity. *} |
|
653 |
||
654 |
theorem mult_le_trans: |
|
655 |
"K <= M ==> M <= N ==> K <= (N::'a::order multiset)" |
|
656 |
apply (unfold le_multiset_def) |
|
657 |
apply (blast intro: mult_less_trans) |
|
658 |
done |
|
659 |
||
11655 | 660 |
theorem mult_less_le: "(M < N) = (M <= N \<and> M \<noteq> (N::'a::order multiset))" |
15072 | 661 |
by (unfold le_multiset_def, auto) |
10249 | 662 |
|
10277 | 663 |
text {* Partial order. *} |
664 |
||
665 |
instance multiset :: (order) order |
|
666 |
apply intro_classes |
|
667 |
apply (rule mult_le_refl) |
|
15072 | 668 |
apply (erule mult_le_trans, assumption) |
669 |
apply (erule mult_le_antisym, assumption) |
|
10277 | 670 |
apply (rule mult_less_le) |
671 |
done |
|
672 |
||
10249 | 673 |
|
674 |
subsubsection {* Monotonicity of multiset union *} |
|
675 |
||
676 |
theorem mult1_union: |
|
11464 | 677 |
"(B, D) \<in> mult1 r ==> trans r ==> (C + B, C + D) \<in> mult1 r" |
15072 | 678 |
apply (unfold mult1_def, auto) |
10249 | 679 |
apply (rule_tac x = a in exI) |
680 |
apply (rule_tac x = "C + M0" in exI) |
|
681 |
apply (simp add: union_assoc) |
|
682 |
done |
|
683 |
||
684 |
lemma union_less_mono2: "B < D ==> C + B < C + (D::'a::order multiset)" |
|
685 |
apply (unfold less_multiset_def mult_def) |
|
686 |
apply (erule trancl_induct) |
|
687 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl) |
|
688 |
apply (blast intro: mult1_union transI order_less_trans r_into_trancl trancl_trans) |
|
689 |
done |
|
690 |
||
691 |
lemma union_less_mono1: "B < D ==> B + C < D + (C::'a::order multiset)" |
|
692 |
apply (subst union_commute [of B C]) |
|
693 |
apply (subst union_commute [of D C]) |
|
694 |
apply (erule union_less_mono2) |
|
695 |
done |
|
696 |
||
697 |
theorem union_less_mono: |
|
698 |
"A < C ==> B < D ==> A + B < C + (D::'a::order multiset)" |
|
699 |
apply (blast intro!: union_less_mono1 union_less_mono2 mult_less_trans) |
|
700 |
done |
|
701 |
||
702 |
theorem union_le_mono: |
|
703 |
"A <= C ==> B <= D ==> A + B <= C + (D::'a::order multiset)" |
|
704 |
apply (unfold le_multiset_def) |
|
705 |
apply (blast intro: union_less_mono union_less_mono1 union_less_mono2) |
|
706 |
done |
|
707 |
||
708 |
theorem empty_leI [iff]: "{#} <= (M::'a::order multiset)" |
|
709 |
apply (unfold le_multiset_def less_multiset_def) |
|
710 |
apply (case_tac "M = {#}") |
|
711 |
prefer 2 |
|
11464 | 712 |
apply (subgoal_tac "({#} + {#}, {#} + M) \<in> mult (Collect (split op <))") |
10249 | 713 |
prefer 2 |
714 |
apply (rule one_step_implies_mult) |
|
15072 | 715 |
apply (simp only: trans_def, auto) |
10249 | 716 |
done |
717 |
||
718 |
theorem union_upper1: "A <= A + (B::'a::order multiset)" |
|
15072 | 719 |
proof - |
720 |
have "A + {#} <= A + B" by (blast intro: union_le_mono) |
|
721 |
thus ?thesis by simp |
|
722 |
qed |
|
723 |
||
724 |
theorem union_upper2: "B <= A + (B::'a::order multiset)" |
|
725 |
by (subst union_commute, rule union_upper1) |
|
726 |
||
727 |
||
728 |
subsection {* Link with lists *} |
|
729 |
||
730 |
consts |
|
731 |
multiset_of :: "'a list \<Rightarrow> 'a multiset" |
|
732 |
primrec |
|
733 |
"multiset_of [] = {#}" |
|
734 |
"multiset_of (a # x) = multiset_of x + {# a #}" |
|
735 |
||
736 |
lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])" |
|
737 |
by (induct_tac x, auto) |
|
738 |
||
739 |
lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])" |
|
740 |
by (induct_tac x, auto) |
|
741 |
||
742 |
lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x" |
|
743 |
by (induct_tac x, auto) |
|
744 |
||
745 |
lemma multset_of_append[simp]: |
|
746 |
"multiset_of (xs @ ys) = multiset_of xs + multiset_of ys" |
|
747 |
by (rule_tac x=ys in spec, induct_tac xs, auto simp: union_ac) |
|
748 |
||
749 |
lemma surj_multiset_of: "surj multiset_of" |
|
750 |
apply (unfold surj_def, rule allI) |
|
751 |
apply (rule_tac M=y in multiset_induct, auto) |
|
752 |
apply (rule_tac x = "x # xa" in exI, auto) |
|
10249 | 753 |
done |
754 |
||
15072 | 755 |
lemma set_count_greater_0: "set x = {a. 0 < count (multiset_of x) a}" |
756 |
by (induct_tac x, auto) |
|
757 |
||
758 |
lemma distinct_count_atmost_1: |
|
759 |
"distinct x = (! a. count (multiset_of x) a = (if a \<in> set x then 1 else 0))" |
|
760 |
apply ( induct_tac x, simp, rule iffI, simp_all) |
|
761 |
apply (rule conjI) |
|
762 |
apply (simp_all add: set_of_multiset_of [THEN sym] del: set_of_multiset_of) |
|
763 |
apply (erule_tac x=a in allE, simp, clarify) |
|
764 |
apply (erule_tac x=aa in allE, simp) |
|
765 |
done |
|
766 |
||
767 |
lemma set_eq_iff_multiset_of_eq_distinct: |
|
768 |
"\<lbrakk>distinct x; distinct y\<rbrakk> |
|
769 |
\<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)" |
|
770 |
by (auto simp: multiset_eq_conv_count_eq distinct_count_atmost_1) |
|
771 |
||
772 |
lemma set_eq_iff_multiset_of_remdups_eq: |
|
773 |
"(set x = set y) = (multiset_of (remdups x) = multiset_of (remdups y))" |
|
774 |
apply (rule iffI) |
|
775 |
apply (simp add: set_eq_iff_multiset_of_eq_distinct[THEN iffD1]) |
|
776 |
apply (drule distinct_remdups[THEN distinct_remdups |
|
777 |
[THEN set_eq_iff_multiset_of_eq_distinct[THEN iffD2]]]) |
|
778 |
apply simp |
|
10249 | 779 |
done |
780 |
||
15072 | 781 |
|
782 |
subsection {* Pointwise ordering induced by count *} |
|
783 |
||
784 |
consts |
|
785 |
mset_le :: "['a multiset, 'a multiset] \<Rightarrow> bool" |
|
786 |
||
787 |
syntax |
|
788 |
"_mset_le" :: "'a multiset \<Rightarrow> 'a multiset \<Rightarrow> bool" ("_ \<le># _" [50,51] 50) |
|
789 |
translations |
|
790 |
"x \<le># y" == "mset_le x y" |
|
791 |
||
792 |
defs |
|
793 |
mset_le_def: "xs \<le># ys == (! a. count xs a \<le> count ys a)" |
|
794 |
||
795 |
lemma mset_le_refl[simp]: "xs \<le># xs" |
|
796 |
by (unfold mset_le_def, auto) |
|
797 |
||
798 |
lemma mset_le_trans: "\<lbrakk> xs \<le># ys; ys \<le># zs \<rbrakk> \<Longrightarrow> xs \<le># zs" |
|
799 |
by (unfold mset_le_def, fast intro: order_trans) |
|
800 |
||
801 |
lemma mset_le_antisym: "\<lbrakk> xs\<le># ys; ys \<le># xs\<rbrakk> \<Longrightarrow> xs = ys" |
|
802 |
apply (unfold mset_le_def) |
|
803 |
apply (rule multiset_eq_conv_count_eq[THEN iffD2]) |
|
804 |
apply (blast intro: order_antisym) |
|
805 |
done |
|
806 |
||
807 |
lemma mset_le_exists_conv: |
|
808 |
"(xs \<le># ys) = (? zs. ys = xs + zs)" |
|
809 |
apply (unfold mset_le_def, rule iffI, rule_tac x = "ys - xs" in exI) |
|
810 |
apply (auto intro: multiset_eq_conv_count_eq [THEN iffD2]) |
|
811 |
done |
|
812 |
||
813 |
lemma mset_le_mono_add_right_cancel[simp]: "(xs + zs \<le># ys + zs) = (xs \<le># ys)" |
|
814 |
by (unfold mset_le_def, auto) |
|
815 |
||
816 |
lemma mset_le_mono_add_left_cancel[simp]: "(zs + xs \<le># zs + ys) = (xs \<le># ys)" |
|
817 |
by (unfold mset_le_def, auto) |
|
818 |
||
819 |
lemma mset_le_mono_add: "\<lbrakk> xs \<le># ys; vs \<le># ws \<rbrakk> \<Longrightarrow> xs + vs \<le># ys + ws" |
|
820 |
apply (unfold mset_le_def, auto) |
|
821 |
apply (erule_tac x=a in allE)+ |
|
822 |
apply auto |
|
823 |
done |
|
824 |
||
825 |
lemma mset_le_add_left[simp]: "xs \<le># xs + ys" |
|
826 |
by (unfold mset_le_def, auto) |
|
827 |
||
828 |
lemma mset_le_add_right[simp]: "ys \<le># xs + ys" |
|
829 |
by (unfold mset_le_def, auto) |
|
830 |
||
831 |
lemma multiset_of_remdups_le: "multiset_of (remdups x) \<le># multiset_of x" |
|
832 |
by (induct_tac x, auto, rule mset_le_trans, auto) |
|
833 |
||
10249 | 834 |
end |